Technique for finding sampling distributions for truncated laws with some applications to reliability estimation Текст научной статьи по специальности «Математика»

TECHNIQUE FOR FINDING SAMPLING DISTRIBUTIONS FOR TRUNCATED LAWS WITH SOME APPLICATIONS TO RELIABILITY ESTIMATION

Nicholas A. NECHVAL*, Konstantin N. NECHVAL**

*Mathematical Statistics Department, University of Latvia Raina Blvd 19, Riga LV-1050, Latvia e-mail : nechval@junik.lv

**Applied Mathematics Department, Transport and Telecommunication Institute Lomonosov Street 1, Riga LV-1019, Latvia e-mail : konstan@tsi.lv

ABSTRACT

In this paper, the problem of finding sampling distributions for truncated laws is considered. This problem concerns the very important area of information processing in Industrial Engineering. It remains today perhaps the most difficult and important of all the problems of mathematical statistics that require considerable efforts and great skill for investigation. The technique discussed here is based on use of the unbiasedness equivalence principle, the idea of which belongs to the authors, and often provides a neat method for finding sampling distributions. It avoids explicit integration over the sample space and the attendant Jacobian but at the expense of verifying completeness of the recognized family of densities. Fortunately, general results on completeness obviate the need for this verification in many problems involving exponential families. The proposed technique allows one to obtain results for truncated laws via the results obtained for non-truncated laws. It is much simpler than the known techniques. The examples are given to illustrate that in many situations this technique allows one to find the results for truncated laws and to estimate system reliability in a simple way.

KEYWORDS

Truncated law, Sampling distribution, Unbiasedness equivalence principle, Reliability estimation

1. INTRODUCTION

The truncated distributions have found many applications. Several examples have been given employing the truncated distributions in fitting rainfall data and animal population studies where observations usually begin after migration has commenced or concluded before it has stopped [1-2]. Similar situations arise with regard to aiming errors (range, deflection, etc.) in gunnery and other bombing accuracy studies. For example, in gun camera missions, the view angle of the camera defines a known truncation point for an exponentially distributed random variable, observable as some function of the radial error or the distance from the aiming point to the point of impact [3]. A situation for the truncated Poisson distribution would occur when one wishes to fit a distribution to Poisson-like data consisting of numbers of individuals in certain groups which posses a given attribute, but in which a group cannot be sampled unless at least a specified number of its members have the attribute. For example, the group may be a household of people, and the attribute measles; the specified number would then be one. Other examples arise in life testing and

reliability problems, where if failure is caused by a wear-out mechanism or is a consequence of accumulated wear, then the length-of-life of a system can be expected to be of finite dimension. The object of the present paper is to obtain a sampling distribution for truncated law with a known truncation point and a minimum variance unbiased estimator of the reliability function for this model using the results obtained for non-truncated law. It is known that a sampling distribution for truncated law may be derived using, namely, the method based on characteristic functions [4], the method based on generating functions [5], or the combinatorial method [6]. In this paper, a much simpler technique than the above ones is proposed. It allows one to obtain the results for truncated laws more easily.

U. UNBIASEDNESS EQUIVALENCE PRINCIPLE

Suppose an experiment yields data sample X" = (X1, ... , Xn) relevant to the value of a parameter 0 (in general, vector). Let LX(xn;9) denote the probability or probability density of X" when the parameter assumes the value 9. Considered as a function of 9 for given X"=x", LX(x";9) is the likelihood function. If the data sample X" can be summarized by a sufficient statistic S, one can write LS(s;0) * LX(x";9). Further, for any non-negative function ®(s), ®(s)LS(s;9) is also a likelihood function equivalent to LX(x";9). Suppose we recognize a function ®(s) such that ®(s)LS(s;9), regarded as a function of s for a given 9, is a density function. It can be shown that this is the sampling density of S if the family of recognized densities is complete.

The unbiasedness equivalence principle consists in the following. If

LxCX";9,5)=[W(9,5,)]"LX(X";9), (1)

represents the likelihood function for the truncated law, where w(9,3) is some function of a parameter (9,5) associated with truncation, 3 is a known truncation point (in general, vector), then a sampling density for the truncated law is determined by

g$(s; 9) = w (s)[w(9,3)] ng (s; 9), s g S 3, (2)

where

w (s)[[(9,3)]ng(s; 9)= ^s)Ls(s;9,3) * Lx(x";9,3), (3)

g(s;9) is a sampling density of a sufficient statistic s(X") (for a family of densities {/(x;9)}) determined on the basis of Lx(X";9), w(S) is an unbiased estimator of 1/[w(9,3)]n with respect to g(s;9), sgS (a sample space of a non-truncated sufficient statistic S), KS) is a function of S for a given 9, which is equivalent to unbiased estimator w (S) of 1/[w(9,3)]n, i.e.,

K(S) * w (S) (4)

or

KS) = w (S)[w(9,3)]" g (S; 9)/L, (S; 9,3), (5)

g3(s;9) is the sampling density of a sufficient statistic S (for a family of densities {/3(x;9)}) when the truncation parameter 3 is known, S3 is a sample space of a truncated sufficient statistic S.

V. EXAMPLES OF APPLICATIONS OF THE UNBIASEDNESS

EQUIVALENCE

PRINCIPLE TO FINDING SAMPLING DISTRIBUTIONS FOR TRUNCATED LAWS

Example 3.1 (Sampling distribution for the left-truncated Poisson law). Let the Poisson probability function be denoted by

nx

f (x;0) = e -0, x = 0, 1, 2, ... . x!

(6)

The probability function of the restricted random variable, which is truncated away from some 3>0, is then

f9(x;6) = w(e,3)f(x;6), x = 3 + 1,3 + 2, ... ,

(7)

where

w(e,3) =

e1

V c-e j!

^ 1=3+1 j •

Y1 f 3 ei V1 10 j ! ,

(8)

Consider a sample of n independent observations X1, X2, ..., Xn, each with probability function f3(x;e), where the likelihood function is defined as

Lx (xn ;e,3) = n f$( x ;e) = ke,3)]"Lx (xn ;e) = [(e,3)]n n f (x ;e)

i=1

i=1

and let

= [(e,3)]V

ei=i

n '

i =1

(9)

It is well known that

S = X X, s = n(3 +1), n(3 +1) +1

(10)

S = XX, s = 0, 1, .

i =1

(11)

is a complete sufficient statistic for the family {fx;$)}. A result of [7] states that sufficiency is preserved under truncation away from any Borel set in the range of X. Hence, in the case at hand Sis sufficient for {f3(x;9)}. It can be verified that Sis also complete.

i=1

For the sake of simplicity but without loss of generality, consider the case 3=0. This is at the same time the most important case for applications and the easiest with which to deal. It follows from (2) that

g3(s;9) = w(s)[w{6,3)]ng (s;9) =

esn!

(e9- 1)ns ! s'

Cn, s = n, n +1, ...

(12)

where

g(s;9) = <9 es = 0, 1,

s !

[w(9,3)]n =

1

n

(1 - e )

(13)

(14)

) n ! n w(s) = - C;,

n

(15)

Cn denotes the Stirling number of the second kind [8] defined by

c; =

1 n

1Y (-1)n-1

( n ^

n !

1=0

V J J

js, s = n, n + 1,

s < n,

(16)

1=0

n! nn (n9)s

s=0

n (j9)s

E{w(s)} = Y w(s) g (s;9) = Y ^ Cs" e"

n s !

c—M c—M ' ' J .

f

n

Y (-1)"-i e-(n-J e-J9= (1 - e ~9)

Vj J °'

s=0 s !

(17)

This is the same result that of Tate and Goen [9]. Their proof was based on characteristic functions.

Example 3.2 (Sampling distribution for the right-truncated exponential law). Let the probability density function of the right-truncated exponential distribution be denoted by

f3(x;9) = w(9,3)f (x;9), 0 < x <3, (18)

where

w(9,3) = --^39, (19)

1 - e

f (x;9) = (1/9)e-x/9, x g [0,»). (20)

Consider a sample of n independent observations X1, X2, ..., Xn, each with density f3 (x;0), where the likelihood function is determined as

Lx (xn ;e,3) = n fA ;e) = [w(e,3)]"Lx (xn ;e)

i =1

n 1 -x xt/e

[w(e,3)]nn f (x;e) =[w(e,3)]n —e « .

i=1 e

(21)

It is well known that

S = fjXl, s g [0,«>), (22)

i=1

is a complete sufficient statistic for the family {fx;$)}. It follows from (2) that

e~s/e " f n^

g3(s;e) = w (s)[w(e,3)] ng(s;e) =———— . (-1)"-1 [(s - (n - j )3)+ ] n-1

r(n)e (1 - e ) j=01 jJ

e-s/e ^ f

r(n)en (1 - ^'T ji V j j

where a+=max(0,a),

Zl.C-ir1 [(s - (n - j)3)+ ] n-1, s g [0, n3], n > 1, (23)

-1

- s/e

g(s;e) = —-e"s/e, s g [0,<x>) , (24)

f(n)en

[w(e,3" = (1 * , (25)

(1 - e )

1 ^ f n ^ ,

(s) = —El (-1)"-1 [(s-(n- j)3)+ ] n-1, (26)

s j=0 V j J

(aJ

E{w (s)}=J w (s)g (s; e)ds

XX fnl(-1)"-V(n-j)3/eJ[(s (n J)3]"-1 e-(s-(n-j)3)+ /ed(s - (n - j)3)+ = (1 - e-3/e)n.

=0 V j J

r(n)en

(27)

This is the same result that of Bain and Weeks [4]. Their proof was based on characteristic functions.

Example 3.3 (Sampling distribution for the doubly truncated exponential law). Consider an exponential distribution (20) that is doubly truncated at a lower truncation point (31) and an upper truncation point (32). The probability density function of the doubly truncated exponential distribution is defined as

f (x;9) = w(9,3)f (x;9), 31 < x < 32, (28)

where 3=3,32),

w(9,3) = -3/9 -32/9 . (29)

e -e

Consider a sample of n independent observations X1, X2, ..., Xn, each with density f (x;9), where the likelihood function is determined as

lx (x" ;9,3) = n f. (x ;9) = [[(9, 3)]"Lx (X" ;9)

i=1

" 1 -Yxi /9

[[(9,3)]"n f (x9) =[w(9,»)]"9-e -=1 .

v=1 9

(30)

It is well known that

S = YX-, s g [0,o), (31)

i=1

is a complete sufficient statistic for the family {fx; 9)}. It follows from (2) that

g, (s;9) = w(s)[w(9,3)] "g (s;9)

e~s/9 ^ (n^

r(n)9" (e^9 - e~32/9)" j=0 V jJ

e~s/9 ^ (n^

(-1)"-1 [(s -n31 - (n - j)(32-31)+ ]

n-1

r(n)9" (e-3/9 - e^9)" jf V J J

Y| .(-1)"-1 [(s-"3 -(n- j)(32-31)+ ] "-1, sg[n3,n32],

n > 1, (32)

where a+=max(0,a), g(s,9) is given by (24),

I " =

(eT3'9 -e^9)

[w(9,3)] " = .-3/9 e-3t/9)", (33)

1 ^ (n ^ 1

w(s) = —Yl (-1)"-1 [(s-"31 -(n- j)(32-31))+ ] "-1, (34)

s

=0 V jJ

E{w(s)}= Jw(s)g(s;9)ds = (e^9 -e^9)". (35)

oo

4. VALIDITY OF THE UNBIASEDNESS EQUIVALENCE PRINCIPLE

The theoretical results of this investigation into the validity of the proposed unbiasedness equivalence principle (UEP) for finding sampling distributions for truncated laws are largely contained in the theorem given below. We introduce the following notation and assumptions. Let X be a random variable taking on values xn in a space X3, let A be a cr-field of subsets of X3 and let (0,3) be a parameter associated with truncation, where 3 is a known truncation point. For all values of the parameter 0 in some parameter space 0, let P3 be a probability measure on A; i.e., for any set A in A, P3 (A;0) is the probability that X will belong to A when the parameter has the value 0. Let S=s(Xn) be a statistic on the measurable space (X3,A ) taking on values in a measurable space (S3, B). For each 0e0, let G3 be the probability distribution of S when X has the distribution P3; i.e., for any BeB, G3(B;0) = P3(s-1(B);0), where s-1(B) is the set of points xn in X3 for which s(xn)eB.

W. Assume the family P = {P3: 0e0} of probability distributions of X is

dominated by a totally c -finite measure u over (X3,A), i.e., there exists, for all 060, a non-negative A - measurable functionp3(xn;0) such that

P»( A; 0) = J P3(xn; 0)d/(xn) (36)

A

for all Ae A. (The integrandp3(xn;0) is called the density of P3 w.r.t. (with respect to) u).

(ii) Assume that s(X) is sufficient for P. From the Halmos-Savage factorization theorem [10], s(X) is sufficient if and only if for each 0e0 there exists a non-negative B-measurable function LS(s(xn);0,3) on S3 and a non-negative A - measurable function v on X3 such that

pff(xn; 0) = Ls (s( xn); 0,3)v(xn) (/). (37)

(The symbol (u) following a statement means that the statement holds except on a set of /-measure zero). In (37), we will assume that LS and v are finite (u).

(iii) