N O T I C E 

THIS DOCUMENT HAS BEEN REPRODUCED FROM 

MICROFICHE. ALTHOUGH IT IS RECOGNIZED THAT 

CERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED 

IN THE INTEREST OF MAKING AVAILABLE AS MUCH 

INFORMATION AS POSSIBLE 
CROP AREA ESTIMATION BASED ON REHOTELY-SENSED DATA

WITH AN ACCURATE BUT COSTLY SUBSAMPLE

(NASA-Cfi-176017) CROF AhIA kETIHATION BASED N86-13814

ON RLMOTELY-SENSED DATA VITh AN ACCURATE BUT

COSTLY SUBSAHFLE Final Report, 1 Jan. 1984

- 31 Auq. 1985 (Southern Methcdist Univ.) Uaclas

33 p HC A03/HF A01 CSCL 02C J1/43 04E33

FINAL REPORT

ON

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

COOPERATIVE AGREEMENT NO. NCC 9-9

1 January 1984 to 31 August 1985

Richard F. Gunst

Department of Statistics

Southern Methodist University

Dallas, Texas 75275

NOV 1985 

Atu 84 ,OFD

Doi

^w
ABSTRACT

This report documents research activities which were conducted under

the auspices of National Aeronauttca and Space Administration Cooperative

Agreement NCC 9-9. During this contract period research efforts were concentrated in two primary areas. The first area was an investigation of the

use of measurement error models as alternatives to least squares regression

estimators of crop production or timber biomass. The second primary area

of investigation was on the estimation of the mixing proportion of twocomponent mixture models. This report documents publications, technical

reports, submitted manuscripts, and oral presentations which occurred as a

result of these research efforts. Possible future areas of fruitful

research are mentioned.
Table of Contents

Page

ABSTRACT i

i

I. MAJOR RESEARCH ACTIVITIES 1

1

A. MEASUREMENT ERROR MODELS 2

B. MIXTURE MODELS 3

C. OTHER TOPICS 4

II. PROSPECTIVE FUTURE RESEARCH 5

III. ORAL PRESENTATIONS AND OTHER ACTIVITIES 7

s

' IV. APPENDICES 8

A. COMPLETED RESEARCH 9

B. ORAL PRESENTATIONS 11

C. ADDITIONAL WORK 12

m.
I

I. MAJOR RESEARCH ACTIVITIES

Research supported by this cooperative agreement primarily focussed on

two major topics: regression estimation using models in which both the

response and the predictor variables are subject to measurement errors and

the estimation of parameters in mixture models. The first topic was

investigated as a means of improving satellite remote-sensing estimates of

crop production and timber biomass by combining satellite estimates with

ground truth measurements. The latter topic was explored in an effort to

improve the estimation of parameters in models of crop or vegetation proportions when there are two or more different types of vegetation it a

segment.

Several research publications resulted from the research on these and

other topics. Six publications appeared in referred scientific journals

and another three manuscripts were published as technical reports or proceedings papers. In addition, four manuscripts are currently submitted for

publication, the last of which is included in this report as Appendix C. A

complete list of publications and submitted manuscripts is given in

Appendix A. Ten oral presentations, listed in Appendix B, were given to

further disseminate the results of this research.

A summary of the major topics of research which were undertaken

through the support of this cooperative agreement are presented in the next

three sections.
2

A. MEASUREMENT ERROR MODELS

The primary focus of the research conducted under the auspices of this

cooperative agreement was on the investigation of techniques for improving

the use of satellite remote-sensing estimates of crop proportions and

timber biomass. Specific attention was devoted to the combining of

relatively inexpensive but imprecise satellite estimates with relatively

expensive but highly precise ground-truth estimates. Although least

squares regression estimates had been used for some time in this effort, it

was recognized that both the satellite estimates and the ground truth

estimates are subject to measurement error, thereby invalidating one of the

key assumptions needed for the use of least squares estimation. Thus

intense research activities were directed toward the investigation of

regression estimation with measurement error models.

Denote a true (i.e., error-free) ground-truth measurement by Y and the

corresponding error-free satellite measurement by X. Assume that an

adequate representation of the relationship between the two measurements is

given by a linear model of the form

Y - a+OX.

Because of errors of measurement (e.g., registration errors, irregularity

shaped fields, etc.), the true ground-truth and satellite measurements are

not observed. Rather, one observes

x - X+u and y - Y+v ,

where u and v represent the measurement errors. In this framework the

least squares estimators of a and S are biased since an underlying

assumption which is necessary for unbiased least squares estimation is that

the observable predictor variable x is measured without error.
3

To date, research has concentrated on structural measurement error

models; i.e.. the true predictor variable X is stochastic. It is well

known that if X. u. and v are normally distributed there does not exist

consistent estimators of a and B unless ( a) one or more of the model

parameters it known. ( b) replicated observations are available . or (c)

measurements are taken on additional ( instrumental) variables which are

correlated with the true predictor variable X but not with the errors u and

V.

Assuming independent normal distributions for X. u. and v with

A - var(v)/var(u) known, an investigation was conducted into the effects of

sample size on the precision of the maximum likelihood estimator of 9 and

on the consequences of selecting an erroneous value for A. The results of

this research were reported in publication 1(b) (Appendix A) and technical

reports 2(a) and 2(c). A parallel investigation when the errors u and v

are correlated is reported in publication 1(d). A summary of many of the

properties of these estimators and a comparison with least squares is

reported in the manuscript 3(b).

