MEASURES OF CENTRAL TENDENCY FOR CENSORED WAGE DATA

Sandra West, Diem-Tran Kratzke, and Shail Butani, Bureau of Labor Statistics

Sandra West, 2 Massachusetts Ave. N.E. Washington, D.C. 20212

Key Words: Mean, Median, Pareto Distribution

I. INTRODUCTION

 The research for this paper began in connection with the need

for measuring the central tendency of hourly wage data from the

Occupational Employment Statistics (OES) survey at the Bureau

of Labor Statistics (BLS). The OES survey is a Federal-State

establishment survey of wage and salary workers designed to

produce data on occupational employment by industry for the

Nation, each State, and selected areas within States. The OES

survey provides employment data for approximately 700 detailed

occupations by surveyed industries. Until recently the survey

did not provide any wage information. The wage data that are

produced by other Federal programs are limited in the level of

occupational, industrial, and geographic coverage. In order to

address this critical void in the Federal statistical effort, the OES

program conducted pilot studies in 1989 and 1990 to test the

feasibility of incorporating wage questions into the OES survey.

 The 1992 OES survey collects data in 15 States on

occupational hourly wage by industry in nonagricultural

establishments. The data are collected in eleven intervals, rather

than in exact dollar amounts, with the lowest and uppermost

intervals open.

 Research was conducted by the Office of Research and

Evaluation and the Statistical Methods Division of the Office of

Employment and Unemployment Statistics to find a suitable

estimate of central tendency for the occupational wage data of

the OES survey. It was determined that both mean and median

would be measured. Each has advantages and disadvantages

which will be discussed.

 West (1986) discussed problems with using medians in

household survey data where the distribution of earnings has

many peaks and estimates are compared over time periods. In

this paper, we analyze the medians of occupational wages from

establishment survey data where the resulting distributions are

not as multi-peaked as distributions of data from household

surveys. For the most part of this paper, we investigate

alternative methods of estimating occupational wage means from

grouped data with open intervals.

 The first part of the research focused on the problem of

estimating the overall occupational wage mean for each industry,

according to the OES survey's objective. The second part of the

research explores the best method for estimating the wage mean

of an upper open interval. This would be useful for analyses

such as regression where interval wages are used as dependent

variables.

 The two measures of central tendency are described in Section

II. The empirical studies and results are given in Section III.

The conclusions are presented in Section IV. Section V contains

plans for future research.

II. MEASURES OF CENTRAL TENDENCY

A. MEDIANS

 In elementary theory the median has considerable claims to be

used as a measure of location for unimodal distributions. It is

readily interpretable in terms of ordinary ideas. What gives the

arithmetic mean the greater importance in advanced theory is its

superior mathematical tractability and certain sampling

properties. The median has a compensating advantage in that it

is less sensitive to the configuration of the outlying parts of the

frequency distribution than is the mean. This is especially

important with earnings data and in particular, with the censored

data, the median is a logical choice. However, the median is

sensitive to the way the data are grouped and operating with

medians can lead to misleading results. The latter is especially

true in the case of the many-peaked earnings distribution, such

as arises in data collected from households, where the

respondent often has to approximate the data requested. The

earnings data considered in this paper are collected from

establishments and are, for the most part, obtained from payroll

records. Thus, the resulting distribution should not be multi

peaked, and indeed, are not. This paper concentrates on finding

the best estimator for the mean rather than for the median. Only

the linear interpolation method for the median is tested. The

method first determines the interval that contains the median,

and then linear interpolation is used to estimate the median. The

occupational minimum wage is used as the lower limit of the

lower open interval. If the median falls in the uppermost open

interval, all that can be said is that the median is equal to or

above the upper limit of hourly wage.

B. MEANS

 A measure of central tendency with desirable properties is the

mean. In addition to the usual desirable properties of means,

there is another nice feature that is proven by West (1985). It is

that the percent difference between two means is bounded

relative to the percent difference between subgroup means, if the

proportion of units in each subgroup remains the same for the

two groups. Since the problem considered in this paper deals

with grouped data that have lower and upper open intervals, it is

not possible to compute an exact mean. The problem will be

considered from the point of view of computing a population

mean from right and left censored data. First the problem will be

formulated and two methods for computing the mean will be

discussed. One method results in the Winsorized mean and the

other method uses a classical Pareto distribution.

