Toward Robust Estimation of 

the Components of Forest Population 

Change: Simulation Results

Francis A. Roesch

United States Department of Agriculture

Forest Service

Research & Development, Southern Research Station

e-General Technical Report SRS-194, July 2014
The Author:

Francis A. Roesch is a Research Mathematical Statistician, 

Southern Research Station, U.S. Department of Agriculture 

Forest Service, Asheville, NC 28804-3454. 

July 2014

Southern Research Station

200 W.T. Weaver Blvd.

Asheville, NC 28804

www.srs.fs.usda.gov
Toward Robust Estimation of the 

Components of Forest Population Change: 

Simulation Results

Francis A. Roesch 

TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS iii

CONTENTS

INTRODUCTION............................................................................................................................................................ 1

ESTIMATORS FOR THE COMPONENTS OF CHANGE....................................................................................... 2

Centralized Difference Estimator....................................................................................................................... 3

Exponentially Weighted Difference Estimator................................................................................................... 3

Moving-Window Mean of Ratios Estimator...................................................................................................... 3

Incorporating Outside Information .................................................................................................................... 4

Mixed Estimation .............................................................................................................................................. 4

SIMULATIONS ............................................................................................................................................................... 5

Population Simulations...................................................................................................................................... 5

Sampling Simulations........................................................................................................................................ 6

RESULTS........................................................................................................................................................................ 7

Population 0 ....................................................................................................................................................... 7

Population 1 ....................................................................................................................................................... 9

Population 2 ....................................................................................................................................................... 9

Population 3 ....................................................................................................................................................... 9

Population 4 ....................................................................................................................................................... 9

Population 3—With Outside Information........................................................................................................ 10

CONCLUSIONS........................................................................................................................................................... 10

TABLES......................................................................................................................................................................... 11

FIGURES....................................................................................................................................................................... 16

ACKNOWLEDGMENTS............................................................................................................................................. 79

LITERATURE CITED .................................................................................................................................................. 79

Abstract

This report presents the full simulation results of the work described 

in Roesch (2014), in which multiple levels of simulation were used to 

test the robustness of estimators for the components of forest change. 

In that study, a variety of spatial-temporal populations were created 

based on, but more variable than, an actual forest monitoring dataset, 

and then those populations were sampled under four sets of sampling 

error structure. An estimator modification was shown, to be used when 

extraneously obtained information indicated that a deviation to the 

assumed population model existed. The extraneous information was 

also incorporated into a mixed estimator. The first three approaches, 

without the incorporation of extraneous information, are compatible 

with large monitoring efforts that require intervention-free results. 

The mixed estimation approach accounts for model assumptions that 

sometimes remain latent in other approaches and is amenable to the 

formal incorporation of the extraneously obtained information. All four 

approaches were shown to work well when the sampling error structure 

was unbiased, while some notable differences in performance were 

observed at the temporal extremities of observation, in the presence 

of temporal anomalies, and in the presence of biased sampling error 

structures. Only those results necessary to make the salient points were 

presented in Roesch (2014). Full results are presented here both for full 

disclosure and for the reader interested in a more detailed understanding 

of the effects of realistic sampling errors on temporal estimates.

Keywords: Annual inventories, components of change, forest 

monitoring, sampling error, spatial-temporal sample design.

INTRODUCTION

The robustness of the estimation system for large-scale 

forest inventory and monitoring systems is an extremely 

important consideration. An estimation system is robust 

when it provides significant revelations about the 

conditions being investigated, even in the presence of 

violations of population and sampling error assumptions. 

This report gives the full results of a study that used two 

levels of simulation, the first to create a variety of spatialtemporal populations based on actual forest monitoring 

data and the second to sample those populations under a 

variety of sampling error structures. The author believes 

that there is an education available to the energetic reader 

through a study of this full set of results that could not 

be conveyed in a single journal article. While reviewing 

the full set of results, the reader should keep in mind the 

differing effects of both overt and latent population trends 

in conjunction with a sometimes unknown sampling 

error structure on the assumptions underlying the various 

estimators. Using the components of change in this 

study facilitates understanding for the simple reason 

that the population trends of the components differ from 

each other for a common population and sampling error 

structure. In real inventories and monitoring efforts, the 

true population trends are never known, as they are in 

these simulations. The results will show how reasonably 

argued trend estimators can produce differing results and 

how those differing results can be used to infer underlying 

latent trends. The results also show a difficult reality in 

trend estimation: often the strength that is garnered from 

temporally adjacent observations will result in overly 

smooth estimates of trend. Additionally, it can often 

be beneficial to use a more sophisticated approach and 

include information from alternative sources.

Initially, I examine the robustness of three estimators for 

annual components of forest change when the temporal 

scale of the population estimand of interest is finer than 

the scale of observation in a temporally rotating panel 

sample design, such as the design used by the U.S. 

Department of Agriculture (USDA) Forest Service’s 

Forest Inventory and Analysis (FIA) Program (Bechtold 

and Patterson 2005). The approaches are similar but 

differ in their underlying trend assumptions, leading to 

differences in how measurement intervals that are not 

centered on a target year inform the estimate for the target 

year. Next, a modification to these approaches shown 

in Roesch (2014), for use when extraneously obtained 

information indicates that a deviation to the assumed 

trend model has occurred, is evaluated. Finally, mixed 

estimation is used to package the extraneous information 

with a compatibility model. I show how conflicting results 

in the estimators can be used in an estimation system 

Toward Robust Estimation of the Components of 

Forest Population Change: Simulation Results

Francis A. Roesch 
2 FRANCIS A. ROESCH

to reveal violations of the investigator’s preconceived 

notions of both population trends and relationships and 

the structure of sampling errors.

The two major components of forest change that are of 

interest when monitoring a dynamic landscape are:

(1) Land use change during the period of interest in the 

form of reversion from some other land use to forest, 

and diversion away from forest to some other land 

use, and

(2) Change of trees on land that is forested during the 

entire period of interest.

Due to the cost constraints present in all forest 

monitoring efforts, neither of these components is ever 

fully observed. For a dynamic forest inventory to be 

sufficiently informative, however, it is important to 

acknowledge the distinction between land use change and 

the change that occurs on forest land. Both are important 

factors when evaluating the state of the forests in an area 

or nation, but the full implications of the distinction are 

often lost in aggregations of national forest inventory 

(NFI) data. That is, (1) and (2) above are somewhat 

confounded. The discussion below assumes that there 

exists a method of grouping land into subsets according 

to the temporal period of forest land classification, within 

the temporal period of interest. The methods discussed 

here can then be applied to the subsets individually. Van 

Deusen and Roesch (2009) and Roesch and Van Deusen 

(2012) explored estimation of the change in forest land 

classification, in the context of the NFI in the United 

States. Roesch (2014) and this paper concentrate on 

estimation of the change in the tree population on forest 

land.

Historically, the definitions of the components of growth 

were sample dependent. The resulting quantities could 

easily be calculated from remeasured samples but were 

not truly estimates of population parameters. Eriksson 

(1995) corrected this deficiency with the presentation 

of a set of definitions for the continuous components of 

change. Roesch (2007b) presented a discrete version of 

the Eriksson definitions, which are used here. 

The population components of change are compatible; 

that is:

(1)

where

Yt = the value of interest at time t

Lt,t+1 = growth in the value of interest on live 

trees between time t and time t+1

Et,t+1 = the value of interest on live trees as they 

enter the population between time t and 

time t+1

Mt,t+1 = the value of interest on trees as they die 

between time t and time t+1

Ht,t+1 = the value of interest on trees as they 

are harvested between time t and time 

t+1

Without loss of generality, in this paper, the value of 

interest is cubic meter volume and the entry criterion is 

12.7 cm at 1.37 m above the ground (diameter at breast 

height, or d.b.h.). 

