ON THE "LARGE" OBSERVATION PROBLEM
Donald T. Searls
Institute of Statistics
Mimeograph Series No. 332
August, 1962
iv
TABLE OF CONTENTS
Page
vii
1.0
LIST OF TABLES
2.0
LIST OF FIGURES.
viii
3.0
INTRODUCTION
.
1
4.0
FORM OF THE ESTIMATORS
3
5.0
REVIEW OF LITERATURE
6
6.0
BASIC NOTATION .
9
7.0
DEFINITION OF ESTIMATORS
13
7.1
7.2
13
7.3
7.4
7.5
7.6
Y1'
Y .
2
Y .
3
Y .
4
Y .
5
Y6 .
9.0
10.0
14
15
15
15
Y .
7
CHARACTERISTICS OF Y .
1
16
8.1
Expectation, Variance and Mean Square Error
17
8.2
Comparison of Y and y.
1
19
CHARACTERISTICS OF Y2' . . .
33
9.1
Expectation, Variance and Mean Square Error
33
9.2
Comparison of Y and y.
2
33
CHARACTERISTICS OF Y3' . . .
40
10.1
Expectation, Variance and Mean Square Error.
40
10.2
Comparison of Y3 and y . • • •
4L
10.3
An Optimum Value for! . . . .
42
10.4
Comparison of Bias(Y3) with Bias(y ) and Bias(Y2)
1
43
7.7
8.0
14
17
v
TABLE OF CONTENTS (continued)
Page
11.0
12.0
CHARACTERISTICS OF Y
4
Expectation, Variance and Mean Square Error . .
45
11.2
An Optimum Value for Wand the Relative
a
Efficiency .
45
11.3
A Region for W .
46
11.4
Comp~rison of Bias(Y4) with Biases for Y1' Y2
and -Y
.
3
47
a
CHARACTERISTICS OF Y .
5
Expectation, Variance and Mean Square Error.
48
12.2
An Optimum Value of W for a Given t
b
A Region for W .
b
An Optimum Value for t
49
50
Comparison of Bias(y ) with Biases for Y , Y ' Y3
5
1
2
and Y4 . . . . . . . .
52
12.5
CHARACTERISTICS OF Y .
6
50
53
13.1
Expectation, Variance and Mean Square Error.
53
13.2
The Covariance Between the Sample Mean and the
Maximum Sample Observation
53
Comparison of Y6 and Y
55
13.3
CHARACTERISTICS OF Y . .
7
57
14.1
Expectation, Variance and Mean Square Error.
57
14.2
An Optimum Value for W .
57
14.3
A Region for W .
58
c
c
14.4
15.0
48
12.1
12.4
14.0
45
11.1
12.3
13.0
.
Comparison of Bias(Y7) with Biases for Y and Y
2
6
DISCUSSION OF RESULTS . .
59
60
15.1
Theoretical Examples
60
15.2
Comparisons . . . . .
82
vi
TABLE OF CONTENTS (continued)
P~ge
16.0
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS FOR FURTHER
RESEARCH . .
86
16.1
Summary.
86
16.2
Conclusions.
88
16.3
Recommendations for Further Research
89
LIST OF REFERENCES . . . . . . . . . . .
92
vii
1.0
LIST OF TABLES
Page
2.
3.
4.
S.
6.
7.
8.
The expected percentage of samples from the
exponential distribution with one or more observations greater than ! . . . . . . . .
67
The expected number of observations greater than
t for samples from the exponential distributiqn.
68
Relative effic.iencies (%) of Y for samples from
' 2
ut1on.
the exponential d'1str1' b
69
....
-
Relative efficiencies (%) of Y for samples from
' 3
ut10n
the exponential d'1str1' b
71
Exponential distribution; characteristics for Y
3
when optimum values for t are used. .' .
72
Relative efficiencies .(%) of Y4 for samples from
distributions with selected coefficients of
variation . .
. . . . . . . . . .
73
Optimum weights for YS for samples from the
exponential distribution. . . •.. . .
74
Relative efficiencies (%) of Y for samples from
S
the exponential distribution when optimum weights
are us ed . . .
9.
It
•
•
.'
It
0
•
o·
•
•
•
•
•
•
•
75
•
Pareto distribution; Y6 with n=2.
76
10.
Exponential distribution; Y
7
77
11.
Ratios of the bias to the population standard deviation
for Y , Y3 from theexponentialdistribution . • • • •
2
12.
.
78
Ratios of the bias to the population standard deviation
for 74, from distributions with selected coefficients of
var1at10n. . . . .
..
It
••.•
e
It
••.•.•••
•
79
13.
Ratios_of the bias to the population standard deviation
for Y with optimum weights, from the exponential
s
distr1bution.
. • . • • . . . . ..• • • • • • •
80
14.
Values of~E(Y4)' ~(Y4>' and [Bias(Y4)] for the
exponent1ald1str1but1on, . • . ~ • . . • . . •
-
-
-
2
.
81
viii
2.0
LIST OF FIGURES
Page
1.
Comparison ofMSE(Y3)
and cr2 In for the
exponential distr~bution
70
3.0
INTRODUCTION
Generally in the past the criteria used for evaluating estimators have been those of unhiasedness and minimum variance.
A
large body of statistical theory has been developed based on these
considerations.
In sampling finite populations, the
me~nsquare
mators has been used as a basis of comparison.
error of esti-
This has been due,
in part at least, because it has been found that biased estimators
of the ratio type have achieved considerable reductions in variance
without much bias.
The objective of achieving a small mean square error is closely
related to the achievement of a small expected absolute difference
in magnitude between an estimator and its true value.
Mathemati-
cally, the mean square error is much simpler to work with.
Such a criterion appears to have particular relevance to the
estimation of the population mean for highly skewed distributions.
Positive skewness is frequently encountered and will be considered
in this thesis.
The sample mean is always unbiased even though sampling is
from highly positively skewed distributions.
For small samples,
however, there is considerable variation from one sample to
another.
Also, when one or more observations appear in the ex-
2
treme tail, the estimate is likely to be much greater than the
population mean.
Alternatives to the sample mean, which tend to reduce the
magnitude of estimates greater than I-L without reducing the proportion of estimates smaller than I-L,willbe'biased.
Thisre-
duction in magnitude would appear to be desirable, however, and
for some alternatives it will lead to mean square errors smaller
than the variance of the sample mean.
This thesis will be concerned with alternative estimators to
the sample mean which might be useful in reducing mean square errors.
3
4.0
FORM OF THE ESTIMATORS
For purposes of this thesis extreme observations will be defined as thosehavirtgva1ues greater than somecut-offpoinb, t.
Five classes of estimators are considered depending upon the
weight given the extreme and non-extreme observations.
In a sample
there will be r non-extreme observations and (n- r) extreme observations.
1.
The cut-off point , .!' is predetermined.
Extreme observations are given a weight of zero· with
sample size fixed at n while non-extreme observations receive a
weight ofl/r.
2.
(y1)
Extreme observations are given a weight of zero with
sample size such that r observations less than t areassurec1.
Non-extreme observations receive a weight of 1/r.
Note that in
this caseon1yr is fixed and n is a random variable.
3.
(Y2)
Extreme observations are first weighted inversely pro-
portional to size, i.e. all extremes take on the value ·of!, then
all observations are weighted by lin.
4.
(Y3)
Extreme observations are given the same weight as the
non-extremeobservatioQ.s but·the weight is not necessarily equal
to lin.
5.
(Y4)
This is a generalization of four in which extremeobsef-
vations receive an equal weight which is less than the weight
assigned,:o non-extreme observations, lin.
(YS)
4
In addition, two other estimators are considered where a
different weight is applied to only the maximum observation regardless of its relation to any particular cut-off point.
6.
These are:
A weight of zero is applied to the maximum observation
1
and a weight ofn - 1 is applied to the remaining observations.
7.
(Y6)
A weight less than lin is applied to the maximum obser-
vation and a weight of lin is applied to the remaining observations.
The results presented will pertain (except as otherwise noted)
to a particular 'class of continuous distributions.
Distributions
belonging to this class have finite first and second moments; they
are unimodal; they do not exhibit negative skewness; and they occur
on the range (a, b) where
(4~O .1)
O<a<b<+ro.
Unimodal distributions are defined as distributions having one
mode with the density function decreasing monotonically from the
modal point.
Non-negative skewness, for this class, means that
mode :s.
J.L •
For finite ranges, maximum values for
J.L
max
= (b+a)/2 ,
and
(J
2
max
= (p-a) 2 /12 •
J.L
and a2 are:
5
Derivations will be presented to show that under various condi':'
tionseach of the alternatives achieves a mean square error which
is smaller than the variance -of the sample mean.
Expressions ·for
these conditions will be developed and optimum choices for the cutoff point, !, and the 'appropriate weights will be presented.
6
5.0
REVIEW OF LITERATURE
As noted previously, an extensive bibliography exists regarding
procedures for handling contaminated distributions.
On the
contrary~
nothing was found published regarding the practice of discarding
true sample observations.
Two unpublished documents exist which pertain to this problem.
Hendricks and Huddleston (1960) have examined empirically a procedureof discarding a small percentage of the largest sample observations.