B. MIXTURE MODELS

Mixture models are used to probabilistically characterize the occurrence

of spectral measurements from segments in which two or more crops or other

vegetation are present. If x denotes a (possibly vector-valued) spectral

measurement from a segment in which two crops are present, the probability

density function f 0 (x) can be expressed as

t:

fa(x) - pg 1 (x;o 1 ) + (1-p)g2(x ;92)
4

where g i (x;e ' ) is the density function for x in crop J. e - ( p,# 1 ,0 2 )' is a

vector of model parameters (with e 1 possibly vector-valued), and p is the

proportion of crop 1 in the segment.

Estimation of a presents challenging and, to date, many unsolved

problems. During the course of this cooperative agreement progress was

achieved in the estimation of the mixing proportion p. The manuscript 3(a)

examines maximum likelihood and minimum distance estimation of the mixing

proportion p when the two component distributions g i ( x1 :*1 ) are represented

by three-parameter Weibull distributions. The three-parameter Weibull was

used because of the variety of shapes it can have by specifying values of

the parameters. In manuscript 3(a) the distance measure used was the

Cramer-von-Rises distance W 2 . whereas in manuscript 3(d) (see also Appendix

C) the Hellinger distance was used with normal component distributions.

C. OTHER TOPICS

Additional research which was completed during the duration of this

cooperative agreement is reported in Appendix A. Several papers were

published (1(a), 1(e), 1(f), 2(b)) or submitted for publication (3(c)) on

regression was with collinear variables. One article (1(c)) was published

on the use of quadratic forms in screening procedures.

J
5

II. PROSPECTIVE FUTURE RESEARCH

The research outlined in this report has not only led to the publication of several scholarly articles, it has also identified topics which

offer great potential for further productive research efforts. Some of the

general areas of possible future research activities are now briefly

outlined.

Many important problems remain unresolved when estimating the

parameters of measurement error models. While there is theoretically no

problem with estimating the parameters of measurement error models when the

true predictor variable X follows some nonnormal probability distribution,

little work has been conducted on the implementation of maximum likelihood

(or minimum distance) estimation in this setting. Likewise, no work has

been done on evaluating the effects of sample size on such estimation

procedures.

Another potential area of research is the extension of the results 

reported above to regression models having more than one predictor

variable. Questions relating to the choice of error variance ratios and

the consequences of misspecifying these ratios require theoretical and

simulation investigations. Again, the difficulty with implementing

estimation procedures other than least squares when the predictor variables

are nonnormal must be addressed.

Associated with the estimation of the parameters of measurement error

models is the use of fitted models for prediction and calibration. Only a

few published articles have appeared on prediction and calibration with

measurement error models.
b

The other major topic which was investigated during the course of this

cooperative agreement, mixture model estimation, likewise poses many

F

problems for potentially fruitful future research. In the current research

only the estimation of the mixing proportion p was studied. Much work

remains before acceptable estimation of all the model parameters can be

achieved. So too, estimation procedures for three or more component

distributions need to be developed.

A great deal of work remains to be done on the selection of component

distributions for use with the mixture model. Mi Amum distance estimation

,t

does not necessitate that the component distributions be the "true" ones in

order to satisfactorily estimate crop proportions. During the investiga-

'- Y

tion of the Weibull components it was discovered that several widely

differing sets of parameters could yield mixture distributions which were

virtually identical. It would be extremely useful to identify component

families of distributions for which parameter estimation is computationally

efficient and which are "stable". The stability of estimates would require

that small perturbations of the data would not result in widely differing

parameter estimates and that radically different parameter choices could

not produce virtually identical mixtures.

--
, 7

a

;r

III. ORAL PRESENTATIONS AND OTHER ACTIVITIES

Appendix B lists ten oral presentations of the research conducted

during this cooperative agreement. These oral presentations permitted

rapid -Assemination of the mayor accomplishments of this research.

Presentations (8). (9), and (10) will be acknowledged as "Outstanding

Contributed Paper Presentations" by the Section on Physical and Engineering

Sciences of the American Statistical Association (ASA) during the 1986

Annual ASA Meetings next August.

Funding from this cooperative agreement was used for partial support

of the principal investigator during the summer months of 1984-1985. Also

partially supported during the summer months of 1985 was Professor Wayne A.

Woodward, who led the mixture model research. Three advanced graduate

students were also partially supported during the duration of this

cooperative agreement: Mr. Kelly Cunningham, Miss Miriam Reilman and Dr.

Many Lakshminarayanan. Dr. Lakshminarayanan completed his doctoral degree

requirements while supported by funds from this contract.
IV. APPENDICES

A. COMPLETED RESEARCH

B. ORAL PRESENTATIONS

C. ADDITIONAL WORK
9

A. COMPLETED RESEARCH

I. Publications in Refereed Journals

(a) "Toward a Balanced Assessment of Collinearity Diagnostics,"

The American Statistician, 38 (1984), 79-82.

(b) "Estimation of Parametera in Linear Structural Relationships:

Sensitivity to the Choice of the Ratio of Error Variances,"

Biometrika, 74 (1984), 569-573.

(c) "Screening Procedures Using Quadratic Forms," Communications

in Statistics, A14 (19.0 4), 1393-1404.

(d) "Structural Model Estimation with Correlated Measurement

Errors." Biometrika, 72 (1985) to appear.

(e) "Outlier-Induced Collinearities". Technometrics, 27 (1985).

to appear.

(f) "Selecting Principal Components in Regression," Statistics and

Probability Letters 3 (1985), 299-301.