Consider a sample design that consists of g mutually 

exclusive, spatially disjoint temporal panels in which, 

subsequent to a random areal start, one panel per year 

is measured, in turn, for g consecutive years. After each 

cycle, the panel measurement sequence reinitiates. Such 

a design is used by the USDA Forest Service’s FIA 

Program and discussed in Bechtold and Patterson (2005) 

and Roesch (2007b). Consequently, in a five-panel 

system, the five panels are measured for growth over a 

10-year period. That is, panel 1 is measured in years 1 

and 6, panel 2 is measured in years 2 and 7, etc. Several 

philosophies have emerged as to how data resulting from 

this design should be applied to estimates of growth and 

change because remeasurement of the panels provides 

observations that are spatially disjoint but temporally 

overlapping, and the temporal scale of the population 

of interest is finer than the scale of observation. Roesch 

(2007b) argued that the average annual growth within 

each individual panel is best applied to the center of 

the measurement interval, which is analogous to an 

assumption of linear change between observations. All 

analytical methods proposed to date for this class of 

sample designs for forest monitoring have been predicated 

on this or similar assumptions. This assumption also 

applies to most of the discussion below. However, I do 

explore the effects of a simple nonconforming population 

trend, which can be either latent or overt.

ESTIMATORS FOR THE COMPONENTS 

OF CHANGE

It is often argued that estimators of these components of 

change should be compatible because the components 

themselves are compatible. In this report, I compare

four general approaches for estimating the components 

of forest change. In the first three approaches, each 

component is estimated independently of the others. In 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 3

the fourth approach, the estimates are constrained to be 

compatible. Assume that the estimands of interest are 

the m3 volume/ha of all live trees in a fixed area in each 

component of change category during each year of a 

multi-decadal period. 

The first estimation approach uses a Centralized 

Difference Estimator (CDE), the second uses the 

Exponentially Weighted Difference (EWD) estimator 

of Roesch (2007a), the third uses the (semi-centralized) 

Moving-Window Mean of Ratios (MWMOR) estimator 

in Roesch and Van Deusen (2013), and the fourth uses 

the mixed estimator variant in Roesch (2007a). Because 

we know that compatibility will have a cost in terms 

of squared error loss for one or more of the estimated 

components, the initial results compare only the first 

three approaches. The fourth approach is discussed in 

a situation in which the first three approaches, lacking 

the benefit of extraneous information, are shown to give 

unfavorable results.

I first present three estimators that can be expected 

to yield equivalent results until they are differentially 

affected when an underlying assumption becomes 

tenuous. When an anomaly occurs that can be expected 

to affect these estimators under the given sample design, 

I apply a reweighting scheme in order to incorporate 

extraneously obtained information. Subsequently, the 

extraneously obtained weights are incorporated directly 

into a mixed estimator, and all four estimators are tested 

in the anomalous situation.

Centralized Difference Estimator

The Centralized Difference Estimator is a moving average 

estimator applied to a series of equally weighted withinpanel differences (i.e., a series of change component 

values). In the CDE, a panel difference is applied to the 

center interval (or year) and combined with adjoining 

panel differences. Let:

the annual mean of a remeasured panel 

difference, such as , and

, where m is odd and is the 

number of remeasured panels used in 

the estimator.

Then the CDE for component C in year k is:

(2)

The CDE provides no estimates for m years on each end 

of the time string. In practice, an ad hoc variation would 

have to be used in order to provide some or all of these 

estimates. Here, five panels are used in the estimator when 

there are differences from at least two panels available 

before and after the central panel. Three panels are 

used in the estimator when there is only one additional 

panel difference available on either side of the central 

panel (e.g., for the second and penultimate estimates). A 

single panel difference is used for the first and the final 

estimates.

Exponentially Weighted Difference Estimator

The Exponentially Weighted Difference estimator of 

Roesch (2007a) is similar in concept to the exponentially 

weighted moving average (EWMA) estimator common 

in the quality control literature (Chandra 2000), and the 

econometrics literature (West and Harrison 1989, p. 55). 

In the EWD estimator, a series of differences (i.e., a series 

of change component values) within panels is calculated. 

The EWD estimator gives larger weights to the interval 

observations closest to the interval of interest, allowing 

more temporally local variation than if equal weights are 

used. The panel difference is applied to the central annual 

interval and combined with the m-1 adjoining interval 

differences. The supporting panels are down-weighted 

exponentially with each step away from the central 

interval. In addition to the notation above for the CDE, 

let: 

The EWD estimator for component C in year y is:

Cy = (1 )

1+ 2 r+1

( ) i

dy+r+i,y r+i

i= r

r

EWD (3)

As with the CDE, the EWD estimator also does not 

provide estimates for m years on each end of the time 

string, and I use the same solution for these extraneous 

years. That is, three panels are used in the estimator 

when there are differences from at least one panel 

available before and after the central panel. A single 

panel difference is used otherwise, rendering the EWD 

estimator equal to the CDE when there is only one panel 

difference available.

Moving-Window Mean of Ratios Estimator

Roesch and Van Deusen (2013) proposed an estimator 

that arose from a different perspective from the two 

estimators above that I apply to the current objective, as 

fully explained in Roesch (2014). The idea was simple: 

one stacks the observations on a temporal scale (or a 

function of the temporal scale), and then slices through 
4 FRANCIS A. ROESCH

the stack (say to create annual segments) to determine 

how much of each observation contributes to the estimate 

for each year. For this problem, as in Roesch and Van 

Deusen (2013), time is rescaled relative to the proportion 

of the growing season elapsed within each year. The 

change components are modeled between observations 

to allocate the components to the years in which the 

change is assumed to have occurred. A simple timeadjusted estimator for annual volume growth (within the 

live growth component) is the MWMOR estimator for 

component C in year y: 

(4)

where

ny = the number of plots observing growth in year y

Pi,y = the product of portion of year y growing 

season observed by plot i and the portion of 

plot i area within the area of interest 

ci,y = the value of component C observed on plot i,

assignable to year y.

Incorporating Outside Information

Situations arise within the scope of national-scale forest 

monitoring efforts for which the data obtained from 

the sample design are inadequate. Many approaches to 

incorporating extraneous information in forest inventories 

have been proposed and proven useful for particular 

applications. A weighting method given in Roesch (2014) 

can be used in conjunction with each of the estimators 

above, and with the mixed estimator below to incorporate 

extraneous information. 

Assume that there exists strong external information that 

the expected value of X at time t, E(Xt), differs from the 

expected value of a previous estimate of X, , at 

time t by a factor kw. For the estimators above, this 

information suggests a reweighting of previous estimates 

for all estimates that had relied on the previous estimate 

for time t. Let:

kv = kw + nt – 1 (5)

where

nt = the number of years used for each annual 

estimate, and let

kt- = nt /kv (6)

and 

kt = (nt*kw)/kv (7)

Then weight the previous estimates at times other than 

time t that used the previous time t estimate by kt-, and 

weight previous estimates at time t by kt to form the 

reweighted estimates.

Mixed Estimation 

The mixed estimator (Theil 1963) can be used to draw 

strength from overlapping panels and easily incorporate 

extraneous information into forest monitoring efforts. 

Mixed estimation was first proposed for use in forestry 

when Korhonen (1993) used the method for calibrating 

tree volume functions. Van Deusen (1996, 1999, 2000) 

developed mixed estimators for annual forest inventory 

designs, and Roesch (2007a) used it for components of 

change estimation. In Roesch (2014), mixed estimation 

is used as a convenient way to incorporate both (1) a 

belief in how the individual growth components should 

be related and transition from year to year, and (2) 

extraneous information that suggests that a modification to 

those beliefs is appropriate. To achieve these goals, start 

with the three transition models below, and then adapt 

those models to incorporate the extraneous information. 

The three base models all assume compatibility of the 

total annual change with the components of change; that 

is, for each year t:

(8)

Initially, Model 1 assumes that for each component 

C = L, E, M, or H, at each time t:

(9)

Model 2 assumes that for each component, at each 

time t: 

(10)

while Model 3 assumes that for each component, at each 

time t:

(11)

Formulation of the constraints under each of these models 

is straightforward.

Next, let:

Y = an (nc*nt) row x ns column response matrix

where 

nc = 5, which is the number of growth components 

plus 1
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 5

nt = the number of years in the estimation interval 

ns = the sample size for all np panels.

The columns of Y are arranged in nt successive ordered 

5-tuples of (1) the MWMOR estimate of annual change, 

(2) observed annual live growth, (3) entry, (4) mortality, 

and (5) harvest, for each year in the observation interval 

for a plot and zeroes otherwise.

A user’s level of belief in the underlying model can 

be incorporated into the mixed estimator in several 

ways. Here, as explained in Roesch (2014), I choose 

to preprocess Y by first reweighting according to the 

extraneous information, adapting the constraint matrix, 

and then strictly applying the constraints to ensure growth 

component compatibility. To accomplish this weighting 

for estimates at times other than time t that use estimates 

for time t, weight the time t estimate by kw-1 before 

combining with the non-time t estimates. For estimates 

at time t, weight the non-time t estimates by kwbefore 

combining with the time t estimate. Indicate the outside 

information-weighted response matrix as YOI . Then let:

= an (nc*nt) row x (nc*nt) column variance/

co-variance matrix of YOI

R = an (nR) row x (nc*nt

) column constraint 

matrix, appropriate for a given model

where

nR = the number of constraint rows

= an (nc*nt) row x 1 column predictive 

coefficient vector for strict constraints:

(12)

Let M be an (nc*nt) row x ns column matrix, with each 

column consisting of nt repetitions of the vector 

(ns

-1, ns-1np, ns-1np, ns-1np, ns-1np). Then:

= (13)

SIMULATIONS

The exact details of the simulations used to both create 

the populations and obtain the error-included samples are 

given in Roesch (2014). Here I emphasize the philosophy 

behind the simulations and contend that the approach used 

is unique in the forestry literature in its completeness of 

consideration of errors in forest monitoring efforts. That 

is not to say that a more varied set of error structures for 

both the population creation and sampling simulations 

would not be appropriate, but it is to say that each of 

these steps is usually incomplete. Usually when the focus 

is on comparing either sampling methods or estimators, 

construction of realistic and diverse populations is given 

inadequate consideration. In addition, painstaking efforts 

are usually made to take design-unbiased samples in 

the sampling simulations, whereas in actual monitoring 

efforts a plethora of errors occurs, rendering the sample 

design a mere model of the realized sample. Unless the 

level of effort in each of these simulation steps is at 

least as high as it is in this study, the robustness of an 

estimation system cannot be fully evaluated.

Population Simulations

In this application, I use FIA data collected in South 

Carolina from 1998 through 2011 to construct five 

simulated populations. Although all of the populations 

are plausible, the intention was for the first population 

(Population 0) to be a seed population using the simplest 

possible model for deriving annual values from multiyear 

observations. The seed population allows the simulation 

of a series of populations, some of which we might 

assume to be like the one from which the sample data 

were drawn, and others that might arise from a wider 

diversity of conditions. 

I created the five simulated populations by first selecting 

all remeasured forested plots spanning the 14-year period 

from the South Carolina data. Most of the resulting 

2,430 forested plots had three measurement times (i.e., 

two observed growth intervals for each component). As 

explained more fully in Roesch (2014), the observations 

were converted to annual values for each year in the 

14-year period to create Set 1, and the construction of 

Population 0 was then obtained from 600 varianceinterjected copies of Set 1, resulting in a population of 

1,458,000 ha. Specifically, the variance was interjected 

at two steps. In step 1, to keep the existing trend and 

add variance to the seed, variance was interjected by 

multiplying all values for each component on each ha by 

a random variate, unique for that ha, drawn from an N(1, 

0.025) distribution. In step 2, variance was introduced 

temporally by multiplying the result of step 1, for each 

annual value for each component on each ha, by a unique 

random variate drawn from an N(1, 0.0025) distribution. 

Because the sampling simulation “observes” 5-year 

intervals, rather than the annual values, the manner of 

population construction should not unduly influence the 

results. However, to reduce further any potential linear 

effect, a mild (latent) nonlinear trend was introduced into 

each of the components in Population 1. 
6 FRANCIS A. ROESCH

For Population 2, a mild nonlinear trend was introduced 

into the components of live growth, entry, and mortality, 

and a stronger nonlinear trend was introduced into 

the harvest component in order to simulate increasing 

harvesting pressure. For Population 3, a mild nonlinear 

trend was introduced into the components of live growth, 

entry, and harvest, and a catastrophic high mortality event 

(of four times the mortality rate of Population 1) was 

introduced for the year 2004. Population 4 was initially as 

in Population 1 and then postulated climate change effects 

were simulated by increasing mortality and decreasing 

growth and recruitment, with harvest levels remaining the 

same. Each of the five populations consists of 1,458,000 

forested “hectares” or “elements” with measurable cubic 

meter volume at some time in the 14-year period. The 

five populations were constructed to examine estimator 

performance and robustness in the presence of plausible 

suboptimal population characteristics, for the given 

sample design and sampling error structure. 

Sampling Simulations

Usually, the overriding criterion for selection of an 

estimator in forest inventories is the minimum mean 

squared error for the candidate unbiased estimators. 

Rarely considered is the effect of sampling error in 

the form of bias on the robustness of theoretically 

unbiased estimators. Some notable exceptions have 

been Gertner (1987), Thomas and Roesch (1990), 

Eastaugh and Hasenauer (2013), and Roesch (2014). In 

this investigation, estimator robustness was tested in 

a simulation by sampling each population under four 

different assumptions of total sampling error structure. 

The models are intended to represent all error that would 

not be addressed by the use of an unbiased sampling 

simulation. The unbiased simulation might be the closest 

one can come to the pure sampling error inherent in a 

perfectly observed and measured sample, while in a 

realized sample there could also be “item observation” 

errors, frame errors, and measurement errors to say 

the least. That is, the error structure models used here 

are intended to represent all of the ways that a realized 

sample might differ from its perfectly observed theoretical 

counterpart.

A complete description of the construction of the error 

structures is given in Roesch (2014). In brief, for Error 

Structure 1, sampling error was assumed to consist 

exclusively of a small variance, effected by multiplying 

a unique random normal deviate of mean 1 and standard 

deviation 0.025 by each sampled observation of each 

component. For Error Structure 2, I assumed that 

sampling error consisted of a small variance and a 

positive bias on all change components. I effected Error 

Structure 2 by multiplying a unique random normal 

deviate of mean 1.05 and standard deviation of 0.025 by 

each observation of each component. 

I assumed that sampling error consisted of variance and 

positive bias on volume, live growth, and entry, and a 

negative bias on harvest and mortality for Error Structure 

3. I effected Error Structure 3 thus: each observation of 

live growth and entry was multiplied by a unique random 

normal deviate of mean 1.05 and standard deviation of 

0.025. Each observation of mortality and harvest was 

multiplied by a unique random normal deviate of mean 

0.90 and standard deviation of 0.025. Although the 

level of simulated error is somewhat arbitrary, errors of 

approximately these magnitudes seem reasonable based 

on the work in Thomas and Roesch (1990) and the results 

in Eastaugh and Hasenauer (2013).

Error Structure 4 was similar to Error Structure 3, but 

I assumed that sampling error consisted of a higher 

variance and greater bias, as follows. Each observation of 

live growth and entry was multiplied by a unique random 

normal deviate of mean 1.1 and standard deviation of 

0.05. Each observation of mortality and harvest was 

multiplied by a unique random normal deviate of mean 

0.80 and standard deviation of 0.05.

For Population 3, I also simulated the availability of 

extraneously obtained information that mortality during 

2004 was about four times higher than in surrounding 

years by drawing a random variate from a Normal 

distribution of mean 4, and standard deviation of 0.05, and 

setting it equal to kw for each observation. Then estimates 

for each estimator—the CDE, the EWD estimator, and 

the MWMOR estimator—were reweighted to obtain the 

CDE-OI, the EWD-OI estimator, and the MWMOR-OI 

estimator. The -OI suffix for each estimator indicates the 

inclusion of outside information, as described in Roesch 

(2014). The input values and constraints for the mixed 

estimator models were reweighted analogously.

Each simulation consisted of 1,000 iterations of 1,000 

plots each (without replacement) from each population, 

under each of the four sampling error structures. For 

each year, I calculated the Empirical Bias (EB) and the 

Empirical Mean Squared Error (EMSE) over the 1,000 

iterations, between each estimator and the true population 

values under each of the four error structures. 

That is:

(14)

where

ˆPESi x is the sample estimate of X in population P for 

estimator E, under error structure S for iterate i. Likewise:
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 7

 (15)

Usually, the overriding criterion for selection of an 

estimator in forest inventories is the minimum mean 

squared error for the candidate unbiased estimators. 