FORMULATION OF PROBLEM

 The population of true earnings data are denoted by :

XX X 1 2 N , ,..., ,

and XX X () ( ) ( ) N , , ..., 1 2 denote the ordered X s' . The mean of the

population is desired; that is,

X XN i

i

N = å

=1

.

The data actually observed are the frequencies of each of the

intervals:

II I 1 2 r , ,..., ,

which are mutually exclusive, and exhaustive of the earnings

scale. I I 1 r and are open intervals, where

Ix U 1 j contains all less than some fixed number ( ) 1, and

I x

U

r j

r

 contain all greater than or equal to some fixed

number 

s ( )

( ). -1

For i rI = - i 23 1 , ,...,( ), are bounded intervals containing all

x( )j between some fixed numbers U( ) i-1 and U( )i .
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Let denote the observed frequency of interval 

 Note that i=1

r

f I

for i r f N

i i

i

,

= = 1 2, ,..., . . å

 Letting Mi denote the mid-point of the i-th interval, then the

usual estimator of X is the grouped mean, Xg ::

X Mf N g i

i

r

= i

=

å

1

 .

Since intervals I1 and Ir are not bounded intervals, M1 and

Mr will need to be estimated.

I1 has a natural lower bound, either 0 or the minimum wage; for

this study the minimum occupational wage for each State, W( ) 1 ,

was used. Thus the estimate for M1 is :

$ ( ). M WU 1 11 ( ) = + 2

An obvious estimate for Mr is:

$M U rw r , ( ), = -1

which would lead to the Winsorized group mean:

$ { $ X Mf Mf U f N gw i i , () }

i

r

=+ + r r

=

-

1 1 å -

2

1

1 .

Clearly, $Mr w, will underestimate Mr . With the Winsorized

mean a straight line is used for the missing values; a natural

extension, now to be considered, is to fit a curve for the missing

values.

 For the estimator of Mr , consider fitting a theoretical

distribution to the (r-1) mid-points, and take M$

r as the mean of

the conditional distribution, PX xX U r ( | ). £ 3 ( ) -1 That is,

$ ( | ) (| ) () ()

() ()

M x dP X x X U x f x U dx r r

U

r

r r U

= £3 = -

¥

-

¥

- -

z z 1 1

1 1

where

f xU r (| ) ( ) -1 is the conditional density of X , given that X is

greater than or equal to the fixed number U( ) r -1 .

 Another possibility for M$

r is the median of the conditional

density. Parker and Fenwick (1983) found that this estimator

performed better than the mean, but this was not the case with

the new method and data considered in West (1985).

 Many distributions have been proposed for earnings data, but it

is clear from the literature that the researchers are satisfied with

the Pareto distribution as a fit to the upper portion of the

earnings curve. Consider the Pareto distribution:

F x P X x K x for x K

for x K

() ( ) ( ) ,

.

= £ =- 3 > >

= <

1 00

0

a a

Noting that,

PX xX U PX x PX U U x r rr ( | ) ( )( )( ) () () () 33 = 3 3 = - --

-

1 11

a a

then

f x U dP X x X U dx

U x for x U r r

r r

(| ) ( | )

() , . () ()

() ()

- -

-

- -

-

=- 3 3

= 3 1 1

1

1 a 1

a a

Thus,

$ ( | ) ( ). ( )

M x f x U dx U r () ()

U

r r

r

= =- -

¥

z - -

1

1 1 a a 1

Note that for the mean to be positive, a > 1.

 Many methods exist for estimating the parameter a . The

method for estimating a most used and recommended in the

literature is the quantile method, which is described next.

 Let Mp and Mq denote the p-th and q -th

quantile respectively; that is,

 FM PM M KM p p pp () ( ) ( ) . = £ =- = 1 a

 Similarly, for Mq .

 Letting M M MM $ $ p q pq and be estimators of and 

respectively, leads to the following estimator of a :

$ ln[( ) ( )] ln[ $ $ a ]. pq q p =- - 1 1 p q MM

 Most researchers seem to use this method with either the midpoints of the last two bounded intervals or the last bounded

interval and the open interval. Specifically, if the mid-points of

the last two bounded intervals are used, the estimator of a

becomes:

$a ln[( ) ( )] ln[ ]. c r rr r r = + fff MM - -- 1 21

 This will lead to $Mr q, ,2 and $Xg q, ,2 as estimators for Mr and

Xg , respectively. That is,

$ { $ $ } X Mf Mf M f N gq i i ,, ,,

i

r

2 11 rq r

2

1

=+ + 2

=

-

å .