2. Technical Report and Proceedings Papers

(a) "Sensitivity of Errors-in-l'ariables Estimators to the

Specification of the Ratio of Error Variances," Technical

Report, NASA Johnson Space Center, Houston, ';X (1983).

(b) "Regression Diagnostics and Approximate Inference Procedures 

for Penalized Least Squares Estimators," Department of Statistics

Technical Report No. 181, SMU, Dallas, TX (1983).

(c) "Exploring the Use of Linear Structural Models to Improve

Remote-Sensing Agricultural Estimates. Proceedings of the

NASA Symposium on Mathematical Pattern Recognition and Image

Analysis, NASA Johnson Space Center, Houston. TX (1934).
10

3. Research Articles Submitted for Publication

(a) "Estimating Mixture Proportions for Component Weibull

Distributions."

(b) "Stochastic Regression with Errors in Both Variables".

(c) "Diagnostics for Penalized Least Squares Estimators"

(d) "Minimum Hellinger Distance Estimation of Mixture Model

Parameters: A Re-Examination"
11

i B. ORAL PRESENTATIONS

i

1. "Exploring the Use of Linear Structural Models to Improve RemoteSensing Agricultural Estimates," NASA Symposium on Mathematical

Pattern Recognition and Image Analysis, June 6-8, 1984, Johnson

Space Center, Houston, Tx.

2. "Effects of Misapecifying the Error Variance Ratio in Linear Structural

Relationships." Joint Annual Meetings of the American Statistical

Association and the Biometric Society, Philadelphia. PA. August 13-16.

1984.

3. "Collinearity Assessment with Errors-in-Variables Models." Joint

Annual Meetings of the American Statistical Association and the

Biometric Society, Philadelphia, PA, August 13-16, 1984.

4. "Regression with Collinear Predictor Variables: Implications for Causal

Inference", Department of Quantitative Business Analysis, Louisiana

State University, October 18, 1984; Department of Mathematics,

Northern Arizona University, November 9, 1984.

5. "Regression Estimation with Linear Structural Models," Division of

Mathematical Sciences, University of Texas at Dallas, February 14,

1985.

6. "Regression Models when Both Variableb are Subject to Measurement

Errors". Spring Regional Meeting of the Biometric Society (ENAR),

Raleigh, NC, March 25-27, 1985.

7. "Measurement Error Models," Mathematics Department. General Motors

Research Laboratories, July 22, 1985.

8. "Replication and Instrumental Variables Estimators for Linear Structural

Models." Annual Meetings of the American Statistical Association and

the Biometric Society, Las Vegas, NV, August 5-8, 1985. 

9. "Regression Estimation with Controlled Observations". Annual Meetings

of the American Statistical Association and the Biometric Society,

Las Vegas, NV, August 5-8. 1985.

10. "Maximum Likelihood as Least Squares in Structural Model Estimation."

Annual Meetings of the American Statistical Association and the

Biometric Society, Las Vegas, NV, August 5-8, 1985.

_, !— 0r,^• Q
C. ADDITIONAL WORK

"Minimum Hellinger Distance Estimators

of Mixture Model Parameter:

A Re-Examination"

12
MINIMUM HELLINGER DISTANCE ESTIMATION

OF MIXTURE MODEL PARAMETERS

A RE-EXAMINATION

Wayne A. Woodward

1. Introduction

The problem of using minimum Hellinger ,distance estimation suggested

by Berm ( 1977) for purposes of estimating mixture model parameters was

initially discussed by Woodward and Eslinger ( 1983). The use of the minimum Hellinger distance estimator ( MHDE) is intuitively appealing due to

the fact that it asymptotically efficient and asymptotically normal in

various settings ( see Beran (1977), Stather ( 1981)) yet it has been shown

to be robust to departrues from normality ( see Eslinger ( 1984)). It was

believed that its use in the mixture of normals setting often assumed in

crop proportion estimation could provide efficient estimates when :.ha

model is correct along with robustness to departures from normality.

Woodward, et. al ( 1982, 19b3, 1984) studied the use of minimum distance

estimation of the mixture model pa r ameters using Cramer-von Mises

distance and found that the Cramer -von Mises distance estimator

(MCVMDE) provided results superior to the maximum likelihood estimator
2

(MLE) under departuues from component normality but yielded decidely

poorer results than the MLE when the assumption of component normality was

true.

Woodward and Eslinger ( 1983) showed empirically that the MHDE does

provide estimators comparable to the MLE under component normality which

at the same time show some robustness to departures from normality. The

MADE was not, however, as robust as the MCVMDE, a result also shown by

Eslinger ( 1984) for the two parameter normal. These results are to be

expected since in essence the MHDE is a compromise between the very robust

estimator which is not efficient at the true model and the efficient

estimator such as the MLE which is not robust. However, Woodward and

Eslinger (1983) encountered problems in implementing the MHDE in the

mixture setting. In particular the implementation of the MHDE used by

those authors tended to be extremely sensitive to starting values which

resulted in "failure to converge" problems an unacceptable high percentage

of the time. In particular for mixtures with overlap, as defined by

Woodward et al. (1984) of .1 the MHDE converged on the average about 82%

of the time and converged 88% of the time for .03 overlap.

In this report we re-examine the use of the MHDE in the mixture of

normals setting. We empirically examine the effect of the selection of

the smoothing parameter h used in the kernel density estimation component

of the MHDE, and we examine the use of an alternative maximization

scheme. In this report as in the earlier reports by this author, we will

be concerned only with the estimation of the mixture proportion.
3

In Section 2 we provide a brief description of the MdDE in the

mixture-of-normals setting as implemented by Woodward and Eslinger ( 1983).