This priority might suggest that the presentation of these 

results should focus on EMSE. Because the four error 

structures were devised to examine estimator performance 

over these populations in the presence of plausible 

differentially biased sampling error, the reader should 

pay special attention to the EB in the results. On the 

other hand, the reader should not be overly influenced by 

an extremely low EB result because it can occur when 

sampling bias is being counteracted by estimation bias.

RESULTS

Tables 0 through 4 give the distribution statistics for 

1999-2011, for Populations 0 through 4, respectively. 

When examining the results of Populations 0 and 1, the 

reader should recall that Population 0 was constructed 

from 600 variance-interjected copies of Set 1, in a manner 

that should reduce the artificial linearity introduced by 

constructing the annual populations from multiyear 

observations. That is, Population 0 could be viewed as the 

result of the minimal prima facie effort to build a realistic 

annual population from the data arising from panelized 

multiyear observation intervals. To reduce further any 

potential linear effect in the annual time-series, a mild 

(latent) nonlinear trend was introduced into each of the 

components in Population 1. For Population 2, a mild 

nonlinear trend was introduced into the components of 

live growth, entry, and mortality, and a stronger nonlinear 

trend was introduced into the harvest component, in 

order to simulate increasing harvesting pressure. For 

Population 3, a mild nonlinear trend was introduced 

into the components of live growth, entry, and harvest, 

and a catastrophic high mortality event (of four times 

the mortality rate of Population 1) was introduced for 

the year 2004. Postulated climate change effects were 

simulated in Population 4 by increasing mortality and 

decreasing growth and recruitment, relative to Population 

1, with harvest levels remaining the same. The five tables 

of statistics for the populations can give the reader a 

fuller understanding of the breadth of trend effects and 

anomalies that existed to challenge the robustness of the 

estimators. 

The sampling simulation results are given in three figures 

for each combination of population and sampling error 

structure. The twenty combinations of population and 

sampling error structure are therefore given in sixty 

figures, labeled as figure peM, peB, and peE for the 

mean, bias, and mean squared error, respectively, for 

population p and Error Structure e. Each graph plots the 

corresponding statistics for the CDE, the EWD estimator, 

and the MWMOR estimator, by growth component for 

estimation years 2000 through 2009. These are the years 

that are the center of at least one observation window 

(or panel), for a sample drawn under this design from a 

population spanning 1998-2001. 

Population 0

Figures 0eM, 0eB, and 0eE, with e = 1, 2, 3, or 4, give 

the empirical mean, bias, and mean squared error, 

respectively, over 1,000 iterations of 1,000 samples each 

from Population 0 under Sampling Error Structure e. As 

proposed above, Population 0 could be viewed as the 

result of the minimal prima facie effort to build a realistic 

annual population from the data arising from panelized 

multiyear observation intervals.

Error Structure 1—Recall that, for Error Structure 

1, sampling error was assumed to consist exclusively 

of a small variance, resulting in an unbiased sampling 

simulation, representing the pure sampling error inherent 

in a perfectly observed and measured sample. Therefore, 

figures 01M, 01B, and 01E give the results that would be 

achieved through the usual assumptions made in estimator 

comparisons.

All three estimators, in conjunction with the design, 

are shown to be general smoothers. That is, the design 

observes 5-year windows, thereby providing an average 

annual change or “smoothing” the actual annual change. 

The estimators, while drawing strength from overlapping 

panels, provide further smoothing. In figure 01M, this 

effect can be clearly seen in all four components, under 

this simplest sampling error structure. It is also seen in all 

of the simulation mean graphs of these results.

Five panels of data are used for all of the estimators for 

estimates of the central estimation years 2002 through 

2007. As expected, owing to the nonlinear latent trends, 

the EWD is sometimes slightly closer to the population 

value than the other estimators are in these central years. 

By definition, the CDE and the EWD estimator use the 

observations from a single panel’s remeasurement for 

years 2000 and 2009 while the noncentralized MWMOR 

estimator uses the observations from three panels. The 

CDE and the EWD estimator use the observations from 

the remeasurement of three panels for years 2001 and 

2008 while the semi-centralized MWMOR estimator uses 

the observations from four panels for these years. In the 

temporal extremes, the results for the MWMOR estimator 
8 FRANCIS A. ROESCH

are mixed, with the expected bias due to the use of the 

off-center panels sometimes (such as in live growth and 

mortality for 2009), but not always, being overridden by 

the lower variance from the larger sample. This outcome 

can be verified by comparing the harvest and mortality 

results in figures 01B and 01E. 

Confounded with this variance-bias tradeoff effect is an 

off-center effect in the MWMOR. Drawing the reader’s 

attention to the mortality estimates for the years 2000 and 

2001 in figures 01M, 01B, and 01E, and noting that the 

true population mean shown in figure 01M is unknown 

in an actual inventory, we can discuss how the use of 

multiple estimators can reveal underlying population 

phenomena. During these particular years, the MWMOR 

results appear superior by all measures and that is the 

conclusion that would be drawn by the usual methods 

of estimator comparison. In figure 01M the MWMOR 

estimates for 2000 and 2001 are much closer to the 

population mean than are the other two estimators. More 

important, however, is the relative association of the 

various estimators. The MWMOR estimates for 2000 

and 2001 are lower than and have a lower slope than the 

corresponding estimates for the other two estimators.

Additionally, the MWMOR estimate for 2000 is very 

close to the CDE and EWD estimates for 2001. This is 

because the three estimates use the same information in 

different ways. If all of the assumptions underlying any 

arguments for optimality for these three estimators were 

strictly true, this particular set of facts would not be in 

evidence. That is, the centralized three-panel estimates 

for 2001 for the CDE and EWD would be closer to the 

population mean for 2001, than the off-center three-panel 

MWMOR estimate for 2000 is to the population mean for 

2000. The relative positioning of the estimates, as they 

exist, would tell us, in lieu of a plotted population mean 

line, that the slope of the mortality trend is increasing to 

the left (or decreasing as time progresses.). Therefore, 

the pattern or relative positioning of the estimates from 

the different estimators is revealing something about 

the underlying population that we would not necessarily 

know if we had favored and used a single estimator. As 

the reader progresses through the results for the various 

populations and sampling error structures, she or he 

should note that different combinations of population 

and error structure will result in different rankings of the 

estimators, but regular patterns between the estimators 

always indicate the same thing about the underlying trend. 

Error Structure 2—Recall that in Error Structure 2, a 

positive sampling bias was interjected into all change 

components. The results can be seen by comparing the 

figures for Error Structure 2 with those for Error Structure 

1. For instance all of the estimates in figure 02M are 

higher than the corresponding estimates in figure 01M, 

but the trends for the individual estimators are virtually 

identical between the two figures. What may not be 

intuitively obvious is that sampling bias does not always 

result in estimates of higher bias. A comparison of figures 

01B and 02B shows that for some components in certain 

years the bias created by the smoothing effect of the 

estimators is counteracted by the sampling bias. This 

offsetting effect is also reflected in lower empirical mean 

squared error values for the year 2000 entry estimates 

in figure 02E, for instance, relative to the corresponding 

values in figure 01E.

Error Structure 3—Sampling error consisted of variance 

and positive bias on volume, live growth, and entry, and a 

negative bias on harvest and mortality for Error Structure 

3. The results for live growth and entry are therefore 

virtually identical for those components in figures 03M, 

03B, and 03E to the the corresponding graphs in figures 

02M, 02B, and 02E. Our interest lies in comparing the 

mortality and harvest results in figures 03M, 03B, and 

03E, to the corresponding results in figures 01M, 01B, 

and 01E. The mortality and harvest results vary starkly 

due mostly to the different curvilinear population trends. 

Surprisingly, the overall estimates for mortality are 

slightly better under the negatively biased Error Structure 

3 than they were for the unbiased Error Structure 1. 

The same cannot be said for the harvest estimates. The 

mortality estimates are somewhat better under negative 

bias due to the concave upward shape of the population 

curve. The harvest estimates, on the other hand, are 

worse under a negatively biased error structure due to the 

concave downward shape of that population curve. 