Pearson typelII distributions were considered.
Thedis-
carded observations were replaced by an estimate ·of the expected
value of observations from the upper tail of the distributions.
Their procedure requires the coefficient of variation to be known.
It is utilized along with the sample mean to estilmate the cut-off
point to be employed.
The procedure appeared to work very well
for the examples presented.
Bershad (1961) explored some modifications of the estimated
total of finite populations.
He found that for a distribution of
households per segment, with an average of approximately six
households, the optimum cut-off point was approximately 400 households per segment.
The optimum weight for sample segments of size
greater than the cut-off was approximately one third of the weight
that otherwise would have been used.
The reduction in mean square
error was expected to be between 26 and 40 percent.
The problem of estimation with censored samples is related to
7
the problem considered in this thesis.
Lehman·(196l) conducted an
investigation to determine optimum sample size, minimum required
failures and minimum test time in order to estimate most effectively
the scale parameter in the Weibull distribution.
Themaximumlikeli-
hood estimator was used.
Lehman also presented a review of literature for the problem
of estimation with censored samples including basic papers by
Epstein (1954) and Kao (1956).
Mendenhall (1958) has compiled a
bibliography on life testing research.
Virtually all previous work has been concerned with maximum
likelihood estimation.
There is no indication that any of these
estimators have smaller mean square errors than estimators that
could be formed if the entire sample were available,
Itappears
that one of the major distinctions between the problem considered
in this thesis and the censoring problem is a knowledge of the
values of the outlying or censored observations.
Thenumerical
value of a censored observation is not known.
One estimator considered in this thesis,
Y3~
is essentially
equal to r/n times the maximum likelihood estimator for the exponential distribution when values of the (n-r) observations are
unknown
(r~l).In
this thesis Y3 is compared with the maximum
likelibood estimator for the exponential when all observations
are known, y.
8
Other comparisons of maximum likelihood estimators with those
considered in this thesis are contemplated in future research.
9
6.0
BASIC NOTATION
The following notation will be employed:
~
=
(6.0.1)
E(y)
where yis distributed as f(y) with a
~y ~
b.
(6.0.2)
MSE
-
Mean square error.
C.V.
=
Coefficient of variation,
(6.0.3)
(J
=
(6.0.4)
~
t = a cut-off point where a < t < b.
(6.0.5)
W =
the weight applied to designated
sample observations.
(6.0.6)
t
F(t)
=
a
~t
=
J
f(y) dYe
t
1
F(t) a J
1
F(t) .
(6.0.7)
yf(y) dYe
<
Y
< t)
(6.0.8)
t
a
(a
J
(y -
~)
t
2
f(y) dYe
(a
<
y
< t)
(6.0.9)
10
IJ. tl
=
1
b
J
1 _ F(t)
(t
Y f(y) dYe
t
=
1
1 _ F(t)
IJ.m
Y
<
b)
(6.0.10)
b
J
-2
(y - IJ.tl)
£(y) dYe
t
Ym
<
(t
<
Y
<
b)
(6.0.11)
=
the maximum sample observation.
=
E(y)
m
=
n
(6.0.12)
b
J
y [F(y)Jn-1 £(y) dYe
a
rJ
2
m
(a < Y < b)
(6.0.13)
=
V(y) ,
m
=
n
b
J
a
(y- IJ. )
m
2
[F (y) J
n-1
£ (y) dYe
(a < y < b)
(6.0.14)
The. distribution 0,£ y i s well known.
m
See especially Cramer
(1945) p. 370.
y
=
the sample mean,
n
=
L y.
j=l J
n
(6.0.15)
11
Ya
=
alternative estimators considered.
(a=1, .•• ,7)
(6.0.16)
- f.J..
(6.0.17)
Bias(y)
a
=
E(Y)
a
MSE(y)
a
=
E(y
a
=
2
V(y) + [Bias (y)] •
a
a
=
Relative Efficiency,
R.E.
=
-
f.J.)
V(Y)
MSE(Y )
a
2
,
for this thesis.
(6.0.18)
(6.0.19)
For simplification set:
p
=
F(t).
(6.0.20)
q
=
1 - F(t).
(6.0.21)
From (6.0.8) and (6.0.10),
(6.0.22)
and
(6.0.23)
From (6.0.9) and (6.0.11),
(6.0.24)
12
Rearranging and substituting for
2
at'
=q
-12
(0'
~t'
gives:
2
-2
pa)
- pq (~
. t
(6.0.25)
Expectation taken over r will be
designated~.
When a partial
derivative is taken the letter d will be used instead of
the correct meaning will be apparent.
O.
However,
13
7.0
DEFINITION OF ESTIMATORS
7.1
The practitioner, in order to mipimize the effect of large
observations, may employ the following estimator:
r
E Yj
j=l
=
Define,
r
for r > 1,
where the y. used are those which.are less than the cut-off
J
point t.
The question arises as to what the practitioner does for
r= O.
The values of t chosen (if in the correct general area)
are such that F(t) is near one.
Since the probability that r= 0
is equal to [1 - F(t)]n or qn, this possibility is usually not
considered in practice.
For the sake of completeness in the dis-
cussion, Y will be defined as equal to t forr = O.
l
This estimator is now defined as:
r
E
Yl
=
Yj
j=l
r
(y.
J
< t)
(r ~ 1)
=
t.
(r =0)
(7.1.1)
14
7.2
If the· practitioner wishes to replace the discarded observations to insure
afi~ed
effective sample size, .!', a slightly differ-
ent estimator is formed.
r
2::
Y2 =
j=l
r
Y.
J
< t)
(y.
J
(r fixed)
(7.2.1)
Let~designate
the total sample size necessary to obtain r
observations less than t.
Then n will be distributed as the nega-
tive binomial with parameters rand E..
An alternative estimator which appears to retain some of the
information contained in the number of extreme observations is:
r
2::
Y3
=
j=l
Yj + (n -.r)t
n
. (Y.
J
< t)
(j=l, ... ,r)
(7.3.1)
In a sense, this estimator could be regarded as being formed
. by the process of weighting the extreme observations inversely
proportional to their size.
The weights then are t/ Y for Y > t.
j
j
15
7.4
An estimator could also be formed giving the extreme observations .a constant weight less than the weight given the remaining
observations.
A special case of this arises when all sample obser-
vations are given this constant weight
(i.e. t::;:a;)
The special case will be considered separately and will be
defined as:
n
=
W
a
L:
j=l
(7.4.1)
y ..
J
7.5
The more general estimator will be defined as:
r
.. L:
Ys
=
j=l
n
Yj + Wb
L:
j=r+l
Yj
n
< 1)
b
(7.5.1)
(W
Wherey. < t for j =1, ... ,rand y. > t for j > r.
J
.
7.6
J
Y6
For highly skewed distributions, a gain might be achieved
when the maximum sample observation is discarded arbitrarily.
This estimator is defined as:
16
n
L: Yj-Ym
j=l
n - 1
n-l
=
L: Yl
1=1
<
(YJ..
n - 1
Ym)
(7.6.1)
7.7
If the maximum sample observation is retained but given less
weight, the following is obtained:
n-l
L:
Y7
=
R=1
Yl +Wy
c m
n
(w
c
<
n
L: Y ' -(1 - W)y
c m
j=l J
n
(7.7.1)
1)
17
8.0
8.1
CHARACTERISTICS OF Y1
Expectation,Variance, and Mean Square Error
r
L:
Y1 =
(Y.
J
Yj
j=l
r
=
t
(r
< t)
>
0)
(r= 0)
.
(8.1.1)
For a given r
~
1, Y1 can be regarded as a simple random
sample of size r from the truncated portion of
thedi~tribution
function and as such it provides an unbiased estimate of the mean
(~t)
of the truncated distribution.
That is,
E[y1lr > 0]
For r
.-
=
~t
(8.1.2)
t.
(8.1. 3)
0, Y is a constant, t, so,
l
E[y
. 1 Ir= 0]
=
Also,Eis distributed as the binomial with parameters nand F(t)
or E.
Taking expectation over
E,
n
=
L:
r=O
- I
n!
E(ylr) r!(n-r)!
r n-r
p q
18
=
n
n
q t + J-l
n
(l~q)
=
J-l
r n~r
pq
!:
t
r=l
n
+ q t
t
(8.1. 4)
the variance will be obtained by substracting the square of the
bias from the mean square error.
(8.1.5)
and
(8.1.6)
Now designate
p
r
=
(n) p r qn-r
r.
(8.1. 7)
Then,
n
MSE(Y1)
=
!:
n=l
+
2
Pr (a t /r +(J-l
n
q (t
-
J-l) 2
-
2
J-l t ) ]
,
(8.1.8)
=
(8.1. 9)
From above is obtained,
Bias(y ) =
1
n
(1 - q ) J-l
n
t
+ q t - J-l,
19
(8.1.10)
Subtracting the square of (8.l.l0) from (8.1.9) gives:
=
8.2
n
2
n
2
(1 - q ) [at E(l/r) + q (t - I-Lt ) ]
(8.1.11)
Comparison of Y and Y
l
The following statement and proof establish that when the
proper cut-off is employed Y has a smaller mean square error than
l
Y for n > 2.