where

$ $ ( $ ). M U rq r c c ,, ( ) 2 1 = - - a a 1

The method is referred to as the quantile II method, and $Xg q, ,2

is referred to as the Qnt II estimator in this paper. If the lower

bounds of the last bounded interval and the open interval are

used, then the estimator of a becomes:

 $ ln[( ) ( )] ln[ ]. a o r rr r r () ( ) = + f ff U U - -- 1 12

This will lead to $Mr q, ,1 and $Xg q, ,1 as estimators for M r and Xg ,

respectively. The method is referred to as the quantile I method

and $

Xg q, ,1 is referred to as the Qnt I estimator in this paper.

In the literature the estimator, ,a$

o seems to be the one most

recommended, for example, see Shryock (1975), Parker and

Fenwick (1983).

An alternative estimator for a is a modified maximum

likelihood estimator developed in West (1985). A brief

description of the estimator will be given here. Since the Pareto

distribution is considered a good fit for the distribution of higher

earnings, the parameter will be estimated from the left truncated

distribution. Letting

MM M ss r , , ..., + - 1 1

denote the left truncated mid-points of the bounded intervals,

then the modified maximum likelihood estimator is:

$ [ ] [ ( ln ) ( ) ln ln( )]. a ms i , ( )

i s

r

i

i s

r

i is

i s

r

r sr = -- f f M f M f MU

=

-

=

-

=

-

åå å -

11 1

1

Note that in the case of a Pareto distribution, truncation is

equivalent to rescaling.

 Regarding the selection of the truncated point, Ms , it was

found in West (1985) that if the earnings distribution was

truncated at the mid-point of the interval containing the

truncated mean, then the resulting estimate of a led to the

estimate of the mean that came the closest to the true mean. The

truncated mean is defined as the mean of the data below U( ) r -1 .

 Consistency is easily verified for the quantile estimator and it

is resistant to outliers. Quandt (1966) found that the
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performance of the quantile estimators was not much inferior to

those of the maximum likelihood estimators. A Monte Carlo

study reported by Koutrouvelis (1981) supported that view.

However in West (1985), it was shown theoretically and

empirically that the quantile method depends heavily on the

classification of the population, can lead to gross errors, and at

best does as well as the modified maximum likelihood estimator.

The data used in the empirical studies were relatively small

populations, based on household data. In West (1986), the

quantile estimator and the modified maximum likelihood

estimator were compared over time on data collected from

households in the Current Population Survey (CPS). The results

were similar to the ones in the 1985 study. In this paper the

estimators will be compared on occupational earnings data

collected from establishments. The Winsorized estimator, the

two quantile estimators, and the modified maximum likelihood

estimators, using four different rules for choosing Ms will be

compared. The four rules are to choose Ms to be the:

1. mid-point of the interval that contains the truncated mean.

2 mid-point of the interval following the one that contains 

the truncated mean.

3. mid-point of the interval that contains the median.

4. mid-point of the interval following the one that contains the

median.

 The modified maximum likelihood estimator of a will be

denoted by $a m k, , and the corresponding mid-point and grouped

mean estimators by $Mrmk , , and $

Xgmk , , , respectively, where

k=1, 2, 3, or 4, corresponding to the above rule of truncating.

The grouped mean estimators are also referred to as Max I, Max

II, Max III, or Max IV, corresponding to the rule of truncating.

III. EMPIRICAL STUDIES

 In this section, the methods and results of two empirical

studies are described. The first study concentrated on the overall

industry/occupational wage mean and the second study

concentrated on the upper open interval mean. The same

measures of evaluation as described below were used for each

study.

 We define the error of estimation to be the difference between

the estimated value and the true value, which is assumed to be

known. The relative error is defined as the ratio of the error of

estimation to the true value. We compare different estimators by

looking at the absolute values of the relative errors of their

estimates. The relative errors may be expressed in percentages

and be called percent errors. Absolute percent errors are the

absolute values of percent errors. For example, if the true mean

is 50, the estimated mean is 48, then the relative error is -.04 and

the absolute percent error is 4 percent.

 Note that for the median, the error in estimation is due to

grouping the data; whereas, for the mean, the error is due to

grouping and to the upper open interval estimation. Sampling

error was not considered in the studies since wage data in exact

dollar amounts were used and regarded as the population.