Section 3 provides details concerning new approaches designed to improve

the convergence of the MHDE. In Section 4 we give details of simulations

examining the new techniques.

2. The Minimum Hollinger Distance Estimator

Let Fe - IFe :8E6} denote a family of distributions which will be

referred to as the projection family. These distributions depend on the

possibly vector valued parameter G. We will assume for our purposes here

that the distributions in Fe are absolutely continuous. In our case we

use the mixture of normals projection model

1 x

_

ul 

2 -1(

x-}► 2 )2

f (x) - P

e 2 ( o f ) + (1-p) e 2

°2 (2.1)

e

^ °1 °2

where 6 - ( u19°1,μ2,o2 , p)' are all assumed to be unknown. M

MHDE of 8 is a value 8 which minimizes a "distance" between the data I'

distribution ( whose model is unknown) and the projection model. The

distance measure used in MHD estimation is the Hollinger distance. The

Hellinger distance between two absolutely continuous distributions is

defined to be 

II f112-g1 /

211 where f and g denote the corresponding

density functions and 11 . 11 denotes the L2 norm, i.e.

I1

fl/2 -

9

112

11 -

ff(fl/2-gl/2)2dx)1/2 (2.2)

F
ff 1/2 

i 

1/2 dx.

0 n

(2.3)

I .

where integration is with respect to Lebesgue mature on the real line.

A- The MHD estimator a 

H 

of 0 is defined as a value of 9 which minimizes

lif 1/2_ i

 1/2 11 where i is a nonparametric density estinator. We will 0 n n

employ a kernel density estimator

I p X-Xi

in

 (x) a— r w( h

n 1=1 n

based on the Epanechnikov kernel w(x) - .75(1-x 2 ) for IxIll. It should be

noted that minimizing 11f 1/2_i

 1/2 11 is equivalent to maximizing 0 n

In the earlier report Woodward and Eslinger (1983) examined the use of the

MHDE in this setting by maximizing (2.3) using Newton's method. This

af 1/2

a 

2 f 1/2

recursive method involved the calculation of 0 and 0 the

89 2

forms of which are given in the Appendix of that report.

3. New Implementations

In the earlier report the smoothing parameter h 

n 

in the kernel

density estimator was taken to be c 

n 

a 0 where c n - 2.16n-* 

271 and

s 0 

was set to the starting value estimate of the component with the

larger mixing proportion, i.e.

s a a (0) - [( P I (0) - r 

(.25) )/.6745) 2

0 1 1

if p Z .5 and
5

s082 (0) _ ((r 2.75 )_ 02(0)/.6745J 2

if p ( .5 where μj (0) is the initial estimate of p j and r(q) is the qth

percentile from the jth cluster, j =1,2. We denote this value of h as

h(1) . This value of h was chosen since h - c a with a the median

n n n n n n

absolute deviation, had been shown, Ealinger ( 1983) to be the optimal h 

when using a two-parameter normal projection model. In this report we

examine the impact of two other choices for h n . The first modification to

the earlier work was to take h = c s where c is as before and a is an

n n m n m

initial estimate of the mixture standard deviation given by

sm = WOO 2 (0) + d2 ) + (1-p(0))(82(0 ) 

+ d2)]1/2

where d l = μ1 (0) - μm( 0), d2 02 ( 0) - 0m(0) with

0m(0) = p (0)μ l (0) + (1-p (0))4 2 (0) .

This resulted in larger values for h n , especially in the cases in which

there is substantial separation between the two components.

Parzen ( 1962) has found the h which minimizes the integrated mean

square error between a kernel density estimator and the true density f.

His result shows that the h optimal in this sense is h = a(w)B(f)n-1/5

where

[fw2(y)dy]1/5

a(w) _

[jw(y)y2dy]2/5

2

S(f) _ (f(a x2x>12dxJ_1/5

ax

and

NW

(3.1)
. . 6

For the Epanechnikov kernel, a(w) = 1.71877. In the case in which f is a

mixture of normals as in (2.1), (82 f(x)/8x2 1 2 is given by

2

( a2 f( 2 ) ) 2 -5-- n(μl,ol/32)(zl-1)2

8x 2o1 3s

2

+ (-- n(u2.o214)(z2-1)2

20 2 3x

x-μ 2 ( x-p 22

_ 1 ( 1 ) _ 1 )

+ 22(1-p3 e 2 al e2 02 (z1-1)(z2

l

-1)

2% aa2

x

-1'1 x ^`2 where zl a z2 = a

and n ( p,a) is the normal density with mean

1 2

K and standard deviation a. We examined the use of hn 2) = 1.718775 (

f)n-1/5

where B(f) was approximated numerically using IMSL's numerical integration

routine DCADRE.

As another strategy for improving convergence of the MHDE, an alternative maximization technique was considered. Recall that the MHDE is

defined to be a value 'which minimizes the integral

1 = !(f112_ 8n/2)2dx.

This integral can be approximated using the trapezoidal rule by

k

I = At E ai(fe/2(ti) _ 8n /2 (t i )) 2

(3.2)

i=1

where a l = a = 1/2 and a i =1 for i-2,3,...,k-1 for a partition

tl,t2, ... ,t k of (a,b) a finite interval, in our case taken to be the

support of gn , i.e., a = Y 1 -hn and b = Y + h where Y 
7

e

denotes the ith order statistic. The procedure employed was to minimize

the sum-of-squares in (3.2) using IMSL routine ZXSSQ which employs the

Marquardt -Levenberg algorithm ( 1963).

4. Simulation Results

In this section we describe the results of simulations designed to

test the procedures described in the previous section. Both the Newton

and Marquardt -Levenberg algorithms were examined using h (1) and h(2).

Simulations involve mixtures of normal and of non-normal components using

the parameter configurations employed by Woodward and Ealinger (1983). In

particular, we use mixing proportions .25, .50, and .75 and "overlap" as

previously defined by Woodward, et al. (1984) of .03 and .10. As in the

previous report we consider cases in which a l /a 2 equals 1 and .1 2. In the

present study we consider mixtures of normal and t(4) components while

some results are given for mixtures t(2) components. For each set of configurations considered, 500 samples of size n-100 were generated from the

corresponding mixture distribution. Simulations were performed on the IBM

3081-D24 computer at Southern Methodist University. Starting values were

obtained as discussed by Woodward, et al. (1984). For each sample

simulated, the MLE and MCVMDE were obtained along with several MHDE estimators. MHD estimators employing Marquardt's procedure for minimizing

(3.2) will be denoted MHDEM(i) where i-1 or 2 depending on whether h(1)

or h (2) were used in the density estimation. Likewise MHDEN(1) denotes

n

the estimator using h (1) and employing the Newton's method for

maximizing (2.3). The estimator MHDEN(2) was examined for selected

a-

^^- ^-:., .rte 
8

configurations and was seen to not perform as well ar MEDEN ( 1). Also

shown are results labeled MHDEN' ( 1). These correspond to the MHDEN(1)

type estimates obtained by using starting values for 

a 

and a2 which are

smaller by a factor of 1.2 than those obtained by the straightforward

starting values given by Woodward et al. ( 1984). Woodward and Ealinger

(1983) showed that in the MRDE setting studied there, these smaller

starting values produced bettor results. For means of comparison we have

denoted by MHDE* in the tables the corresponding values obtained by

Woodward and Eslinger ( 1983). These estimators were not obtained on the

same sequence of samples as the current simulations. In Table 1 we present results for simulated mixtures of normal components, in Table 2 we 

show the corresponding results for simulated mixtures of t(4) components

while in Table 4 we show a few results for simulated t(2) mixtures. 

Simulation based estimates of the bias and MSE associated with the various 

estimators are given by 

n

s

Bias

n

E (Pi- P)

s i^l

n

MSEn Es (pi - p)2

s i^l

where n denotes the number of samples ( 500 in our case) and pi

denotes an estimate of p for the ith sample. As in the earlier reports

nMSE is given in the tables where n is the size of each individual sample

(100 in our case). We provide empirical measures of the relative

efficiencies of the various estimators with the MLE, by
I -, 9

M$E (MLE)

MSE (estimator)

An approximate standard error of a tabled WE is (.0632) (WE). Also

listed is the percentage of the 500 samples for which that estimator

converged to "reasonable" values. For a discussion of "reasonable" values

see Woodward, et al. (1984). If such convergence was not obtained, for

purposes of the simulation study the estimate of p was taken to be the

starting value.

Examination of the tables shows that indeed the MHDE does appear to

behave as expected, i.e.provides fairly efficient estimators under component normality (as evidenced by E values near 1) along with estimates

more robust than the MLE for simulations of mixtures of non-normal components (as evidenced by E ) 1).

The percentage converging information is summarized in Table 3 where

the value tabled for a given overlap is the average percent convergence

obtained for the 10 configurations of normal and t(4) components in Tables

1 and 2 for that overlap. All of the techniques proposed in this report

produced estimators with higher rates of convergence than the estimators

in the earlier study, especially for the .03 overlap. It is very clear

from the table that the Marquardt based estimators are far superior to

those using Newton's method in terms of percentage convergence obtained.

It should be noted that convergence was almost always obtained in these

settings by the MLE and MCVMDE.
10

Concerning MSE performance, initial examination shove that MHDEN(1)

provides smaller MSE's than the other MHDE techniques for .10 overlap yet

MH)EM(2) seems to provide the best estimates at the .03 overlap. However,

due to the fact that in the .10 overlap case the starting values tend to

outperform all other estimators and that about 16% of the "MHDEN(1)"

results are actually starting values, this has a tendency to deflate the

nMSE for MHDEN(1). As in the earlier study, the MHDEN'(1) (using the

scaled starting values) seem to perform better than MHDEN(1).

It should be pointed out that about twice as such time is required to

produce MHDEM(i) estimates ..,s the MHDEN(i) estimates. It was mentioned

by Woodward and Eslinger (3983) that the time required for the MHDEN was

comparable: to that for the Cramer-von Mises estimator.

In Table 4 we show results for simulated t(2) components for

configurations with o f /a 2 - 1. It is seen that again, in this

extremely heavy tailed departure from component normality, the MHDE

produced results markedly better than the MLE yet usually not as robust as

the MCVMDE. Convergence in this setting was sometimes a problem for the

MLE procedure which used the EM algorithm. However, the MHDEM(2)

estimates converged at an extremely high rate.

S. Concluding Remarks

As a result of the present study it seems that the MHDE could indeed

be a useful estimator of the mixing proportion of a two component

mixture. Using the Marquardt procedure, the convergence problems no

J
11

longer exist although computer time is doubled. Work should be done in

order to determine whether or not the time required to calculate the

MHDEM(i) estimates can be decreased.