Statement I:
For distributions of the class considered and a given
n> 2, a region for t exists such that
2
< a In
or
Proof:
(8.2.1)
Since MSE(Y ) is composed of sums and products of continuous
l
functions of ! it is, for a given £' a continuous function of !
Sokolnikoff, 1939, p. 35).
Also, evaluatingMSE(Yl) at t
= Q.
=b
(See
gives:
2
n
(8.2.2)
Thus the statement will be proven i f it can be shown thatMSE (Y )
l
2
is approaching a /n from below as t approaches b.
DifferentiatingMSE(y ) with respect to !'
l
20
+ q n (t - J.J.)
+ nq
n-l
2]
(8.2.3)
,
2 dn
n
(t - J.J.)~+ 2q (t - J.J.),
(8.2.4)
Evaluating the individual components gives:
q = 1 - F(t) ,
.i9.
=
dt
- f(t)·
'
(8.2;5)
21
~t
L(II )
dt
~t
1
t
= _ !1tl.)
F(t ~ t + _(1.)
F t (tf(t)] ,
- !ill..
_
- F(t) (t
2
crt
:t(cr~)
1
=
F(t) af
t
;~~~ cr~
= -
~t)
(8.2.6)
.
(y- ~t)
2
fey) dy ,
+ Ftt) [(t -
2
= !1tl.
F(t) [(t - ~t)2 - cr t )
1
(1 - q n ) E[-)
r
(a < y < t)
= F(t) a J y fey) dy ,
=
n 1 (n.) r n-r
~ p q
r=l r r
n
+
~
r=l
~t)2
(a < Y < t)
f(t») ,
22
- f(t)n
n-1
n 1
r n -r-1
~ (~.) p q
r=l
= f'(t) [(l-f> - n(l - qn-l)
=f'~t>
{(I _
qn) _ np(l_ qn-l)
E[~ I n
E[~
- 1,
In - 1,
PJ],
PJ}
(8.2.8)
substituting the components in (8.2.4) gives:
£ft'
=~
p
{
. E[-]
1 (t
. - f.L) 2 - a 2 ]
(1 - q n ) [
a2+
. t
r
t
t
n-1
- npq
(t n
+2q (t. - f.L)
f.L
t
) (t + f.L
t
-
(
2f.L)J
(8.2.9)
23
Designate the term in large brackets as [AJ.
Then
Evaluating (8.2.10) at t = b gives:
i J'
I
~t [MSE (y1) J
_= f (b) [ [A J
t=b
t-b
where
[AJ
n
I
- - - (J
n-l
2
t=b
=
(8.2.11)
There are three possible cases to consider:
1.
f(b) > O.
2.
f(b) = 0, b of +
CD
3.
f(b) = 0, b =+
CD
Case 1.
f(b) > O.
The derivative will be positive if [AJ evaluated at t=b
is positive.
Maximum values for I,..I , (J2, and [2n-1
n-l J will minimize
[AJ evaluated at t = b.
For the class of distributions considered
~=
(a + b)/2 and (J2
= (b - a)2/ l2 .
max
max
mizes (2n - l)/(n - l'~
For n >2, n = 3 maxi-
24
Substituting these values in (8.2.11) gives:
I ;:
[AJ
(1/3) [(b- a)2j4 - 5(b - a)2 /24J
t=b
>
(8.2.12)
O.
Thus the derivative is positive.
This indicates that MSE(y ) is
l
2
approaching a /n from below since from (8.2.2) MSE(Y
at t
=b
t
evaluated
2
is equal to a In.
Cas e 2 .
At
r)
f (b) = 0, b :f:.
+
CD
•
= b the first derivative vanishes.
Statement I can be
proven for this situation by showing that a maximum for MSE(y ) is
l
attained at this point.
is positive.
A maximum exists if the second derivative
If the second derivative vanishes also, the procedure
used will be to investigate the first derivative to the left of t
Taking the second derivative,
d
2
dt
2
Ldt [AJ. + f (t) [AJ
p
r ) J = !..ttl.
P
i
[MSE(y
f(t)
2
p
n
+ 2q
2
[AJ
-
2nq
n-l
f(t) (t - IJ.)
(8.2.13)
= b.
25
Evaluating at t = b gives:
d
2
dt
I
2 [MSE (Y1) J
=
f I (b)
t=b
With f(b) = 0,.
~
max
and cr
2
max
[rA,J
L J.
(8.2.14)
are not achieved so that for
this case
I
[AJ
> (b - a)2/ 72 .
(8.2.15)
< 0.
(8.2.16)
j:=b
Now,
f
t
(b)
When f'(b) < 0, the second derivative is negative and a maximum
exists at t=b, proving the statement.
the second derivative vanishes.
However, when f'(b) = 0,
Investigating the first derivative
to the left of t = b it is noted that
!i!L and
n
2q (t -
~)
are
p
positive.
[AJ has a positive value at!.
~t
"
fAJ
I
If
:f ro
t=b
[AJ must at some point less than b become and remain positive.
Differentiating [AJ with respect to ! gives:
26
- np
- n(l - q
n-1
1
d2
) E[~ In,. 1, p] [dt (p crt)]
(8.2.17)
Evaluating the derivative of pcr~ gives:
(8.2.18)
27
Since f(b) = 0, this gives:
1
~t[A]'
= 2 (b -
~)/n,
t=b
(b =I-
=1-00.
Thus for! approaching
£,
d
CD)
(8.2.19)
-
dt[MSE(y )] is positive and the proof
1
of the statement is complete for case 2.
Case 3.
f (b) = 0, b = +
00.
Theconditionn )2 is relaxed so that the proof includes
n = 1 and n= 2.
+
CD
Evaluating the first derivative for! approaching
gives as before
!.l!l > 0 ,
P
n
2q (t - ~)
>
0,
and [A] ultimately becomes and remains positive since the term
(, t - ~)2E[1:.]
will·dominate.
t
. r
tive as t approaches +
Thus the first derivative is posi-
00.
This completes the proof of Statement I.
Note ,that the proof of Case 3.
of distributions studied.
is not restricted to the class
28
Statement I, (8.2.1), is true for n= 2 if
1
(a
f(y) =f (b-a)
<y
<b)
(8.2.20)
Proof: - From (8.2.4) some algebra gives:
2
+ 2q (t -
I-L).
(8.2.21)
Designate the term in curled brackets as [AI].
Then
(8.2.22)
Evaluating at t = b giyes:
(8;2.23)
where,
[A I]
J
t=b
Gasel.
f(b) >0.
=
[(b - I-L) 2 - 3c?]/2.
(8.2.24)
29
Substituting maximum values for
~
and a
2
in (8.2.24) indicates
that:
[A Y]
I
~ 0
(8.2.25)
t=b
For positive values of [Ai] evaluated at t =b, the right hand side
of (8.2.23) is positive and the proof holds as above.
at t = b is zero for maximum values of ~ and a
fey) = l/(b-a).
2
[Ai] evaluated
which occur when
In this case the first derivative ofMSE(Y1)
vanishes.
Taking the second derivative,
2
L
dt2
[MSE(y)]
1.
=
f(t) L[Ai] +f'(t) [Ai]
dt
- 4qf(t)(t -
. 2
~)
+2q
(8; 2.26)
whether the second derivative is positive or negative for t = b will
d
depend or the sign of dt [A I]
I
since f I (t) is zero.
t=b
Taking the derivative of [AI] ,
d
d
2
2
dt[A'] = (2 - 3p) dt (at) - 3 f(t) at
d
- ~) [1 - -(J.L)]
t
dt t
30
- (3/2) f(t) [(t - f.L )
t
2
2
- at
(8.2.27)
To evaluate (8.2.27) at t=b, the following components are
required:
~t(f.Lt)
I
-
f (b)
(b - f.L) ,
(8.2.28)
t=b
(8.2.29)
Evaluating the derivative at t= b,
£dt [Ai].
I
=
(b -
~)
,
t=b
> O.
(8.2,30)
31
Thus for this situation MSE(Y ) is a minimum at t = b and the
l
exception is established since MSE(y )
1
I
=
a
2
~
.
t=b
Case 2.
f(b) =0, b
=F
+co.
Similar to the proof of Case 2. above.
For n= 2, [see
(8.2.25) ]
[A' ]
I>
0, instead of (b - a) 2/72, however, this
t=b
does not affect the argument.
Case 3.
b= + co .
Identical with proof of Case 3. above.
Statement I, (8.2.1), also holds when n
Proof:
=
1 for all three cases.
For n= 1,
(8.2.31)
Differentiating with respect to t,
= f(t) [(t - J-L t )
2·
+ (Il - J-L t )
2
2(J-L - J-L ) (t - J-L
t
t
- (t - J-L)2] +2q(t - J-L) ,
= 2q(t - J-L)
(8.2.32)
)
32
Thus,
d
-
dt[MSE(Y1)] > 0
for t >
~
and the same argument
holds as above.
The optimum choice of
t
for any particular density function
can be found by equating (8.2;9) to zero and investigating the roots.
2
kquating (8.1.9) to cr /n, solving for !, and evaluating (8.2.9) at
these points will determine the region for which Statement! holds.