Error Structure 4—Error structure 4 had higher variance 

and greater bias than Error Structure 3, with bias of the 

same sign for each component as Error Structure 3. The 

strong sampling bias effect of Error Structure 4 is easily 

observable by comparing the simulation mean graphs of 

figure 04M with those of figure 01M. However, note also 

in those comparisons that the trends for live growth, entry, 

and mortality would appear almost identical in those two 

figures if the population mean lines were not present, as 

they would not be in an actual inventory, and the y-axes 

were not labeled. That is, under both error structures, the 

conclusions that one would draw about trend would be 

the same, and those estimators appear robust for trend in 

those mildly curvilinear cases. Again, the same conclusion 

could not be drawn for harvest. The strong concavedownward curve resulted in most of the estimates having 

negative bias under unbiased sampling error structure, 

while the effect was greatly exacerbated under the strong 

negatively biased error structure.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 9

Population 1

Figures 1eM, 1eB, and 1eE, with e = 1, 2, 3, or 4, give 

the empirical mean, bias, and mean squared error, 

respectively, over 1,000 iterations of 1,000 samples each 

from Population 1 under Sampling Error Structure e, by 

growth component for estimation years 2000 through 

2009.

Population 1 was constructed similarly to Population 

0 except that a mild (latent) nonlinear trend was 

introduced into each of the components to reduce further 

any potential linear effect resulting from the method 

of construction. By examining the results figures for 

Population 1, the reader can verify that the results for 

Population 1 are almost identical to the corresponding 

results for Population 0.

As with the same case in Population 0, the three figures 

for Error Structure 4 in general reflect the bias and 

greater variance of Error Structure 4 over Error Structure 

1 for Population 1. The estimators do not appear to be 

differentially affected by the bias in Error Structure 4. I do 

note that occasionally the bias in the temporal extremities 

shown in figure 11B for the MWMOR estimator is offset 

in figure 14B by the counteracting bias in the sampling 

error structure, resulting in lower empirical mean squared 

error in figure 14E relative to figure 11E. This reinforces 

a point made in Eastaugh and Hasenauer (2013): often the 

implementation of theoretically unbiased sample designs 

will result in biased samples for a myriad of reasons, and 

estimators should therefore be evaluated in consideration 

of that possibility. With respect to robustness, note 

that if we were examining only the outcome for one 

or more of these estimators, and did not know the true 

population values (that is, the population line was missing 

from figures 11M and 14M), we would draw the same 

conclusions about the trend in each of the components. 

This consistency suggests that all of these estimators are, 

at least in this regard, robust.

Population 2

For Population 2, a mild nonlinear trend was introduced 

into the components of live growth, entry, and mortality, 

and a stronger nonlinear trend was introduced into the 

harvest component, in order to simulate increasing 

harvesting pressure. The increased harvesting pressure 

throughout the estimation period had subtle effects 

on the other change components. Close examination 

of the results for live growth, entry, and mortality for 

Population 2 should convince the reader that they are 

virtually identical to the corresponding results figures for 

Population 1. Although the increased harvesting pressure 

is seen in the Mean graphs for Population 2, the concave 

curvilinear shape of the population trend once again is the 

biggest factor in the results, and the resulting conclusions 

mimic exactly those reached in the previously discussed 

population graphs.

Population 3

For Population 3, a mild nonlinear trend was introduced 

into the components of live growth, entry, and harvest, 

and a catastrophic high mortality event (of four times the 

mortality rate of Population 1) was introduced for the year 

2004. 

Figures 34M, 34B, and 34E give the empirical mean, bias, 

and mean squared error, respectively, for 1,000 iterations 

of 1,000 samples each from Population 3 under Sampling 

Error Structure 4, for the CDE, the EWD estimator, 

and the MWMOR estimator, by growth component and 

estimation year. The most notable outcome for Population 

3 under all error structures is that all three estimators 

give no indication of the extreme anomaly for mortality 

in 2004, due to the smoothing effects of both the sample 

design and the estimators applied to the outcomes of 

the design. The anomaly in 2004 is spread out over the 

estimates for the surrounding years, so the estimators 

reflect very high empirical bias and empirical mean 

squared error for mortality in 2004 in figures 34B and 

34E, respectively. The sample design itself makes this 

single-year anomaly particularly difficult to evaluate, and 

impossible to definitively separate from a possible spatial 

effect.

Population 4

Population 4 was initially as in Population 1 and then 

postulated climate change effects were simulated 

by increasing mortality and decreasing growth and 

recruitment, with harvest levels remaining the same.

The empirical mean, bias, and mean squared error, 

respectively, for 1,000 iterations of 1,000 samples each 

from Population 4 under Sampling Error Structure 3, 

for the CDE, the EWD estimator, and the MWMOR 

estimator, by growth component and estimation year, are 

given in figures 43M, 43B, and 43E. Although Population 

4 has a greater diversity of conditions than Population 1, 

the differences in estimator performance between figures 

43M, 43B, and 43E, relative to the corresponding graphs 

in figures 11M, 11B, and 11E, appear to correspond 

to those that could be expected under the more severe 

Sampling Error Structure 3.
10 FRANCIS A. ROESCH

Population 3—With Outside Information

Figures 3MM, 3MB, and 3ME give the empirical mean, 

bias, and mean squared error, respectively, for 1,000 

iterations of 1,000 samples each from Population 3 under 

Sampling Error Structures 1 through 4, for the estimators 

incorporating outside information, for the mortality 

component by estimation year. Although it is true that the 

outside information was not perfect, all of the estimators 

incorporating the outside information benefitted in the 

form of improved mortality estimates for the year 2004 

and the surrounding years that used observations spanning 

2004. Figure 3MM shows that under each of the sampling 

error structures each of the estimators exhibits a pattern 

very similar to the mortality trend for the population. 

The patterns differ from the population trend predictably 

by sampling error structure. Figure 3MB shows more 

clearly than figure 3MM that the order of the estimates 

for each year remains constant through the different 

sampling error structures, indicating that sampling error 

structure did not differentially affect the estimators. Some 

interaction between the outside information and sampling 

error structure is indicated by the position of the group of 

estimators for 2004 relative to the groups of estimators 

for the surrounding years. Figure 3ME shows that the 

empirical mean squared error results for the mixed 

estimator models are often, but not always, higher than 

they are for the other models. This effect could be viewed 

as the cost of compatibility.

CONCLUSIONS

I explored some special problems that arise in estimation 

of the components of change when the temporal scale 

of the population estimand of interest is finer than the 

scale of observation under both biased and unbiased 

sampling error structures. In the example simulations, 

the temporal scale of observation was 5 years while 

the temporal dimension of the population of interest 

was 1 year. All of the approaches worked well in the 

temporal mid-range of observations in the presence of 

smooth population trends, under unbiased sampling error 

structures. The smoothing aspect of the sample design 

and the estimators that did not use outside information 

was problematic when the sign of the slope of a trend 

changed. The maximum year and years surrounding 

it were under-predicted, while the minimum year and 

years surrounding it were over-predicted under unbiased 

sampling error. By interjecting a single year anomaly 

into Population 3, I presented an especially difficult 

(but realistic) situation given the sampling frame. The 

results of these more thorough simulations support the 

simulations and conclusions of Roesch (2007a) with 

respect to comparisons between the EWD estimator and 

this application of the Mixed Estimator. The simulations 

also showed the variance/bias trade-off encountered 

when the MWMOR estimator was used in the extremity 

years of observation. Although the MWMOR estimator 

is sometimes biased in the presence of trend in the 

extremity years, the empirical mean-squared error was 

often lower than it was for the other estimators. Of the 

four general approaches for estimating the components 

of forest change from this annually rotating five-panel 

sample design (the Centralized Difference Estimator, 

the Exponentially Weighted Difference estimator, the 

Moving-Window Mean of Ratios estimator, and the 

Mixed Estimator), the first three approaches are very 

compatible with large monitoring efforts that require 

intervention-free results. These three simple approaches 

were shown to be amenable when outside information 

suggests an adaptation to the weighting scheme. The 

fourth approach, the mixed estimator, is also amenable 

to the incorporation of extraneously derived information 

and can easily incorporate complex models. No single 

estimation approach has (or could have) been shown to be 

a panacea, and some notable differences in performance 

were observed at the temporal extremities of observation, 

in the presence of temporal anomalies, and in the presence 

of biased sampling error structures. With respect to trend, 

all of the estimators are robust, but the CDE, the EWD 

estimator, and the MWMOR estimator are somewhat 

nonresponsive to highly variable trends, and all estimators 

are subject to the smoothing effect of the sample design 

considered here. This particular deficiency in the 

design was shown to be readily corrected through the 

incorporation of outside information. 