A. OES RESEARCH

 This part of the research is called the OES research because it

is designed to meet the objective of the Occupational

Employment Statistics (OES) survey, that is, to find the best

estimators for the median and for the mean of occupational

hourly wage for each industry. Estimates, therefore, were

computed for each industry/occupation level.

1. Data

 The data used for the research are from the 1989 and 1990

White-Collar Pay (WCP) surveys. The WCP survey collects

wage data from establishments employing 50 or more workers in

industries throughout the United States, except Alaska and

Hawaii. The WCP survey collects actual dollar amounts for

wages.

 We selected nine industries from different major industry

groups to study. Each industry is classified by a standard

industrial classification (SIC) code. The nine industries and their

SIC codes were: oil and gas extraction (SIC 13); food and

kindred products (SIC 20); chemicals and allied products (SIC

28); stone, clay, and glass products (SIC 32); transportation by

air (SIC 45); miscellaneous retail (SIC 59); security and

commodity brokers (SIC 62); hotels and other lodging places

(SIC 70); and educational services (SIC 82).

2. Method

 We adjusted the collected WCP data by their weights and the

weighted data were considered the true population, from which

we computed the true mean and median. We then grouped the

data into wage interval categories by industry and occupation.

For each set of grouped data, we estimated the median by the

linear interpolation method and estimated the mean by the

methods outlined in Section II. For the modified maximum

likelihood method, we only used the interval that contained the

truncated mean as the left truncated point. That is, the only

estimator examined was $

Xg m, ,1 (Max I).

 For the research, the 1989 Federal minimum wage of $3.35

was used as the lower limit of the lower open interval.

3. Selecting the Interval Categories

The pilot OES interval categories specified $35.00 as the lower

limit of the upper open interval. It was found that this figure was

too low to provide good estimates. Out of a total of 792

occupations over the nine industries chosen, 81 percent (639) did

not have records in the upper open interval. Of the 153

occupations that did have some records in this interval, as many

as 53 did not have records in any other interval. For these 53

occupations, there were no data to fit any distribution; the

Winsorized mean was the only alternative for computing a mean.

For these cases, the Winsorized mean could underestimate the

true mean as much as 53 percent. Many other occupations had

only a few observations in the previous bounded interval and the

rest in the upper open-interval. For these occupations, there

were not enough data to fit a Pareto distribution. In order to use

the Pareto distribution, the wage distribution should be a

decreasing function after a peak and should have a small "tail."

The "tail" for our purpose is the upper open-interval itself. Of

the 153 occupations that have records in the upper open-interval,

only 29 (19%) occupations have "tails" less than ten percent; 8

(.5%) occupations have "tails" of ten to twenty percent; 24 (16%)

occupations have "tails" of twenty to fifty percent; and 92 (60%)

occupations have "tails" over fifty percent. Attempts to fit a

Pareto distribution to these occupations of "large tails" lead to

gross errors. In addition, there could be no median estimates for

the occupations with "tails" over fifty percent.

 Based on the above observations and on the distribution of the

true mean of the upper open interval over industries, a modified

version for the interval categories was proposed and accepted for

future OES surveys. The modified interval categories increased

the lower limit of the upper open interval from $35 to $60 and

widened the middle ranges, while leaving the two lowest

intervals the same as before. The number of interval categories

was kept at eleven for administrative purposes. Now more
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Pareto distributions could be fitted, allowing different methods

for computing the mean for the upper open interval. The pilot

versus the future interval categories are given below.

 Interval Category Pilot Future

 A <5 <5

 B [5-6.5) [5-6.5)

 C [6.5-8) [6.5-9)

 D [8-10) [9-12)

 E [10-12) [12-16)

 F [12-14) [16-20)

 G [14-17) [20-25)

 H [17-21) [25-35)

 I [21-25) [35-45)

 J [25-35] [45-60)

 K >35 3 60

4. Results

 The modified interval categories gave us 23 occupations with

observations in the upper open interval. Of these, twelve had

"tails" between ten and fifty percent, and six had "tails" of fifty

percent or more. There were no median estimates for these six

occupations. Four of these six occupations only had

observations in the upper open interval and did not have

alternative measures of the mean besides the Winsorized

method.

 The estimates for overall means improved significantly from

the pilot interval categories as the following tables show. We

were able to estimate more medians using the new interval

categories.

 An error profile for all occupations using pilot interval

categories is displayed in Table I. The numbers in the body of

the table denote the number of times a specific method resulted

in errors that fell in a specified error range. The error ranges are

absolute percent errors (percent errors in absolute values).