33
9.0
9.1
CHARACTERISTICS OFY
2
Expectation, Variance, and Mean Square Error
r
2:
=
Y2
J=l
Y
J
r
(y. < t)
J
(r fixed)
(9.1.1)
For any!, y
behaves as the sample mean of the truncateddistri-
but ion.
Taking expectation,
(9.1. 2)
Also,
(9.1. 3)
Obtaining the mean square error,
,
(9.1.4)
(9.1.5)
9.2
The sample size
Comparison ofY2 and Y
~
required to obtain Eobservations smaller
than t is distributed as the negative binom:l.al with parameters .E.
and.Po.
If the sample mean is computed from the !!, observations,
-
.. 2.
E(y - 1-1-)
2
=crE(l/n) .
(9.2.1)
34
It appears more reasonable for purposes of comparingMSE(Y2)
with V(y) , to use E(n) as an equivalent sample size thus eliminating
the inflation in the variance of y due to the fluctuation in sample
size.
This substitution gives:
(9.2.2)
A statement for Y2 will be formulated and proven using both
representations of the variance ofy.
Statement IIa.
given
For distributions of the class considered and a
£' a region fort exists such that
MSE (y2)
< V(y)
,
(9.2.3)
where V(y) is taken to be (9.2.1).
Proof:
Let g(t) = V(Y)MSE<Y).
(9.2.4)
Then, since g(t) = 0 at t =b [or g(b).= 0], the statement can be
proved by demonstrating that g(t) is positive as ! approaches
£.
Differentiating with respect to !'
where
1 n-1
rn-r
n (n-r) p q
Evaluating individual terms,
fro
E r .(n-1) r-1 n-r f(t.)
n n-r p
q
n==r
(9.2.6)
35
+00
_~. ~ (n-1)
n
n=r
n-r
r n-r-1 £(t) ,
p q
(9.2.7)
fnd from (9.1.5)
=. 1:.n .lli2.
[( t
p
- i-L ) 2 - 2n (i-L - i-L ) (t t t
Thus,
g I (t)
=-
.lli2.[l.
p
n
2
[t - i-L)2
- 2n(i-L
- i-L)(t
- i-L)
t
.
t
t - cr t J
11 . r + 1,
- ncr 2 [E [-n1 J - E [-n
Designq,te the term in cutledbrackets as . [BJ.
gl(t)
p]]
}•
(9.2.9)
Then·
= _ f;t)[BJ,
(9.2.10)
and
gl t (t)
=
-ffCt)
p
L
dt
[BJ
+ f 'et) [B] _ f(t)2[BJ} .
P
p
2
(9.2.lJ:)
The statement is proven if
a,
g I (0)
<
g'(b)
=0
or i f
and
36
gIl (b) > O.
There are three caSes to consider:
.f(b) > O.
L
Case 1.
2.
f(b) = 0, b ::j:
+
00 •
3.
f(b) = 0, b .= +
00 •
f(b)·>O.
g'(b) will be less than zero if [B] evaluated att = b is
greater than zero.
Evaluating '[b] at t = b gives:
I
[B]
=
1:.
r
[(b - 1-!)2 _ 2r+1 a 2 ] •
r+1
(9.2.12)
t=b
2
This expression is minimized by sub,stituting I-!max' a max and letting
rbecome large.
This gives the result:
I
[B]
> (b-a) 2 /12 ,
t=b
>
(9.2.13)
0 •
The proof of case 1 is complete.
Case 2.
f (b) == 0, b::j: +
00
•
For this case gi(b) = O.
Also, f'(b)
then g"(b) > 0 and the case is proven.
gl '(b) = O.
:s O.
When f'(b) <0,
However, when f.(b) = 0,
The case can still be proven for this situation by
showing that gl(t) is negative as! approaches
~.
Investigatingg'(t) to the left of t= b, ,it is noted that
- f(t)!p is negative.
Since [B] evaluated at t= b is positive,
37
it will also be positive to the left of t = b if
I
d
dt [B]
:fOO •
t=b
Differentiating [B] with respect to ! gives:
~t
[B] -
~[
- ra2
Since f (b) =
° and b :f co,
: t
I
rB]
=
2(t -
~t) [1
r: t [E(~)] :t [E(~
-
Ir+l, p)]].
(9.2.14)
this gives:
2 (b - IJ.) / r
,
t=b
:f co .
(9.2.15)
Thus for! approaching £, gl(t) is negative and the proof of
case 2 is complete.
Case 3.
For
f(b) = 0, b = +co .
! approaching£, g I. (b) will ultimately become and remain
negative since the term (t - IJ. )2 will dominate.
t
proof for case 3.
This
complete~ the
38
Statement lIb:
given
£
For distributions of the class considered and a
a region for t exists such that
(9.2.16)
where V(y) is taken to be (9.2.2).
(9.2.17)
Then, sinceg(t) =0 at t = b, the statement can be proved by
demonstrating that get) is positive as ! approaches~.
Differenti-
atingwith respect to t,
2
gl(t) =f(t)cr
r -
~t
(9.2.18)
[MSE(Y2)J
where
= -:-r1
fft'[
~
p
2
(t - j.J.) 2 - 2r (j.J. - j.J.) (t _ j.J.) _ cr ]
t
t
t
t .
:(9.2.19)
Thus,
2
j.J.t)- crt-pcr
(9.2.20)
Designate the term in curled brackets as [13J •
gl(t)
and
gOO
Then
= _.lltl.
. [BJ
rp
(t) --P;t
l
d
dt
(9.2.21)
[BJ
+
£1 (t)
P
[B J
£1J'
39
- [ f (t
2
)f
[B]J.
(9.2.22)
p
The proof procedes in a similar manner as the proof for Statement IIa. after noting:
I=
[B]
(b - IJ.)
2
- 20'
2
,
t=b
(9.2.23)
>0
d
+2r(t - IJ. t ) dt
0-\)
d
2
-"dt(O' t ) - f(t) a
2
(9.2.24)
~t [B]
I
=2(b - IJ.)
t=b
f(b)=O
: /: +00
(9.2.25)
The optimum choice for! for any particular density function
can be found by
~quating
gating the roots.
(9.2.9) or (9.2.20) to zero and inves1ci-
The region for which get) > 0 determines the
region for which Statement II holds.
40
CHARACTERISTICS OF Y3
10.0
10.1
Expectation, Variance, and Mean Square Error
r
=
+ (n-r)t
L: y.
j=l J
(r=O, ... ,n)
(y. < t)
J -
n
(10.1.1)
where r is distributed as the binomial with parameters nand p.
Taking expectation,
O[t!.]
n
E
=
P I-l
t
+
I
E [y. Y .<t ]]
J
+
J
(10.112)
qt .
Also,
t
2
V(r)
r
2t Cov [ L::
0
j=l
(10.1.3)
Considering the separate terms,
V(r)
= npq
;
(10.1.4.)
=
vh[~y.lrJ]+
EorV[.~
[" J=l J
J=l
=
np ( q I-l t + at)
2
(10.1.5)
41
r
r
Cov [
L:.
j=l
Y. , r ]
J
E
=
[r'L:
j=l
222
[E (r ) - n p ]
~t
=
(10.1.6)
Substituting in (10.1.3) gives:
(10.1.7)
Obtaining the mean square error,
Bias(Y3)
MSE(Y3)
=
=
p
~t
-
q(~t'
+
2
~
,
- t)
£. [0 + q(t
n
t
10.2
-
qt
(10.1.8)
- ~t)l
+
Comparison of
q2(~t' -
13
and
t)2
.
(10.1.9)
y
From the previous results it appears intuitive that MSE(Y3)
is for some region of ! less than the variance or
demonstrated by the following:
Differentiating with respect to !'
= .ilil
n
y.
This is
42
2q f(t) (I-ltl -t)
+
t
2
(10.2.1)
Differentiating I-l t ' ,
d
1
dt [-
q t
f
b
(t<y<b)
y f(y) dy]
:lli.2.
=
(10.2.2)
q
Substituting (10.2.2) and previously derived derivatives in
(10.2.1) gives:
d
2q[£ (t - I-l ) - q (I-l t , - t)] .
t
n
dt
The derivative is positive as ! approaches
£
(10.2.3)
since the second
term in brackets is approaching zero while the first term is
approaching a positive constant.
10.3
An Optimum Value For t
Evaluating the derivative at t = a ,
.
~t
[MSE(Y3)]
I=
-
2(1-l - a) ,
t=a
<
0
(10.3.1)
43
Therefore a mimimum exists between t
=a
and t
= b.
Equating (10.2.3) to zero and solving for an optimum value,
t
o
t
~tol
nqo
o
+ Po
nqo +
~to
(10.3.Q)
Po
where
p
o
F(t )
o
1 - F(t )
o
The region of
!
for which MSE(Y3) < V(y) is found by equating
2
(10.1.9) to cr In and investigating the roots.
This will be shown in
Fig. 1., Chapter 10, for the exponential.
The biases of the first three estimators are negative.
From
(8.1.10), (9,1.4), and (10.118) the absolute values of the biases
are:
/BiascYl) I
(10.4.1)
(10.4.2)
=
/Bias(Y3)I
=
q( ~t' - t)
Since t> ~t' IBias'cY )/ < !Bias cY )/
1
z
Also, !Bias(y ) I '<
3
IBias
cY1 ) I .
(10.4.3)
(10.4.4).
(10.4.5)
44
(10.4.6
Proof:
-
Ilt)
< f-l
f-l - Pf-l t
-
qt
n
< (f-l :-' f-l t )- q (t - f-l t )
q(f-l ,
t
-
t)
<
-
n
q (t - f-l )
t
f-l - q(t
n
(f-l - f-l t ) - q (t - f-l t )
(10.4.7)
(10.4.8)
. (10.4.9)
Thus the biases can be ranked:
(10.4.10)
45
11.0
11.1
CHARACTERISTICS OF Y4
Expectation, Variance, and Mean Square Error
n
W
L;
a
j=l
Y.
J
(11.1.1)
The y. observations are those obtained from a simple random sample
J
of size n .
Taking expectation,
E(y )
4
= nWa fl
V<Y )
4
-,
(11.1.2)
Also
2
nW
a
0
2
(11.1.3)
Forming the mean square error,
Bias (Y4)
MSE(Y4)
11.2
-
fl( 1
-
nW )
a
(11.1,4)
2
2 2
= nWa o + fl (1 - nWa )2
(11.1.5)
An Optimum Value for Wand the Relative Efficiency
a
Differentiating (11.1.5)-with respect to W ,
a
2n [W 0 2 - fl2(1 - nW)]
a
a
(11.2.1)
Equating (11.2.1) to zero and solving for W will yield a
a
minimum because the second derivative is positive for all values
of W
a
46
Taking the second derivative,
idW2
2n [a
=
[MSE(Y4)]
2
+ n
2
I-L ]
(11.2.2)
,
a
> 0
(11.2.1) to zero and solving for W gives:
Equating
W
ao
=
a
J..
n+(C.V.)
2
(11.2.3)
where (C.V.)2 is the square of the coefficient of variation.
It is interesting to note that .this value of Wa would give for
the exponential distribution, regardless of value of the parameter,
n
L:
Y4
= ;=1
y.
J
n+1
(11.2.4)
Substituting (11.2.3) into (11.1.5) gives:
MSE[Y4· Iwao ]
=
a
2
!
. 2
n+(C.V.)
.
(11.2.5)
The relative efficiency at the Optimum is:
R.E. = n+(C.V.)
2
(11.2.6)
n
11.3
A Region for W
a
2
Equating (11.1.5) to a In ,
2
a In,
47
(11.3.1)
Solving the separate factors for W yields the region:
a
.!. f!!.:(c.v.)2
n In+(C.V.)2
11. 4
J
< W'
<
a
1:-
n
Comparison of Bias (Y4) with Biases for
(11.3.2)
Y1~2 ,and
Y3
For nW < 1, Bias(Y4) is negative and from (11.1.4) is obtatned:
a
=
f.l(1 - nW )
a
(11.4.1)
From (10.4.1) and (11.4.1),
(11.4.2)
if
nW
a
>
(11. 4.3)
From (10.4.2) and (11.4.1),
(11.4.4)
if
nW
a
(11.4.5)
From (10.4.3) and (11.4.1),
(11.4.6)
if
nW
a
(11. 4.7)
48
12.0
CHARACTERISTICS OF 12.0
YS
12.1 Expectation, Variance, and Mean Square Error
n
n
L: Y +
j
j=l
YS =
where
>
and
L:
Yj
j=r+1
(12.1.1)
n
for j - 1, ... ,r,
Y·J ( t
Yj
Wb
t
for j
>
r,
0 (W ( 1.
b
Taking expectation,
E(yS)
and Bias(ys)
=
P!-Lt + Wbq !-L t '
=
P!-Lt +Wbq !-Lt '
=
-
(12.1.2)
-
!-L ,
( 1-w )q !-L '
b
t
(12.1.3)
Now
V<Ys!r)
=
2 2
2 2
2
r a/n + Wb (n-r) at,/n
=
r !-L/n +
Wb(n-r)
and Bias(YS/r)
n
!-Lt'
-
!-L
(12.1.4)
.
(12.1.S)
Hence,
MSE(yS)
=
2
2
2
[p at + q W at' ] /n
b
(12.1.6)
49
(12.1. 7)
Subtracting the square of (12.1.3) gives:
12.2
An Optimum Value of W for a Given t
b
Differentiating (12.1.7) with respect to W ,
b
d
dW
[MSE (ys)]
b
(12.2.1)
Taking the second derivative,
d
2
2
dW
b
2
~ [a 2, + p !-t 2,] + 2q 2
!-tt'
t
n
t
[MSE(yS)]
(12.2.2)
-,0":.,
> 0
(12.2.3)
Since the second derivative is positive, an optimum exists.
Equating (12.2.1) to zero and solving for W yields an optimum,
b
2
nq !-tt' + P !-tt !-tt'
2
2
(nq+p) !-tt' + at'
222
(nq+p) !-tt' + at' > nq !-tt' +
or
2
P!-tt'
+
2
at' > p !-tt !-tt'
(12.2.4)
p!-tt !-tt'
50
Since
>
~t'
~t
' the verification is complete,
12,3
The region of
A Region for W
b
Wb for which
(12.3.1)
is found by solving (12.3.1).
Multiplying (12.3.1) through by n gives:
2
pat
+
(12.3.2)
Substituting (6.0.24) for a
2
and dividing through by -q,
(12.3.3)
If W < 1, division by (l-W ) will not reverse the inequality,
b
b
(7.20)
This gives:
2
(nq-p) ~ti
.,
Wb
>
+
2
2p ~t ~t' - at'
2
2
(nq+p) ~tl + at'
12.4
(12.3.4)
An Optimum Value For t
In line with previous results a minimumshou1d exist between
a and
£.__
51
Differentiating (12.1.7) with respect to t
+
(12.4.1)
..
D~'ff erent~at~ng
qat'2 an d q
,
~t'
2
~(qat' )
dt
=
- (~t' - t)
=
- t
2
(12.4.2)
f(t)
(12.4.3)
fO:) .
Substituting for all derivatives, equating (12.4.1) to zero, and
solving for! gives:
2[~
t
o
=
+ (1 - Wb ) (n-1)
q ~t']
o
(12.4.41)
52
12.5
Comparison of Bias(Y5) with Biases
Bias<Y5) is negative.
forY1~2~3
and Y4
From (12.1.3) the absolute value is:
(12.5.1)
From (10.4.1) and (12.5.1)
/Bias(Y1)I
n-1
if
>
+
I-l t
q
(12.5.2)
(t-I-l)
.
. t
(12.5.3)
I-l ,
t
From (10.4.2) and (12.5.1),
(12.5.4)
if
(12.5.5)
From (10.4.3) and (12.5.1),
(12.5.6)
(1 2 .5.7)
if
From (11,4.1) and (12.5.1),
(12.5.8)
if
nW
a
<
(12.5.9)
If nW > W , saynW = W + A, then (12.5.8) still holds if
b
a
a
b
<
1
(12.5.10)
53
13.0
13.1
CHARACXERTSTICS OF Y6
Expectation, Variance, and Mean Square Error
(13.1.1)
n
L: y. - Y
= ;=1 J
m
. (y
,
n - 1
m
=
maximum sample observation)
Taking expectation,
(13.1.2)
=
Also,
V(Y6)
=
1 2 [na
2 + am2 - 2nCov(Y'Ym
-)J .
(n-1)
(13.1.3)
Obtaining the mean square error,
Bias(y )
6
_.1_.
n-1
;=
(~
m
-~)
(13.1.4)
- 2nC ov <Y,
. ym) J
13.2
(13.1.5)
The Covariance Between the Sample Mean and
The Maximum Sample Observation
To evaluate MSE(Y6), an expression is required for Cov(y,y ).
m
Ost1e and Steck (1959) examined the correlation between sample
means and sample rangeS.
A similar approach will be employed below.
n
Cov(y'Ym) =
1
-n E [(L:
j'=1
Y.) YmJ J
~ .~m
(13.2.1)
54
If
i
is used to denote the order of the draw there are n identical
terms in the expectation so that,
Cov(y,y)
m
=
~ ~
E(y.y) J m
(13.2.2)
m
It is helpful to consider the case when j=l,
(13.2.3)
The conditional distribution of Ym given Y is obtained in
I
the following manner:
(y<w)
0
p
bms y/YI = vi] =
[F(w) ]n-l
(y=w)
[F(y)]n-l
(y>w)
(13.2.4)
Then,
b
J
w[F(w)]n-1 + (n-l)
y[F(y)]n-2 fey) dy
w
(13.2.5)
In order to simpli fy this expression consider:
b
b
J
(n-l)
J
y[F(y)]n-2 fey) dy = (n-I)
y[F(y)]n-2 fey) dy
a
w
w
J
- (n-l)
y[F(y)]n-2 fey) dy
a
(13.2.6)
55
Integrating the negative term by parts,
Let:
u = y
du = dy
v = [F(y) ]n-1
dv = (n_1)[F(y)]n-2 f(y) dy .
And,
w
f y[F(y)]n-
(n-1)
2
f(y) dy = w[F(w)]
n-1
w
J
-
[F(y)]n-1 dy .
a
a
(13.2.7)
Substituting in (13.2.5) gives:
w
J
fJ.,+
m
[F(y)]
n-1
dy,
(13.2.8)
a
Where fJ. ; is equal to the expectation of the maximum sample
m
observation for samples of size n-1.
This gives:
w
b
E(yy )
m
n-1
dy] f(w) dw,
f w[ J [F(y)]
= fJ. fJ. ,+
m
a
(13.2.9)
a
w
b
Cov(yym) = - fJ.(fJ.,m -fJ.m ,) +
J
f [F(y)]n-1 dy] f(w) dw.
w['
a
a
(13.2.10)
13.3
Comparison of
Y6 and y
Whether or not
(13.3.1)
56
will depend on Cov(y,y) .
m
Substituting (13.1.5) into (13.3.1) gives:
---L
(n-1) 2.
2
J a
[ ncr 2
+2
crm + (j.lm - j.l) 2 - 2nCov(y,y)
.. m <n
(13.3.2)
Solving the inequality for Cov(y, Ym)'
Thus the inequality (13. 3.1) is true only for those
functions for which (13.3.3) holds.
distribution
57
14.0
CHARACTERISTICS OF Y
7
Expectation, Variance and Mean Square Error
14.1
n
L.
j=l Yj
=
-
(1 - Wc ) Ym
(W <1)
c
n
(14.1.1)
Designate (1 - W ) =
c
~~
.
Taking expectation,
ECY7 ) =
(14.1.2)
Also,
2
o /n +
~~ 0
2
m
/n -
2~~
Cov(y,y )/n . (14.1.3)
m
Obtaining the mean square error,
=
=
=
flm
- (1 - W )
n
(14.1. 4)
c
222
2
o /n + (W"(/n) (0 m + f""m
II
)
-
2Wk CovCy,ym)/n
(14.1.5)
14.2
An Optimum Value for W
c
Differentiating (14.1.4) with respect to Wk,
d
d~~ [MSE(Y7) ]
2
(2/n) [Wk(o m
+
2
m )/n - Cov(Y,ym)J.
f1<
(14.2.1)
58
Taking the secend derivative,
d2
dW*2
=
[MSE(Y7) ]
2
(2/n ) (a
2
m
+
fl
2
m
)
> 0
(14.2.2)
Hence equating (14.2.1) to zero and solving for W1(,
nCov(y,y )
W*
m
(14.2.3)
o
Since Cov(y, y . ) > 0,
m
~'(
0
> 0 and hence Wco < 1.
nCov(y,y )
1 _ ---",_ _---,m_.
W
co
2
am +
(14.2.4)
2
flm
Substituting (14.2.4) in (14.1.5) gives:
(14.2.5)
Thus,
[Cov(y,y) ] 2
=
14.3
m
(14.2.6)
2
2
am + f lm
A Region for W
c
Solving the inequality
(14.3.1)
for W gives a region,
c
59
WI
c
14.4
2nCov(y,y )
m
>1
(14.3.2)
Comparison of Bias(Y7) with Biases for Y2andY6
General comparisons of the biases o.f 'Y6 ana Y7 with the first
five alternative estimators are' not too fruitful.
involving
~m
~)
(dependent upon
and
~t
or
~tl
Conditions
(dependent upon !)
are obtained.
For illustrative P4rposes the comparison of Bias(y ) with
7
Bias(Y2) gives:
(14.4.1)
if
W
c
>
n(~ -
1 -
~ )
t
(14.4.2)
A comparison of Bias(y ) with Bias (Y6) gives:
7
(14.4.3)
if
1 - (n-1)~
m
(14.4.4)
60
15.0 DISCUSSION OF RESULTS
15.1 The<:>retical Examples
The exponential distribution,
f(y)
= -1
fJ.
e
-y/fJ.
(o<y<+co)
(15.1.1)
was chosen for the calculation of appropriate examples for three
reasons:
1.
The exponential distribution is a good example of the class
of distributions under consideration since it is highly positively
skewed.
2.
The sample mean is also the maximum likelihood estimator
and uhe best linear unbiased estimator of the population mean.
3.
The computations involved, though laborious, are simpler
than for most other distributions.
Three tables are not concerned with the exponential distribution
directly.
Tab~es
i6 and 12 pertain to Y4 which depends only upon the
coefficient of variation of the population.
For these tables, data
was calculated for coefficients of variation from 1 to 5.
Table 9 is concerned with
Y6'
The distribution presented here
is the Pareto,
f(y)
= ex
ex - (ex +
yoY
1)
(y <y<+co)
o
(ex > 2)
(15.1.2)
61
For the
ex~mple
presented y
o
was chosen equal to one.,
Although the exponential distribution does not be19ng to the
subclass for whiFh
for n
= 2, Y6
y
6
is appropriate, it is of interest to note that
has a relative efficiency of 100% for that distribution.
As n increases the relative efficiency drops.
Tables 1 and 2 present information which is useful in evaluating
the remaining tables involving the exponential distribution.
Samples
having one or more observations larger than some point ! have a
probability of occurence of 1 - [F(t)]n.
These probabilit~es
Table 2
expressed as expected percentages are presented in Table 1.
presents the expected number of observations greater than!.
expected number is equal to
Values of
t/~
The
n [1 - F(t)]
which occur in many of the tables simply present
t values as multiples of the population mean.
Since
~
=0
for the
exponential distribution an alternative expression for these values
could be presented in terms of the population mean plus multiples of
the standard deviation.
t/~
or t
as
=2
= 2~
could have been expressed
t = ~
+
t/~ =
3
as t
For example:
=~+
0
20 , etc.
62
In table 3, relative efficiencies [MSE(Y2)/V(y)] are presented
for Y2'
The sample size refers to the number of observations
retained in the sample.
Relative efficiencies for Y1 are expected
to be similar and are not presented.
of the relative efficiency as
t/~
The characteristic rise and fall
increases can be seen.
in size of the relative efficiency for a given
t/~
The decline
as n increases
is also apparent.
Figure 1 shows the comparison of MSE(Y3) and (/ /n for n= 5
and n = 10.
The ordinate of the graph presents the mean square error
2
of Y3 for values of (t/~), in terms of a In.
The two horizontal
2
2
lines represent the constant values of a /5 and a /10.
the figure shows that for an optimum choice of
t/~
For n = 5
(about 2) MSE(y )
3
2
is approximately equal to a /16.
The relative efficiencies for Y3 are presented in Table 4.
same basic patterns are present that existed in Table 3.
The
The two
tables are not strictly comparable since the sample sizes in Table 3
refer to the number of observations less than the cut-off point
rather than the sample size required to obtain this number.
certain comparisons can, still be made.
Even so,
For instance, it appears that
the relative efficiencies for Y3 hit a higher peak sooner as t/~
increases but they also decline faster than the corresponding values
63
Table 5 presents characteristics for Y3 when the optimum!
value is used.
The optimum cut-off point increases with sample
size, however, it increases slowly enough that the expected number
of observations greater than t
o
and the expected percentage of
samples with at least one observation greater than t , also
o
increase.
Relative efficiencies of Y4 for selected coefficients of variation are listed in Table 6.
Relative efficiencies decrease with
sample size and increase with increases in the Coefficient of Variation.
The large app.arent gains for small samples from populations
with large Coefficients of Variation may be misleading.
For example
the relative efficiency of 600% for samples of size 5 does not appear
so large when it is noted that a constant estimator of zero would
have a relative efficiency of 500%.
have more utility.
The smaller indicated gains may
The first row (C.V.
=
1) presents relative
efficiencies for the exponential distribution.
Tables 7 and 8 refer to Y5'
Table 7 gives optimum weights and
Table 8 presents the relative efficiencies when the optimum weights
are employed.
The weights increase with sample size but decrease
with increases in
t/~
and are approaching zero for
t/~
= 10.
A
feature of Tab1e.8 is that no relative efficiencies are less than
100%.
This is expected from the results for
Y4'
Y3 appears to
achieve about the same maximum relative ·efficiencies as y5 for each
sample size presented.
64
Characteristics of
Table 9.
Y6
are given for the Pareto distribution in
This table is presented primarily to exhibit an example
of a distribution which satisfies the condition on the covariance
between
Yand
the maximum sample observation, (13.3.3), required for
Table 10 presents optimum weights and other characteristics for
Y7 ' when the parent population is the exponential.
weight increases with
~he
The optimum
sample size and is independent of parameter
size.
Ratios of bias to standard deviation are presented for
Y4 and
YS
in Tables 11, 12 and 13.
sample size for
Y '
S
Y2
and Y3'
Y2 ,
Y3'
These ratios are independent of
In an overall comparison of
Y3 has the lowest ratios for values of
t/~
Y2' Y3 ,
and
greater than 5.
Table 14 presents the components of MSE(Y4)'
Certain conclusions can be drawn from the tables presented
regarding a choice of estimator and cut-off point when sampling is
from the exponential distribution.
A sample size of five will be
assumed.
One estimator, Y , can be eliminated from consideration immedi6
ately.
The exponential distribution does not belong to the class of
distributions for which it achieves a gain in accuracy.