Parameter estimates using the MHDE conform to predictable patterns,

i.e. the estimates are more efficient than the Cramir-von Mises estimates

t-ndrr component normality, yet are not as robust as the Cramir-von Mises

results. Because of the increased percentage of convergence, the results

obtained here provide a clearer picture of actual estimator performance

over the results given earlier by Woodward and Eslinger (1983).

Whether the MHDE could be successfully used as an alternative to

maximum likelihood estimation of a crop proportion based on remote sensing

data remains to be determined. However, the results shown here imply that

its successfully application is a possibility.
12

REFERENCES

Baran, R. (1911). "Minimum Mellinger Distance Estimates for Parametric

Models," Annals of Statistics 5, 445-463.

Eslinger, P.W. (1983). "Minimum Hollinger Distance Estimation," Ph.D.

Dissertation, Statistics Department, Siuthern Methodist University.

Dallas, Texas.

Marquardt, D. W. (1963). "An Algorithm for Least-Squares Estimation of

Nonlinear Parameters," Journal of the Societ for Industrial

Engineers 11, 431-441.

Parzen, E. (1962). "On the Estimation of Probability Density Function and

Mode," Annals of Mathematical Statistics 33, 1065-1076.

Stather, C. R. (1981), "Robust Statistical Inference Using HollingerDistance Methods", Ph.D. Dissertation, Department of Mathematical

Science, School rf Physical Science, LaTrobe University, Australia.

Woodward, W. A. and Eslinger, P. W. (1983). "Minimum Hellinges Distance

Estimation of Mixture Model Parameters," NASA Technical Report

SR-63-04433, July 1983.

Woodward, W. A., Parr, W. C., Schucany, W. R. and Gray, H. L. (1983).

"Minimum Distance Estimation of Mixture Model Parameters-Asymptotic

Results and Simulation Comparisons with Maximum Likelihood," NASA

Technical Report SR-63-04427, June 1983.

Woodward, W. A., Parr, W. C., Schucany, W. R., and Lindsey, H. (1984).

"A Comparison of Minimum Distance and Maximum Likelihood Estimation

of a Mixture Proportion," Journal of American Statistical Association

79, 590-598.

Woodward, W. A., Schucany, W. R., Lindsey, H. and Gray, H. L. (1902). "A

Comparison of Minimum Distance and Maximum Likelihood Techniques for

Proportion Estimation," NASA Technical Report SR-62-04376.
0

40

PA

M

O

41

u

N c

d

e4

a w

1^ d N

>

O C ^ U

N N -- (E

41 N

N

­4

.1 •-1

N O

41 (W

wa

ww

E o

0

^ w4!

.O

^ W

x x

O

N

N

(t0 t0

i+41

9 d

UG4)

00

eo 41

GO

U

I d 11 0% 00 Go v^ 00 00 00 O

%aNO in I-%D%D n

Ln " -a cp% w X ao 1Oddddddd W 1 N O% O% d 430 f+ %a C4

yC

Co d O ^o d ^n ^r1 ^r1

.y•

C

V

O

O

U

r-1

Oe

0

ae

w

0

41

w

a.+KM

x

WO

w

a

a

d

a

G

O

N

O

7

.400 Go -e N d 00

O 0 0 0 0 0 0 8888888

1 1

I 1

o0 %a00%D I 0000^000^o i

BS^^oo^^ g00(070% call%88^^

1 -+ .-• . + re 1 1

I 1

1 i

I 1

1 1

^OOO^Dr`T 1 0 1

u1 .-. O O t^ I 00 d M ., O. I

I I

I 1

i

^?Md^1^0-+C0 1 NOOMt,"1p0NMt^ 1

Ln 0 L pJ1ps 1 .-.v1aDr,.r40^t O I

N0C%C14 r- NoNMCOMB I

N f1 00 11; L' 1D M, 14 M M N N N 14 N I 

1 I 1 I

I I

1 1

1 1

1 1

1 1

0%MN0MM•-4M 1 0%t+1Lf'1 rd ^1 1

40%0 T OO r- Ctco 1 00000000 i

00 --- 000 1 00000000 1

I 1 1 1

1 1

1 1

I 1

1 1

I 1

00 00 1D .D 0 1 co cc co 1 I Ototo+ 1

OO^naoO 0 1 OOMr^ 1 OO^^O OL D 1 OOCtcllc•c%ao 1

1I

01

O

F

N

b

O

tw W ^GAQG G^

vOxx x^

anN a

i. 1

di W ...• ..mss 1

VA

IO W^ AIr A A A Q I

LAxx^ I

O
00 0% 00 u) <W .-4 O O N in %O .O d

O ao a0 tT O

%b 44 eJ711 O .r .r O

11

00 O% r-y -Q N .4 r-. O W n n eh p er1 -es ^D Q` I

u^.a00-tsa •aT^M I

^D N eh N O^ t10 q0 ^O

e'1 0 O% n %a N 00 MI) C^ -e► t•1 d 1!•e! e^ n I

• • I

N V-4C4

l d .-e ^

•e en o .0 .0 0%

b

C,•t

u

G

O

as

41

.D

F

t^eo 080880-8fo888880"198008881" . . . .. . , ... .

eh ,^ ^ ^ 11 1 I l l l l l l t l l l l l l

O OA 1 1

1 1

I I

r ^ ^

i t

etl d I I

.-1 U 1 I

M G 1 I

tD 0 ^. I 1

00 . . . . .