This study suggests that one should probably not attempt 

to choose a single estimation approach in order to address 

the widest range of estimation objectives. The estimators 

considered here were shown to differ in their respective 

robustness to different, but realistic, underlying error 

structures, aspects of which may not always be known. 

Rather, an investigator would be well served to embrace 

the philosophy behind the Thomas and Roesch (1990) and 

Eastaugh and Hasenauer (2013) articles: when real data 

are involved, there is value in making estimates using as 

many theoretical approaches as possible. When different 

approaches produce similar results, there is strong 

evidence for those results. However, it can be even more 

informative when different (but defensible) approaches 

produce varied results. In that situation, it is incumbent 

upon the analyst to figure out why the varied results have 

occurred. Quite simply, an analyst should look at the 

data from as many angles as she or he has the time and 

energy for, in an attempt to understand fully the natural 

phenomena that are being imperfectly observed. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 11

Table 0—Distribution Statistics for Population 0

Year

Component Statistic 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

 Volume

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 4.48 10.78 16.45 21.54 25.98 30.52 33.88 36.37 38.89 40.42 41.86 42.69 44.04

Median 55.26 63.07 69.78 75.28 80.49 83.95 86.51 88.65 90.57 93.02 96.14 99.99 103.70

Mean 81.92 87.45 92.68 97.18 101.35 105.13 108.23 110.63 112.65 114.66 117.31 120.53 124.62

3rd Quartile 125.85 133.47 139.69 145.20 149.49 153.86 157.93 160.80 163.55 166.30 169.10 174.00 180.14

Maximum 583.18 591.50 609.80 625.30 638.04 648.09 655.38 659.95 663.75 664.22 661.10 664.47 789.74

Live growth

Minimum -5.72 -6.80 -42.60 -20.45 -10.48 -13.15 -22.14 -31.07 -23.06 -9.42 -10.28 -6.06 -11.56

1st Quartile 0 0.42 1.08 1.52 1.88 1.84 1.69 1.22 0.48 0 0 0 0

Median 2.86 3.65 4.21 4.54 4.77 4.66 4.28 3.79 3.31 2.76 2.24 1.98 1.83

Mean 6.36 6.70 6.88 6.77 6.54 6.15 5.71 5.30 5.05 4.98 5.02 5.22 5.42

3rd Quartile 8.99 9.34 9.56 9.34 9.10 8.56 8.06 7.72 7.40 7.41 7.53 7.86 8.11

Maximum 69.48 65.71 68.33 64.76 72.13 60.43 53.66 78.84 103.90 127.03 150.16 173.29 196.42

Entry

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0.09 0.13 0.12 0.09 0.02 0 0 0 0 0

Median 0.19 0.29 0.34 0.38 0.38 0.38 0.33 0.27 0.19 0.13 0.04 0 0

Mean 0.99 0.97 0.89 0.84 0.81 0.75 0.67 0.60 0.57 0.56 0.58 0.61 0.65

3rd Quartile 0.81 0.89 0.93 0.92 0.91 0.87 0.79 0.72 0.67 0.62 0.60 0.61 0.64

Maximum 64.29 66.40 67.40 63.08 70.45 40.92 30.22 11.06 12.71 14.28 15.85 17.42 18.99

Mortality

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0.04 0.10 0.09 0.06 0 0 0 0 0 0

Mean 1.46 1.31 1.17 1.04 0.93 0.84 0.77 0.74 0.74 0.77 0.82 0.87 0.92

3rd Quartile 0.56 0.62 0.70 0.82 0.87 0.83 0.73 0.57 0.44 0.38 0.35 0.30 0.30

Maximum 159.91 122.92 85.92 62.07 55.37 48.66 41.95 35.25 28.54 27.41 30.57 33.40 35.90

Harvest

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0 0 0 0 0 0 0 0 0 0

Mean 0.36 1.14 2.11 2.41 2.63 2.96 3.21 3.13 2.86 2.13 1.57 0.87 0.33

3rd Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Maximum 462.97 469.93 478.33 471.11 474.23 483.09 470.31 573.27 573.37 568.45 574.04 571.40 579.96
12 FRANCIS A. ROESCH

Table 1—Distribution Statistics for Population 1

Year

Component Statistic 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Volume

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 4.68 11.1 16.78 21.82 26.3 30.81 34.43 36.86 39.3 40.89 42.37 43.28 44.62

Median 56.14 64.25 71 76.6 82.02 85.6 88.27 90.41 92.59 95 98.34 102.38 106.3

Mean 83.06 88.69 94.05 98.69 103.0 106.94 110.14 112.71 114.87 117 119.88 123.33 127.69

3rd Quartile 127 134.71 141.72 147.24 152.21 156.87 161.05 164.54 167.4 170.31 173.11 178.22 184.87

Maximum 599.56 602.77 620.56 636.28 650.25 661.39 669.52 674.7 679.09 679.78 676.46 684.32 831.11

Live growth

Minimum -5.66 -6.86 -43.60 -21.13 -10.62 -13.44 -22.80 -32.27 -24.60 -10.21 -10.94 -6.57 -12.45

1st Quartile 0 0.46 1.12 1.58 1.95 1.931 1.78 1.29 0.51 0 0 0 0

Median 2.91 3.72 4.30 4.68 4.96 4.88 4.51 4.02 3.53 2.95 2.408 2.13 1.99

Mean 6.49 6.84 7.06 6.99 6.79 6.44 6.02 5.61 5.37 5.32 5.36 5.56 5.76

3rd Quartile 9.10 9.50 9.80 9.63 9.45 8.95 8.49 8.18 7.89 7.91 8.06 8.42 8.67

Maximum 70.95 67.54 70.22 66.27 74.08 63.73 56.33 83.58 110.91 134.04 157.17 180.30 203.43

Entry

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0.09 0.13 0.12 0.10 0.02 0 0 0 0 0

Median 0.19 0.29 0.35 0.39 0.40 0.39 0.35 0.28 0.20 0.13 0.043 0 0

Mean 0.99 0.98 0.90 0.86 0.84 0.78 0.71 0.63 0.60 0.60 0.62 0.65 0.69

3rd Quartile 0.82 0.90 0.95 0.95 0.94 0.91 0.84 0.76 0.71 0.67 0.64 0.65 0.68

Maximum 65.92 67.49 69.45 43.39 73.56 33.3 32.30 11.83 13.74 15.31 16.88 18.44 20.01

Mortality

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0.04 0.11 0.10 0.06 0 0 0 0 0 0

Mean 1.48 1.33 1.19 1.07 0.97 0.88 0.82 0.78 0.79 0.82 0.87 0.92 0.97

3rd Quartile 0.56 0.63 0.72 0.84 0.90 0.87 0.77 0.61 0.47 0.40 0.37 0.32 0.31

Maximum 162.27 125.27 88.28 63.89 57.18 50.48 43.77 37.07 30.36 29.23 32.39 34.55 37.63

Harvest

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0 0 0 0 0 0 0 0 0 0

Mean 0.37 1.14 2.132 2.47 2.73 3.14 3.34 3.3 3.06 2.23 1.66 0.93 0.35

3rd Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Maximum 461.93 493.15 478.92 486.63 493.17 495.13 497.93 575.7 591.30 590.59 602.60 589.36 628.30
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 13

Table 2—Distribution Statistics for Population 2

Year

Component Statistic 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Volume

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 4.68 11.13 16.78 21.74 26.16 30.62 34.15 36.32 38.42 39.84 41.33 42.10 43.36

Median 56.13 64.25 71.04 76.52 81.91 85.37 87.90 89.73 91.90 94.25 97.37 101.10 104.79

Mean 83.07 88.72 94.05 98.63 102.91 106.77 109.87 112.23 114.24 116.25 119.01 122.40 126.71

3rd Quartile 126.99 134.77 141.73 147.21 152.17 156.79 160.95 164.30 167.08 169.98 172.70 177.70 184.12