 A similar error profile using the future interval categories are

displayed in Table II.

TABLE I

Error profile for all occupations using pilot interval categories

Error Range Qnt I Qnt II Max I Winsorized Median

 0 - 4.99 591 596 608 602 570

 5 - 9.99 84 81 83 108 110

 10 -14.99 21 24 20 24 18

 315.00 43 38 28 58 3

 N/A* 53 53 53 0 91

TABLE II

Error profile for all occupations using future interval categories

Error Range Qnt I Qnt II Max I Winsorized Median

 0 - 4.99 626 626 634 631 566

 5 - 9.99 112 112 113 115 156

 10 -14.99 41 41 38 40 56

 315.00 9 9 3 6 8

 N/A* 4 4 4 0 6

* Estimates could not be computed.

 The quantile I and quantile II methods performed similarly.

These estimates were not as good as the maximum likelihood or

the Winsorized estimates. They had more errors in the "315%"

range. Additionally, the errors by the quantile methods in this

range were extremely high. The Winsorized estimate had more

errors in the "315%" range than the maximum likelihood

estimate, however most of these errors resulted in the cases in

which the maximum likelihood estimator could not be computed.

We recommended the Winsorized mean for the OES survey

since it is easy to understand and to implement.

 The median also performed well. Under the pilot interval

categories, the median for 91 occupations could not be computed

compared to six occupations under the future interval categories.

 Empirical results also suggested that the absolute percent error

tends to be high when the number of unweighted workers in an

occupation is small. For the median, 19 percent of the

occupations that have less than ten unweighted workers have

percent errors of the magnitude ten percent or higher compared

to three percent of the occupations that have at least ten

unweighted workers. For the mean, the comparison is 15 percent

versus .8 percent. Based on these results, we recommended that

publishability criteria include the provision that all published

figures come from occupations with at least ten unweighted

workers.

5. Validation of other data sets

 The White Collar Pay surveys are not completely

representative of the OES survey. The survey does not cover

Hawaii and Alaska and does not include small establishments of

less than 50 workers. It does not target production workers and

therefore its wage distribution is not the same as the wage

distribution of the OES survey which includes all occupations.

 However, we feel that the mean and median estimators are

robust and should work as well on OES data. We validated the

recommended estimators on two additional sources of wage data:

the Alaska data and the Industry Wage Survey data.

 In the State of Alaska a Wage Rate survey was conducted. The

hourly wage data from this survey was sent to the Bureau of

Labor Statistics to be tested with the recommended procedures.

The staff in Alaska indicated that the following industries should

be tested: metal mining (SIC 10); general building contractors

(SIC 15); food and kindred products (SIC 20); depository

institutions (SIC 60); hotels and other lodging places (SIC 70);

and engineering and management services (SIC 87). The

minimum wage of $4.75 was given to us by the staff in Alaska

and was used in the research.

 Since we recommended that all publishable means and

medians have at least ten unweighted workers, we only

considered the percent errors of the occupations with at least ten

workers. Out of 89 occupations, five (6%) had absolute percent

errors for the median exceeding ten percent and two (2%) had

absolute percent errors for the mean exceeding ten percent.

These numbers are thought to be small enough to be acceptable.

 The Industry Wage Survey (IWS) collects wage data for

production workers only. Wage data from this survey do not

have as many observations in the upper interval categories. In

order to see the effect of the future interval categories on data

sets with lower wages, we tested the procedures on two IWS data

sets using both sets of interval categories. The data sets

available were the 1987 Men's and Boys' Shirts (SIC 2321) and

the 1984 Millwork (SIC 2431). The Federal minimum wage of

$3.35 at that time was used as the lower limit of the lower open

interval.

 There were 31 occupations in the Millwork data and 30

occupations in the Shirts data to be tested. For each data set, the
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two sets of interval categories basically gave the same error

distribution.

6. Compare Percent Change Across Time or SICs

 Of interest to the OES survey is the comparison of

occupational hourly wages across time. As mentioned in Section

II, the median or functions of median should not be used for

these purposes. The difference in means of a characteristic of a

population was found to be bounded relative to the means of its

subpopulations when compared across time periods, provided

the proportional size of the subpopulations tend to stay the same

over time.