1 1

a: ►+ 00 ^ N 00 n 1

. . . . . . . 1 000000000 ' d O O c, CPS O O 00 1 O tT 0% O ^D I

> •..•, .,•.• I o$o^o+$oo. 1 80 MLM" C I .... .^ .., 1 O Olk 0P O 00

O I I 1.41.0 r. .•r

— — — — — — — — — — — — — — — — — — — — — — — — —

1 1

OOpON4n I

1 00Lf)dQYON

1

I -7 00O^

.^ c+'1 er1 O n 1 OD d .7 .• tT 1 ^D .r 1.r tT 00

1

1

I

11

On00 wntTO^m

1

1

I %D-a.-.-70%n t+1 M

I

1 -tt t1'1NO^u1N en .r

W -•+.-.N M NO ^D 00 1 ^Dw 000 NN^O I O^O^^T ^D t•"1N•7.-i

tN) tT.-^N v1./100'^^7 I O^ ^1O+nn N^DO^ 1 NO^nt^1 NQnn

. . . . • . .

G Nom.-+.+NN -+ 1 ANN.-+1-+NN.-1

1

I

I -tT^Tn •Q•Q^1^.sy

1

I

O 1 I

.. 0 pp N ^Dt>0 1 N.7 u1t-1N•r .0M I .r1nO%MNOu9%0 p

•-+Oa0 c^ OONO I O^ r+OcvN00^"^ 1 ^Dn1-+OaOnnN r rn 00000000 100000000 1 .•.O.-+^ 000.-^ '

+^ I I 1 I 1 1 t l l l l l l l l l i

-•t ^ I I

tL 1 1 I

tU I I

u I 1

a^i OOO^DO0ao^D i 00.00000 i 9O0101a ^oO 017 1 I

^+ O O 00 rr tT O^ ^D 1 p O N '0 O O N .T M C^ 0, .•r

ae d O O n OO tT ON n 1 O 0 ON 0 0 O^ 1 $ ON 00 00 ON OT co

oU 1

1

1 I

t

1

1

I ^

1

1 ^ I

.+ v .-^ N 1 --^ ^ ^--^ N 1 .-+ v r-. N

rn W v- ^.^. I W W ^- ..v 1 W W v- ..^y

Wo 0A 1 AW A 1 aaw>QAAA0

Z

A

x

0x 00 li

t h X X X X X X I N X X X Z I N X X 11

I O

I

I Ln

N 1 u1 I n I
A

LI

Q

OI R

0

v

w

0

0

dw

r

z

w

0

w

to

7

a

d

a

0

44

Aj

e0

7

6

N

+1O

r

w

d

O

r

CL

t0

Ir

d

N

d

.-1

.D

F

r

8 mc

-* 044

r w

m u

N +I

to 0.

Ol

d a

p' w

It o

eA

to W

d

e

7

z

e

1i =

1 1

tV N^oM0% 1 nd OoN 1

tW IT 000od -: 1 u'► O..O%OA

1

1 1

1 1

1 1 NMOp .rN^D^DO I dM.••iM%n M 1 W O N N 0 00 1 in r^ 0o ^4 en

(CA.•+00-4 -4dU)M U1 I &M-- w llU^lldd ee►►17M

^ 1 1

1 1

1 1

1 1

1 1

1 I

00 U en LM Cn O a 1 •p+ M .^ .-r rp

r dp I ta°00 00000000 1 SSSSS S 1

iq 1 1 1 1 1 1

1 1

I I

1 1

m i i

u 1 1

a I I

e 0)

1 1

Io O d 0o O O eo I % O 0 N 0 0 %o I

d . 1 1 . . . . . . . . . . . . .

aS^0Soao1 0S^a88^^ o .r r. .r 1 

U 1 I

1 1

1 1

I 1

00 U11000%D 1 d W-aMO 1

ennONe+1 1 0` In u1 I

.N 1

I I

t 1

. 1 I

I 1

1 I

W oo U1 s ^ 0 0 00 ^D I N n v1 U1 ^D ^C ^G ^O 1 ttn .•+Nd NDM%C.- n 1 O^NUINNO%Itd 1

^n^NndN.T 1 O.T00 U1MN1Dn

C

. t 1 . . . . . . . . . . . . . .

..+U1d MN^Y^7N 1 •-+^^^'^1M^l1Mnr 1

1 1

1 1

1 1

1 I

1 i

I I

u) mC,%000Mn e+1 .-r I N •OEn.n M.r^DN 1 <RI ^o ^o in en m 17 U1 1 O N p .-+ r..-. O 1 ++ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

1 I 1 I 1

I 1

I 1

I I

Ol 1 I

U 1 1 sONN0.7O i -arO0o0oO^od i

K 00 00 C; C+4^1T I O%O%COr all 1

w O+Onr+CAMI- I 0%0000000100 1

> I I C I I O 1 1

U 1 1

I I 1 I

I 1

N N W...- ^.v 1 m W v^ 

^'y

^ 1

I 1

I I 1 I

I i

CL N I 1

• 1

i
r •

' 1 I 1 1 1 1

1

^ •0 CD N.O n n 1 O O, O% O 1+1 1 .-+ ao CO %D n

M CO CO ^0 n I Ct! O •- ••• CO ( M CO CO M n

1 I 1

1 1 1

40 NIr1NdasOk%C.%0 1 %.0%01 ONm0►

1 W0%^ONN <'rt •Cm 1 CO ^MMO+^p^ 1 n an r•CO•.rdN 1 n •.rO^O^O^CDCn^ 1

^1 ^OrnNM1M1d ( n d NdMdNM t ^O^/r1v1U1t^1NO I

1 1 1

I 1 1

1 f I

a 1 ^ 1 «r Co%a .-^ .-. M .^ rn CO 1 %a C^ %a %D aD N Co 1 0^ d 0% oes oo e+^ d Cn 1

0 00000000108888880°0088800

1 1 I l l l t l l l l l l l

1 1 1

I 1 I

I 1 1

1 1 1

1 1 1

1 1 1

u 1 1 1

d 1 I 1

^D O^oNOON I W00N000 1 C QC C400 N 1 d . . . . . 1 • . . . . . . 1 . . . • . . . 1

c ^S^

.