Maximum 595.09 603.74 622.28 638.03 651.04 661.42 668.94 673.73 677.89 678.48 675.11 691.20 828.74

Live growth

Minimum -5.71 -6.83 -43.71 -21.18 -10.77 -13.48 -22.79 -32.38 -24.33 -10.20 -11.00 -6.54 -12.55

1st Quartile 0 0.46 1.12 1.58 1.95 1.93 1.78 1.29 0.50 0 0 0 0

Median 2.91 3.71 4.30 4.68 4.96 4.88 4.51 4.02 3.52 2.95 2.41 2.13 1.99

Mean 6.49 6.84 7.06 6.99 6.79 6.44 6.02 5.61 5.37 5.32 5.36 5.56 5.76

3rd Quartile 9.10 9.49 9.80 9.63 9.45 8.96 8.49 8.18 7.88 7.91 8.06 8.42 8.67

Maximum 71.19 67.80 70.52 67.14 74.87 63.18 56.71 83.66 111.09 134.23 157.36 180.49 203.62

Entry

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0.09 0.13 0.12 0.10 0.02 0 0 0 0 0

Median 0.19 0.29 0.35 0.39 0.40 0.39 0.35 0.28 0.20 0.13 0.04 0 0

Mean 1.00 0.98 0.90 0.86 0.84 0.78 0.71 0.63 0.60 0.60 0.62 0.65 0.69

3rd Quartile 0.82 0.90 0.95 0.95 0.94 0.91 0.84 0.76 0.71 0.67 0.64 0.65 0.68

Maximum 66.43 67.37 69.43 42.65 73.55 33.63 32.12 11.68 13.49 15.06 16.63 18.20 19.76

Mortality

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0.04 0.11 0.10 0.06 0 0 0 0 0 0

Mean 1.48 1.33 1.19 1.07 0.97 0.87 0.82 0.78 0.79 0.82 0.87 0.92 0.97

3rd Quartile 0.56 0.63 0.72 0.84 0.90 0.87 0.76 0.61 0.47 0.40 0.37 0.32 0.31

Maximum 162.00 125.00 88.01 63.83 57.13 50.42 43.71 37.01 30.30 29.03 32.19 34.57 37.65

Harvest

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0 0 0 0 0 0 0 0 0 0

Mean 0.34 1.17 2.19 2.51 2.81 3.23 3.55 3.45 3.18 2.35 1.72 0.99 0.38

3rd Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Maximum 459.38 481.46 485.90 497.54 505.01 504.85 514.68 577.39 586.59 589.27 599.83 594.19 676.00
14 FRANCIS A. ROESCH

Table 3—Distribution Statistics for Population 3

Year

Component Statistic 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Volume

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 4.95 11.35 16.92 21.96 26.34 30.84 33.85 36.19 38.53 39.87 41.12 42.26 43.58

Median 56.68 64.91 71.49 76.92 82.19 85.65 86.53 88.33 90.98 93.18 96.48 100.53 103.69

Mean 84.02 89.49 94.64 99.10 103.25 106.97 108.10 110.65 112.81 114.99 117.83 121.32 125.72

3rd Quartile 127.27 135.26 142.04 147.53 152.35 156.86 157.92 161.43 164.42 167.13 170.33 175.53 181.97

Maximum 667.96 609.09 628.10 644.38 658.01 668.73 667.17 672.10 676.24 676.75 673.31 688.97 822.51

Live growth

Minimum -5.68 -6.76 -43.76 -21.12 -10.72 -13.35 -22.78 -32.17 -24.85 -10.08 -11.08 -6.56 -12.57

1st Quartile 0 0.47 1.12 1.58 1.95 1.93 1.78 1.29 0.50 0 0 0 0

Median 2.93 3.73 4.31 4.68 4.96 4.88 4.51 4.02 3.52 2.95 2.41 2.13 1.99

Mean 6.53 6.85 7.06 6.99 6.79 6.44 6.02 5.61 5.37 5.32 5.36 5.56 5.76

3rd Quartile 9.16 9.51 9.80 9.63 9.45 8.95 8.49 8.18 7.88 7.91 8.06 8.42 8.67

Maximum 71.03 67.42 70.88 66.71 74.77 63.07 56.87 84.05 111.66 134.79 157.92 181.05 204.18

Entry

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0.09 0.13 0.12 0.10 0.02 0 0 0 0 0

Median 0.19 0.29 0.35 0.39 0.40 0.39 0.35 0.28 0.20 0.13 0.04 0 0

Mean 0.99 0.99 0.91 0.86 0.84 0.78 0.71 0.63 0.60 0.60 0.62 0.65 0.69

3rd Quartile 0.82 0.90 0.95 0.95 0.95 0.91 0.84 0.76 0.71 0.67 0.64 0.65 0.68

Maximum 66.93 67.97 69.15 43.37 72.30 44.83 32.57 11.77 13.61 15.17 16.74 18.31 19.88

Mortality

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0.04 0.11 0.38 0.06 0 0 0 0 0 0

Mean 1.68 1.53 1.39 1.27 1.17 2.96 0.81 0.78 0.78 0.81 0.84 0.89 0.94

3rd Quartile 0.58 0.65 0.75 0.87 0.97 3.13 0.76 0.60 0.46 0.40 0.36 0.32 0.30

Maximum 161.99 132.67 124.81 116.96 109.11 101.25 43.73 37.02 30.31 28.49 31.57 34.65 37.73

Harvest

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0 0 0 0 0 0 0 0 0 0

Mean 0.37 1.16 2.11 2.43 2.74 3.13 3.36 3.31 3.02 2.27 1.65 0.91 0.37

3rd Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Maximum 455.44 477.77 483.53 484.98 489.37 501.90 506.03 581.44 578.92 584.22 582.56 601.94 628.32
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 15

Table 4—Distribution Statistics for Population 4

Year

Component Statistic 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Volume

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 7.55 13.69 18.64 21.62 25.20 27.87 30.13 31.57 32.68 33.18 33.70 34.31 35.19

Median 60.42 67.19 72.32 76.17 79.53 80.75 80.88 80.38 80.30 80.88 81.91 83.56 85.78

Mean 86.40 91.11 95.22 98.23 100.65 102.47 103.42 103.70 103.65 103.61 104.24 105.51 107.70

3rd Quartile 130.65 137.25 143.28 146.07 148.62 150.60 151.03 151.21 151.33 151.43 152.03 152.77 155.92

Maximum 608.21 602.18 615.03 625.44 634.34 640.85 645.19 647.47 649.17 648.63 645.73 640.58 674.22

Live growth

Minimum -4.73 -5.33 -32.68 -15.17 -7.41 -9.00 -14.86 -20.59 -15.24 -6.27 -6.59 -3.93 -7.37

1st Quartile 0 0.49 0.92 1.20 1.37 1.33 1.17 0.83 0.31 0 0 0 0

Median 2.47 3.05 3.31 3.43 3.49 3.33 2.97 2.58 2.22 1.83 1.46 1.28 1.15

Mean 5.69 5.83 5.64 5.29 4.89 4.44 3.97 3.61 3.39 3.34 3.40 3.59 3.81

3rd Quartile 7.69 7.87 7.65 7.08 6.71 6.12 5.60 5.25 4.97 4.94 4.99 5.24 5.50

Maximum 67.08 60.22 60.14 52.60 57.23 45.53 37.36 53.90 69.99 93.12 116.25 139.38 162.51

Entry

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0.08 0.11 0.10 0.08 0.01 0 0 0 0 0

Median 0.19 0.27 0.31 0.33 0.33 0.32 0.28 0.22 0.15 0.10 0.03 0 0

Mean 0.94 0.84 0.80 0.76 0.70 0.61 0.58 0.50 0.47 0.47 0.49 0.52 0.56

3rd Quartile 0.78 0.81 0.84 0.81 0.80 0.74 0.67 0.60 0.56 0.52 0.50 0.51 0.54

Maximum 66.89 62.86 49.51 53.90 24.77 13.09 31.69 9.36 10.71 12.28 13.85 15.41 16.98

Mortality

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0.05 0.12 0.11 0.07 0 0 0 0 0 0