 Also of interest to the OES survey is the comparison of

occupational hourly wages across populations and across their

subpopulations. The population could be a major occupational

group such as "Engineers." The subpopulations could be

different pay levels (which reflect the different expertise and

experience levels within one major occupational group) or

different detailed occupations (which reflect the many different

but related jobs, for example, different kinds of engineers). The

size of the subpopulations relative to the size of their parent

population may not be the same in different industries.

 We looked at the "bounded property" of true and estimated

means and of true and estimated medians in the case of

comparing different occupational pay levels. We found that true

values or their estimates may or may not be bounded.

Furthermore, the existence or lack of "bounded property" of true

values does not carry over to their estimates. That is, bounded

true means or bounded medians do not lead to bounded

estimated means or bounded estimated medians, and vice versa.

For these comparisons, the median as well as the mean should be

used with caution.

B. INTERVAL MEAN RESEARCH

 This part of the research expands on the OES research by

concentrating on estimating the mean of the upper open interval.

1. Data

 Data from the same 1989 and 1990 White-Collar Pay surveys

were used. The following nine industries were added: coal

mining (SIC 12), special trade contractors (SIC 17), apparel and

other textile products (SIC 23), electronic and other electric

equipment (SIC 36), communications (SIC 48), wholesale trade

(SIC 50), automotive dealers and service stations (SIC 55), real

estate (SIC 65), and legal services (SIC 81). These additional

industries allowed additional coverage of industry groups.

2. Method

 As discussed in the OES research, the Pareto distribution is not

always suitable for industry/occupational wage data. With

occupational data by industry, using either set of interval

categories, there is a high percentage of occupations with large

frequency in the upper open interval ("large tails" for Pareto

distributions). In order to conduct research on estimating the

mean for the upper open interval, we need data that are more

suitable to fitting the Pareto distribution. To bring smaller

"tails," we grouped all the industrial occupations into

professional, technical, and clerical occupational types as

described in the White-Collar Pay survey bulletin published by

the Bureau of Labor Statistics. We estimated the median and the

mean by industry/occupational type with the methods mentioned

in Section II. When using the maximum likelihood method on

these data sets, the four rules for choosing the left truncated

point were applied. Since there were 18 industries, each with

three different occupational types, a total of 54 populations were

considered.

3. Selecting the Interval Categories

 When we grouped the occupational type data according to the

future OES interval categories, there were not enough data in the

upper open interval (in percentages) to do meaningful research.

Based on the frequency distribution of workers in each

occupational type, we decided to use the pilot OES interval

categories with different lower bounds for the upper open

intervals: $35 for the professional type data, $21 for the

technical type, and $17 for the clerical type. These lower

bounds were not chosen arbitrarily. They were lower bounds of

the pilot OES intervals, but not necessarily of the upper open

interval. These lower bounds were chosen with the aim for

"tails" of less than ten percent. It was found from the research

that the Pareto distribution gives best estimates with "tails" of

this size. This gave the percentage of workers in the upper open

interval from .4 to 6.5 percent in the professional occupations

(with one exception of 18 percent), from .05 to 7.6 percent in the

technical occupations, and from .04 to 3.4 percent in the clerical

occupations.

4. Results

 The estimated median and estimated truncated mean usually

fell in the same interval. Therefore, the maximum likelihood

estimator derived from using either the interval containing the

median or the interval containing the truncated mean as the left

truncated point usually was the same. For this reason, we will

only discuss the left truncated point as in the interval containing

the estimated truncated mean (Max I) or in the interval following

the one that contains the estimated truncated mean (Max II). For

the professional occupations, the quantile II method performed

well, followed by the maximum II method. For the technical

occupations, the Winsorized performed well, followed by the

quantile II and maximum II methods. For the clerical

occupations, the maximum II method is the best. Although the

Max II estimate is not always the best over the occupational

types, it is consistently one of the best.

 A third of the 54 occupational type populations did not have

records in the upper open interval. An error profile for the two

thirds that had upper open interval estimates is shown in Table

III.

TABLE III

Error Profile for Estimating the Mean of the Upper open Interval

Error Range Qnt I Qnt II Max I Max II Winsorized

 0-4.99 17 23 14 18 4

 5-9.99 12 7 12 15 10

 10-14.99 3 5 5 2 13

 3 15 4 1 5 1 9

As the table indicates, the Winsorized was not the best method.

The quantile II method could be good, but had more large errors

than the maximum II method. The quantile I, maximum I, and

Winsorized methods were not recommended. They seemed to be

sensitive to different data sets and gave more errors in the

"3 15%" range. The Max II is the one recommended.