ChSo^. 1 ^S^aS8^1 ^a^^88^1 p .r .+ I .-+ .-..-+ 1 .•+ ^••^ 1

1 1 1

I 1 1

I 1 1 C4 MNON C*4 nn I OM N M O 1 tn ^ 1DO.7 1 '4O

C4

1

<W 1 1 1

.-+ N N N N N C14 .-^ I

I I I

1 1 1

1 I I 1 1 1

W OOCO0oO%01000 1M 0 w M 0 L %.Q 1 %aMrnCO M.rr.. I

(W O a0 •D.-+0 T nM 1 O^ M r^ CO OCOCO-4 1 000 •1 1 

COOO^N000100 1 109C^P+.-+C^ 7 C 11 O0J1e^ON 1

• . . . . . . . 1 . . . . . . . . 1 . . . . . . . . 1

U'1^TNNM^TN I ^+v1.•rMM^TMN I M00^7M .Q v1^1.7 1 1 I 1

I 1 t

1 1 I 1 1 I 1 1 t

di 0) r- n- %T 1 N-7t LnMMM C•COu1MN00^00+ 1 (M LnTry 1 OONNO drld 1 Mr-rnIT.C• NMO% I

•ri 00000000 1 00 00 000 1 .-.0000 000 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I i 1

I I 1

t 1 1

1 I 1

G 00 ^OOO000 1 .70%0000%0 1 OOC4 N-4 I t11 . . . . . . . 1 . . . . . . . 1 . . . . . . . 1

°o °o^°h°o°o^°- 1 ^08w 1 ^S^°^^^ 1

c

r0 1

VZ

1 1 1

I t I 1 1 I

.r v .-r N 1 .-^ ^. .^ N 1 .^ v .-+ N 1

N N W ^r- v...^ 1 m W vy

v

^ v 1 m W v- ^r^

w O W W W Et I amL + 0 W W W 1 W ^y W W Wl 1

I I 1

1 1 1

V) 1 O I u1 1

(^. N 1 ^1 1 f` 1

I I I

M

0

r

.i

w

m

eV

t:

,a

ar

C

O

V

N

tl)

.i

.0

etl

F

O.r

M

O

Lil d

O
Table 3 - Percentage Convergence Obtained

for Normal and t ( 4) Components

Overlap

.10 .03

MHDEN(1) 82.4 95.8

MHDEN' ( 1) 85.6 96.8

MHDEM ( 1) 99.5 100.0

MHDEM( 2) 99.7 100.0

MHDE* 82.1 88.1

-

fin..--.^^. r •
aNdCOOU eh N

a+r

w

o

a

0

o

A

u1 r••1

A

W a, r m

a+ .-+

C

.F4

M •r01

d A

M N U O r1 +1 w y .-^a a d a a m A O a " wm

•-1

2

^O

6

en

raA4

O> aJ

ti>

^o

AF

<W

1 I i •Q eP1 O0 1 V^ u1 An 1

tiO+M I 0+O0^ ^ 1 1

.elN L i NNIV 1 1 1 I 1 w 1 1 M N u1 00 n O% 1 O er•1 N

.y eft " N 00

1 N .4 O0 .r 00 1

CM U1CnLnN 1 d 00N

•

e

► N 1 I i 1 1 a 1 1

(A Ln 0% .et N %D 1 .y 

•Q

d C4 .p+ N

•d O %1ON ".• 1

Q

p 1 iO 00000 ^ 00000 11 1 I 1 ; 1 1 1 1 d 1 1 U I 1 G I 1 d 1 t

baI

1

W OONO 1 N00N00 1

0

C c 008 11 Go^^^ 1 U 1 1 1 1 1 1 1 1 1 1

en s O0 1 u1 r^ -+ I

r+p

M I LA-4at I

ew

I

t N N r• 1 W N N 1 11 1 w I ^ N Ln I 1 N %0 Nt` -4 1

een e•'i %D ? O W 1 N r+ r< O ^D 1 O '0' rn n I -^ O ^ • ^D ^ I 1 1 1 I

fn n N O^ u1 r^ I e• t ^0 ^1 N

eel+ BOO n .7 o+ 1 OOOO N I 0 0 0 0 0 1 0 0 0 0 0 1 I 1 1 1 1

1

asUd O ^ O ^

ale 00fr O^ O^ .-• 0'^ d 00O•t^^ COV

00 N co %D

U1P (

•1 0+ 00O,N( 1

N I .+ N

1

vi W.... I

a W.... 

I

N

u C2S I ^.+ Ay.z 

LTy 1 ^ a^CW ^ WAA I ^ W>Q A

^+ N .^ U S O I y «a L) xx

1

b tI, Z Z 

z 11

Lo z Z Z

1I 1 1

LF) 1 O I N 1 ^ I

a11