If the only information known about the distribution sampled
is that it is exponential,
case:
Y4
and
Y
7
can be employed.
For this
65
5
L:
= j=:1
Y4
Yj
6
(15.1.3)
5
L:
and
Y7
=
j=lY f(' 342) Ym
5
(y
m
= max. obs.)
(15.1.4)
Y4 has a slight edge in
rela~ive
efficiency, 120.0% to 118.5%.
Greater gains in mean square error may be achieved if some
information is available concerning J.. L.
theuptimumvaluefor! is about 2J...L.
For example, from Table 5,
A
An estimate, J.. L, maybe availalJle
which the practitioner is reasonably certain exceeds J.. L but does not
exceed 1.5J...L.
A
Inserting the estimated value, J.. L, into the formula
for the estimator, Y3' gives
r
L:
Y3 =
J=l
A.
(y.<2J...L)
. J
5
(15.1. 5)
For this situation the relative efficiency, as shown by Table 4,
would be between 183% and 140%.
If ~ is between J.. L and 2J...L the relative efficiency would be
between 183% and 117%.
~
Larger ranges for J.. L could lead to smaller
gains than the sure gain provided by Y4'
Also, the possibility
of seriously underestimating J.. L, and thus obtaining a large mean
square error,
sho~ld
efficiency is 83%.
not be ignored.
'" = J...L, the relative
If 2J...L
66
Results with Y3 for other sample sizes are similar with the
exception that the gains become smaller and the consequences of
poor choices more serious as n increases.
With
Y4
a gain is
always achieved; however, as sample size increases it becomes
negligible.
67
Table 1.
The expected percentage of samples from the exponential
distribution with one or more observations greater than t
Sample Size
Values
of till
5
10
50
100
500
1
89.9
99.0
100.0
100.0
100.0
2
51.7
76.6
99.9
100.0
100.0
3
22.5
40.0
92.2
99.4
100.0
4
8.8
16.9
60.3
84.3
100 00
5
3.3
6.5
28.7
49.1
96.6
6
1.2
205
11. 7
22.0
71. 2
7
.5
.9
4.4
8.7
3605
8
.2
.3
1.7
3.3
1504
9
01
.1
.6
1.2
6.0
10
.0
.0
.2
.5
202
68
Table 2.
The expected number of observations greater than
samples from the exponential distribution
Sample Size
Values
of t/fl
5
10
1
1.84
3.68
18.39
36.79
183.94
2
.68
1. 35
6.77
13.53
67.67
3
.25
.50
2.49
4.98
24.89
4
.09
.18
.92
1.83
9.16
5
.03
.07
.34
.67
3.37
6
.01
.02
.12
.25
1.24
7
.00
.01
.05
.09
.46
8
.00
.00
.02
.03
.17
9
.00
.00
.01
.01
.06
10
.00
.00
.00
.00
.02
50
100
500
.t for
69
Table 3.
Relative efficiencies (%) of Y2 for samples from the
exponential distribution
Sample Size (r)
Values
of t/f.L
5
10
50
1
35.7
18.2
3.7
1.9
.4
2
112.9
68.9
16.7
8,6
L8
3
151. 5
126.6
54.6
31. 9
7.4
4
135.6
130.6
100.7
78.4
28.2
5
119.0
118.1
112.0
105.2
70.7
6
109.4
109.3
108.3
lOr? . 0
97.7
7
104.6
104.5
104.4
104.1
102.4
8
102.2
102.2
102.1
102.1
10L8
9
100.6
100.6
100.6
100.6
100.6
10
100.4
100.4
100.4
100.4
100.4
100
500
70
.25a
2
2
------:I+---------------------__='
.20a
n
.15a
=
5
2
2
.10a
----T---------============n = 10
.05a
2
.00a 2
_
o
Fig. 1.
1
2
3
4
2
5
6
7
Comparison of MSE(Y3) and a jn for the exponential
distribution
8
71
Table 4.
Relative efficiencies (%) of )73 for samples from the
exponential distribution
Sample Size
Values
of t/I-l
5
10
50
1
83.4
44.1
9.2
4.6
.9
2
183.1
153.3
66.7
39.1
9.1
3
140.5
138.0
120.6
104.2
49.9
4
117.0
116.7
114.9
112.7
97.7
5
107.2
107.2
106.9
106.7
104.7
6
103.1
103.1
103.0
103.0
102.7
7
101. 3
101. 3
101. 3
101.3
101. 3
8
100.5
100.5
100.5
100.5
100.5
9
100.2
100.2
100.2
100.2
100.2
10
100.1
100.1
100.1
100.1
100.1
100
500
72
Table 5.
Sample
size
Exponential distribution; characteristics for )73 when
optitnum values for t are used
Values
of t/fl
Exp.
no:>t
0
Exp. % of
samples
with one
or more
>t
Ratio of
bias to fl
(or a)
Relative
eff. (%)
0
5
1.9
.75
55.5
.150
184.9
10
2.2
1.11
69.1
.111
156.7
50
3.2
2.04
87.5
.041
121. 3
100
3.6
2.73
93.7
.027
113.7
500
4.9
3.72
93.5
.007
104.7
73
Relative efficiencies (%) of Y4 for samples from
distributions with selected coefficients of variation
Table 6.
Sample Size
C.V.
5
10
50
1
120.0
110.0
2
180.0
3
100
500
102.0
101,0
100.2
140.0
108.0
104.0
100.8
280.0
190.0
118.0
109.0
101,8
4
420.0
260.0
132.0
116.0
103.2
5
600.0
350.0
150.0
125.0
105.0
74
Table 7.
Optimum weights for )75 for samples from the exponential
distribution
Sample Size
Values
of t/fJ.
5
10
50
100
500
1
.724
.836
.961
.980
.996
2
.529
.666
.900
.946
.989
3
.356
.462
.768
.864
.969
4
.245
.303
.566
.706
.917
5
.184
.209
.366
.492
.804
6
.148
.158
.231
.307
.611
7
.126
.130
.160
.195
.394
8
.111
.112
.124
.138
.236
9
.099
.100
.104
.110
.151
10
.090
.091
.092
.094
.1l0
75
Table 8.
Relative efficiencies (%) of )75 for samples from the
exponential distribution when optimum weights are used
Sample Size
Values
of t/t+
5
10
50
100
500
1
143.7
122.2
104.5
102.2
100.4
2
180.5
146.3
1l0.5
107.2
101.1
3
171. 5
153.4
117.7
109.6
102.1
4
140.9
136.6
120.0
112.8
103.3
5
120.6
119.8
115.3
111. 9
104.3
6
110.0
109.9
108.9
108.0
104.3
7
104.8
104.7
104.6
104.4
,103.3
8
102.2
102.2
102.2
102.2
101. 9
9
100.6
100.6
100.6
100.6
100.6
10
100.5
100.5
100.5
100.5
100.4
76
Table 9.
Pareto distribution; Y6 with n
Value
of
ex
Value of
2
r Cy, Ym)
=
2
Relative
efficiency
(%)
Ratio of
bias to
mean
3
.964
250.0
.200
4
.947
197.5
.143'
5
.938
150.0
.111
6
.931
137.5
.091
10
.918
118.8
.053
77
Table 10.
Exponential distribution;
Sample
size
Optimum
weight
Value of
2
r (y, Y )
m
Y7
Relative
efficiency
(%)
Ratio to
bias to
mean
(or 0)
2
.571
.900
147.4
.322
3
.612
.823
131.1
.237
4
.639
.762
123.2
.188
5
.658
.712
118.5
.156
78
Table 11.
Ratios of the bias to the population standard deviation
for Y2' Y3 from the exponential distribution
Values
of t/IJ.
Y2
1
.582
.368
2
.313
.135
3
.157
.050
4
.075
.018
5
.034
.007
6
.015
.002
7
.006
.001
8
.003
.000
9
.001
.000
10
.000
.000
Y
3
79
Table 12.
Ratios of the bias to the population standard deviation
for Y4 from distributions with selected coefficients
of variation
Sample Size
C.V.
5
10
50
100
500
1
.033
.018
.004
.002
.000
2
.089
.057
.015
.008
.002
3
.129
.095
.030
.016
.004
4
.152
.123
.048
.028
.006
5
.167
.143
.067
.040
.010
80
Table 13.
Ratios of the bias to the population standard deviation
for Y5 with optimum weights, from the exponential
distribution
Sample Size
Values
of t/JJ.
5
10
50
100
500
1
.203
.121
.029
.015
.003
2
.191
.136
.041
.022
.004
3
.128
.108
.046
.027
.006
4
.069
.064
.040
.027
.008
5
.033
.032
.026
.021
.008
6
.015
.015
.013
.012
.007
7
.006
.006
.007
.006
.004
8
.003
.003
.003
.003
.002
9
.001
.001
.001
.001
.001
10
.000
.000
.000
.000
.000
81
Table 14.