5. Effects of Grouping Data

 The quantile II method uses the two bounded intervals

preceding the upper open interval in estimation. When the

boundary point of these two closed intervals was changed from

$25 to $28 for the professional type data, the quantile II

estimator did considerably worse. The other estimators were not
-6-

affected much by the change. This is in agreement with West's

findings (1986) that the quantile method is sensitive to the way

the data are grouped.

 When we lowered the lower bound of the upper open interval

of the technical type data from $21 to $17, allowing larger

"tails," there was a uniform decrease in the percent errors in the

"0-4.99%" range and an increase in the higher percent errors

across industries. The quantile II method was the best this time.

 The quantile I method was affected most when Pareto

distributions were fitted to "large tails." For example, when data

of the gas and oil industry were grouped according to the pilot

OES interval categories, the largest absolute percent error was

over 400 percent for the quantile I method and just over 100

percent for the quantile II and for the maximum I methods. Most

of these occupations have their truncated means fall in the

bounded interval preceding the upper open interval and could

not have the Max II estimates computed.

IV. CONCLUSIONS

 When data are grouped such that there are no suitable "tails" to

the Pareto distribution, such as when the SIC/occupation wage

data were grouped according to the OES pilot interval categories,

errors of the overall mean and of the upper open interval were

large. In particular, the maximum likelihood and quantile

methods could lead to gross errors. Furthermore, these estimates

could not be computed when the upper open interval was the

lone interval category. In such cases, the Winsorized estimate

may be the best (although technically, all that can be said is that

the mean is 3 U( ) r -1 ). In our research data, while the other

estimates could have absolute percent errors in the hundreds, the

highest error for the Winsorized estimate was just over 50

percent with the pilot OES interval categories.

 When the objective is to compute an overall mean, then all

estimators produce large errors if there are "large" (10% or

more) tails, and all estimators perform satisfactorily if there are

"small" (less than 10%) tails. When "tails" are large, the Pareto

distribution is not suitable for estimation purposes. When "tails"

are small, the overall error is not affected much by large errors in

the small "tails." This was the case when the wage data were

grouped according to occupational types: professional,

technical, or clerical. Given the nature of the occupational data

by industry and the OES's objective of computing an overall

occupational wage for each industry, the Winsorized estimator

was the one recommended with the modified interval categories.

 For the upper open interval, the Winsorized estimate is biased

downward. Depending on the lower limit of the upper open

interval, this bias could be large or small. The advantage of this

method is that it is simple and the direction of its bias is known.

For "small-tailed" distributions, the quantile II method performed

well most of the time. However, when it did not perform well, it

had much larger errors than the maximum likelihood method.

Although the maximum II estimator is not always the best over

the occupational types, it is consistently one of the best. This

estimator is recommended because it is more robust than either

the quantile I or the quantile II method. The Winsorized

estimator does not perform well for upper open interval

estimation in "small-tailed" distributions.

V. FUTURE RESEARCH

 Variance estimators for the mean, using the Pareto tail with the

modified maximum likelihood estimator for the parameter, will

be derived and evaluated in the next study. Also, in this study it

had been planned to fit theoretical distributions (mixture

distributions) to the occupational wage data. Unfortunately the

data were too thin to accomplish this. We hope a larger

appropriate data set will be available for future research.

REFERENCES

Koutrouvelis, I. A., (1981) "Large Sample Quantile Estimation

in Pareto Laws," Communications in Statistics, Theory and

Methods, A10, 189-201.

Parker, R. N., and Fenwick, R. (1983), "The Pareto Curve and

Its Utility for Open-ended Income Distribution in Survey

Research," Social Forces, 61, 872-885.

Quandt, R. E. (1966), "Old and New Methods of Estimation

and the Pareto Distribution," Metrika, 10, 55-82.

Shryock, H. and Siegel, J. (1975), The Methods and Materials

of Demography, Washington, D.C., U.S. Government

Printing Office.

West, Sandra A. (1986), "Measures of Central Tendency for

Censored Earnings Data from the Current Population

Survey," Proceedings of the Section on Business and

Economic Statistics, American Statistical Association, 751-

756.

West, Sandra A. (1985), "Estimation of the Mean From

Censored Earnings Data," Proceedings of the Section on

Survey Research Methods, American Statistical Association,

665-670.