Mean 1.56 1.42 1.28 1.17 1.07 0.99 0.94 0.92 0.93 0.96 1.00 1.03 1.07

3rd Quartile 0.63 0.70 0.78 0.93 1.01 0.99 0.88 0.71 0.55 0.48 0.44 0.38 0.34

Maximum 167.89 130.90 93.90 69.67 62.96 56.25 49.55 42.84 36.13 34.56 37.64 40.72 43.80

Harvest

Minimum 0 0 0 0 0 0 0 0 0 0 0 0 0

1st Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Median 0 0 0 0 0 0 0 0 0 0 0 0 0

Mean 0.36 1.15 2.14 2.46 2.71 3.11 3.33 3.24 2.97 2.22 1.62 0.89 0.36

3rd Quartile 0 0 0 0 0 0 0 0 0 0 0 0 0

Maximum 474.55 479.35 482.29 486.63 485.76 501.16 496.55 542.87 559.86 553.63 552.81 554.08 618.84
16 FRANCIS A. ROESCH

Figure 01M—The mean over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error Structure 1, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 17

Figure 01B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error 

Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
18 FRANCIS A. ROESCH

Figure 01E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 0 under 

Sampling Error Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 19

Figure 02M—The mean over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error Structure 2, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
20 FRANCIS A. ROESCH

Figure 02B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error 

Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 21

Figure 02E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 0 under 

Sampling Error Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
22 FRANCIS A. ROESCH

Figure 03M—The mean over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error Structure 3, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 23

Figure 03B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error 

Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
24 FRANCIS A. ROESCH

Figure 03E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling 

Error Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 25

Figure 04M—The mean over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error Structure 4, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
26 FRANCIS A. ROESCH

Figure 04B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling Error 

Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 27

Figure 04E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 0 under Sampling 

Error Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
28 FRANCIS A. ROESCH

Figure 11M—The mean over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error Structure 1, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 29

Figure 11B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error Structure 

1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
30 FRANCIS A. ROESCH

Figure 11E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 1 under 

Sampling Error Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 31

Figure 12M—The mean over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error Structure 2, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
32 FRANCIS A. ROESCH

Figure 12B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error 

Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 33

Figure 12E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 1 under 

Sampling Error Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
34 FRANCIS A. ROESCH

Figure 13M—The mean over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error Structure 3, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 35

Figure 13B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error 

Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
36 FRANCIS A. ROESCH

Figure 13E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 1 under Sampling 

Error Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 37

Figure 14M—The mean over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error Structure 4, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
38 FRANCIS A. ROESCH

Figure 14B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 1 under Sampling Error 

Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 39

Figure 14E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 1 under 

Sampling Error Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
40 FRANCIS A. ROESCH

Figure 21M—The mean over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error Structure 1, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 41

Figure 21B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error 

Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
42 FRANCIS A. ROESCH

Figure 21E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 2 under 

Sampling Error Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 43

Figure 22M—The mean over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error Structure 2, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
44 FRANCIS A. ROESCH

Figure 22B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error 

Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 45

Figure 22E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 2 under 

Sampling Error Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
46 FRANCIS A. ROESCH

Figure 23M—The mean over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error Structure 3, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 47

Figure 23B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error 

Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
48 FRANCIS A. ROESCH

Figure 23E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 2 under 

Sampling Error Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 49

Figure 24M—The mean over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error Structure 4, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
50 FRANCIS A. ROESCH

Figure 24B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 2 under Sampling Error 

Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 51

Figure 24E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 2 under 

Sampling Error Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
52 FRANCIS A. ROESCH

Figure 31M—The mean over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error Structure 1, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 53

Figure 31B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error 

Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
54 FRANCIS A. ROESCH

Figure 31E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 3 under 

Sampling Error Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 55

Figure 32M—The mean over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error Structure 2, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
56 FRANCIS A. ROESCH

Figure 32B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error 

Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 57

Figure 32E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 3 under 

Sampling Error Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
58 FRANCIS A. ROESCH

Figure 33M—The mean over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error Structure 3, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 59

Figure 33B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error 

Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
60 FRANCIS A. ROESCH

Figure 33E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 3 under 

Sampling Error Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 61

Figure 34M—The mean over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error Structure 4, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
62 FRANCIS A. ROESCH

Figure 34B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error 

Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 63

Figure 34E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 3 under 

Sampling Error Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
64 FRANCIS A. ROESCH

Figure 41M—The mean over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error Structure 1, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 65

Figure 41B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error 

Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
66 FRANCIS A. ROESCH

Figure 41E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 4 under 

Sampling Error Structure 1, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 67

Figure 42M—The mean over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error Structure 2, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
68 FRANCIS A. ROESCH

Figure 42B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error 

Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 69

Figure 42E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 4 under 

Sampling Error Structure 2, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
70 FRANCIS A. ROESCH

Figure 43M—The mean over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error Structure 3, for 

the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the Moving-Windows 

Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 71

Figure 43B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error 

Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the 

Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
72 FRANCIS A. ROESCH

Figure 43E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 4 under 

Sampling Error Structure 3, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 73

Figure 44M—The mean over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error Structure 4, 

for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and the MovingWindows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
74 FRANCIS A. ROESCH

Figure 44B—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 4 under Sampling Error 

Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) estimator, and 

the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 75

Figure 44E—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 4 under 

Sampling Error Structure 4, for the Centralized Difference Estimator (CDE), the Exponentially Weighted Difference (EWD) 

estimator, and the Moving-Windows Mean of Ratios (MWMOR) estimator, by growth component and estimation year.
76 FRANCIS A. ROESCH

Figure 3MM—The mean over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error Structures 

1 through 4, for the Mortality component by estimation year, for the estimators incorporating outside information (CDE-OI, 

EWD-OI, MWMOR-OI) and the Mixed Estimator (ME) under Models 1 through 3. 
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 77

Figure 3MB—The empirical bias, over 1,000 iterations of 1,000 samples each from Population 3 under Sampling Error 

Structures 1 through 4, for the Mortality component by estimation year, for the estimators incorporating outside information 

(CDE-OI, EWD-OI, MWMOR-OI) and the Mixed Estimator (ME) under Models 1 through 3. 
78 FRANCIS A. ROESCH

Figure 3ME—The empirical mean squared error, over 1,000 iterations of 1,000 samples each from Population 3 under 

Sampling Error Structures 1 through 4, for the Mortality component by estimation year, for the estimators incorporating outside 

information (CDE-OI, EWD-OI, MWMOR-OI) and the Mixed Estimator (ME) under Models 1 through 3.
TOWARD ROBUST ESTIMATION OF THE COMPONENTS OF FOREST POPULATION CHANGE: SIMULATION RESULTS 79

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LITERATURE CITED

ACKNOWLEGMENTS

This manuscript was improved by comments offered by Dr. James Westfall, U.S. Department of Agriculture Forest 

Service, Northern Research Station, and Dr. Christopher Oswalt, U.S. Department of Agriculture Forest Service, 

Southern Research Station. 

Roesch, Francis A. 2014. Toward robust estimation of the components of forest population change: 

simulation results. e-Gen. Tech. Rep. SRS-194. Asheville, NC: U.S. Department of Agriculture Forest 

Service, Southern Research Station. 79 p.

This report presents the full simulation results of the work described in Roesch (2014), in which multiple 

levels of simulation were used to test the robustness of estimators for the components of forest change. 

In that study, a variety of spatial-temporal populations were created based on, but more variable than, an 

actual forest monitoring dataset, and then those populations were sampled under four sets of sampling 

error structure. An estimator modification was shown, to be used when extraneously obtained information 

indicated that a deviation to the assumed population model existed. The extraneous information was also 

incorporated into a mixed estimator. The first three approaches, without the incorporation of extraneous 

information, are compatible with large monitoring efforts that require intervention-free results. The mixed 

estimation approach accounts for model assumptions that sometimes remain latent in other approaches 

and is amenable to the formal incorporation of the extraneously obtained information. All four approaches 

were shown to work well when the sampling error structure was unbiased, while some notable differences 

in performance were observed at the temporal extremities of observation, in the presence of temporal 

anomalies, and in the presence of biased sampling error structures. Only those results necessary to make 

the salient points were presented in Roesch (2014). Full results are presented here both for full disclosure 

and for the reader interested in a more detailed understanding of the effects of realistic sampling errors on 

temporal estimates.

Key Words: Annual inventories, components of change, forest monitoring, sampling error, spatial-temporal 

sample design.

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2 FRANCIS A. ROESCH

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