Values of MSE(Y4)' V(Y ) and [Bias (Y4)]2 for the
4
exponential distribution
Sample
Size
MSE<Y )
4
V(Y4)
[:Bias (y4)]2
5
.1667
.1389
.0278
10
.0909
.0826
.0083
50
.0196
.0192
.0004
100
.0099
.0098
.0001
500
.0020
.0020
.0000
82
15.2
Comparisons
Biases for the seven estimators are:
Bias
Bias
Bias
cY1 ) =
cY2 ) =
cY3 ) =
~
~
~
-
Bias (Y4)
Bias (Y5)
Bias
Bias
fl )
(fl
fl ),
=
-
t
n
(t
-
t
-
q (fl t i
-
fl( 1
(1
cY6 )
cY 7 )
+ q
(fl
-
fl ) ,
t
(8.1.10)
(9.1.4)
t),
(10.1.8)
nW ),
a
(1l.1.4)
W ) q flti
b
1
n-1 (flm - fl),
flm (1 - W ).
--'c
n
(12.1.3)
(13.1.4)
(14.1. 4)
In terms of absolute magnitude, the first three biases can be
ranked:
Bias (Y3) < Bias (Y ) < Bias (Y )
1
2
(10.4.10)
Also,
Bias
cY4 )
< Bias (Y )
3
(1l.4.6)
if
nW >
a
P fl t
+ qt
fl
(11.4.7)
Bias (Y ) < Bias (Y )
5
3
(12.5.6)
Wb > tlfl t
(12.5.7)
if
I
and
Bias
<Y5 )
< Bias (Y )
4
(12.5.8)
if
nWa
~
Wb
(12.5.9)
83
or if
= W
b
+ A and
(12.5.10)
Ignoring sign, a comparison of biases for Y6 and Y7 gives:
Bias (Y7)
< Bias (Y6 )
(14.4.3)
> 1 - n(l-Lm - l-L)
(14.4.4)
if
W
c
(n-1) l-L
m
No general conclusions can be drawn for the other comparisons
since they will depend on the cut-off points, weights, sample sizes
and the particular densities sampled.
Optimum cut-off points and weights are presented below:
Estimator
Optimum
Solution of equation (8.2.9) gives
optimum cut-off point.
Solution of equation (9.2.9) or (9.2.20)is
optimum cut-off point.
t
= nqo l-L t i + Po l-L t
o
0 .
0
(10.3.2)
W
ao
=
1
- 2
n+ (C.V.)
(11.2.3)
(12.2.4)
84
[ 2 fl
t
o
+ ( 1- Wb ) (n- 1) q
fl
t
J
0'
= -----------
(12.4.4)
No optimum involved.
density for use:
Condition on
(13.3.3)
n Cov<y',y )
W
co
m
= 1 -
2
am +
fl
2
m
(14.2.4)
The mean square errors of the estimators are:
Estimator
MSE
(1_ qn)[a 2 E[!J + (fl-fl )2 J + qn(t_~)2
t
r
t
(8.1.9)
(r fixed)
(9.1.5)
(10.1.9)
(11.1.5)
+ (1 - Wb )
2
q
2
2
t
fl ,
(12.1.7)
85
1
(n-1)2
[na
2
2
m
+ a +
(~ -~)
m
2
-
- 2n Cov(y,y
(13.1.5)
2
Q. +
n
2~'( _ Cov(y,y )
n
m
(W>'~ =
1 - W)
c
(14.1. 5)
Comparisons between mean square errors will depend on cut-off
points, and weights chosen as well as the density function .
.e
~
m
86
16 .0
SUMMARY, CONCLUSIONS ,AND RECOMMENDATIONS FOR FURTHER RESEARCH
16.1
Sunnnary
This thesis was concerned with seven alternative estimators
to the sample mean.
Extreme observations were defined as those
having values greater than some cut-off point, t.
The seven esti-
mators are:
1.
Extreme observations are given a weight of zero with
sample size fixed at n while non-extreme observations receive a
weight of 1/r.
2.
(Y1)
Extreme observations are given a weight of zero with
sample size such that L observations less than t are assured.
Non-extreme observations receive a weight of 1/r.
3.
(Y )
2
Extreme observations are first weighted inversely
proportional to size, i.e. all extremes take on the value of !,
then all observations are weighted by lin.
4.
(Y3)
Extreme observations are given the same weight as the non-
extreme observations but the weight is not necessarily equal to
lin.
(Y4)
5.
A generalization of four in which extreme observations
receive an equal weight which is less than the
non-extreme observations, lin.
6.
~eight
assigned to
(Ys)
A weight of zero is applied to the maximum observation
and a weight of1/(n-1) is applied to the remaining observations.
87
7.
A weight less than lin is applied to the maximum observa-
tion and a weight of lin is applied to the remaining observations.
(Y7)
Results were developed primarily for a particular class of
continuous distributions.
Distributions belonging to this class
have finite first and second moments; they are unimodal; they do
not exhibit negative skewness; and they occur on the range (a, b)
where
O<a<b <
+00.
In chapters three through nine expressions were developed
for the expectation, variance and mean square error of each estimator considered.
These expressions were used in the derivation
of conditions under which the estimators achieve a mean square
error which is smaller than the variance of the sample mean.
Expressions were also developed for optimum cut-off points and
weights where appropriate.
Comparisons of the biases were made in sections 10.4, 11.4,
12.5 and 14.4.
These comparisons are summarized in section 15.2.
Chapter 15.0 contains a discussion of results.
Guidelines
for the choice ·of estimator, optimum weights and optimum cut-off
points when the parent distribution is exponential are presented
in the tables of section 15.1.
88
16.2
Conclusions
From the standpoint of mean square errors alone and under the
proper conditions, all seven of the considered estimators are superior to the sample mean.
For most of the estimators, however, the
proper conditions are complex functions of the population parameters.
An exception is Y4 which requires only a knowledge of the population coefficient of variation.
The examples in section 15.1 indicate that the regions for
the proper conditions can be quite large.
In particular, for sam-
pling from the exponential distribution, the use of Y3 requires
=5
that for n
and 10, the cut-off point, !, should be approxi-
mately equal to
figure is
~
+
~
+a (or
2~)
2a; and for n
or greater; for n =50 and 100, the
= 500,
~
+
4a.
All seven of the estimators are negatively biased.
The first
three can be ranked:
(10,4.10)
Relativerankings for the other estimators depend upon choices of
cut-off or weights and the particular density sampled,
From the examples presented, conclusions concerning the exponential distribution are:
1.
Two estimators do not require any knowledge of population
parameters:
89
n
L: y.
J
Y4
=
;=1
n+l
(lL2.4)
n
L: y.
and
Y7
=
j=l
J
-
(l-W
co
)y
m
(16.2.1)
n
W can be calculated for any sample size.
co
For n = 5, W = .658.
co
Also for n=5MSE<Y4) and MSE(Y7) are approximately equaL
The
respective relativeefficiences are 120 and 118.5 per cent.
2.
The highest relative efficiency for n = 5 is attained
by Y (185%) with a cut-off point of
3
3.
~ + .9a.
A gain inMSE can not be achieved by arbitrarily dis-
carding the maximum sample observation.
16.3
Recommendations for Further Research
A number of items could be explored in future research.
Approximations from sample data for optimum cut-off points
and optimum weights would be useful.
For the cut-off points this
would involve developing tests to determine whether or not an
observation should be considered an extreme.
Tests would also be
helpful that would discriminate between alternative estimators.
Weights·determined from the sample data may provideestimators which have smatler mean square errors than estimators
employing a fixed optimum weight.
i
An example for Wis:
C
90
n
L: y.
j=2 J
n
L: y.
j=l J
/\.
W
c
smallest observation)
(16.3.1)
Y7 with this weight was computed for 100 samples of size five drawn
from a random number table
(Ha1d~
1952).
The empirical results
indicated a relative efficiency of 106%, which is higher than expected for the uniform distribution when Y7 with the optimum weight
is employed.
A
When W was applied to the maximum observation in the maximum
c
likelihood estimator, a relative efficiency (as compared with the
regular ML estimator) of 125% was obtained.
Estimators for a
2
analogous to those considered for
thesis should be investigated.
~
in this
It is the writer's opinion that
the mean square errors of such estimators would, under proper
2
conditions, be substantially smaller than the variance of s .
It should be noted that the alternative estimators considered
are but a few of the total number of possibilities.
Future re-
search could profitably be spend in developing and investigating
other alternatives.
Perhaps the first alternative to be considered
should be the maximum likelihood estimator for the exponential
distribution which is similar to y :
>
3'~
91
y.
=
]
+ (n-r)t
.~ (Y.
J
r
j=l
!~
(r
< t)
2:
1)
(16.3.2)
For r= 0, the maximum likelihood estimator does not exist.
Another alternative which is similar to YS is:
r
n
.6 y. + W
.6
b j=r+1 Yj
A.
j=l J
=
1J. 2
r + (n-r)W
b
where
Yj < t for j = 1,
Yj > t for j
and
o
•
0
,
(16.3.3)
r,
> r,
° < Wb < 1.
Also an alternative which is similar to Y is:
7
n
=
.6 y. - (l-W)y
i=l J
c m
n -
(l-W )
c
(W <1)
c
(16.3.4)
Alternatives should be considered for other parameters in
2
addition to IJ. and a .
.
Future investigations should also consider the problem of
assigning confidence limits which incorporate alternative estimators.
92
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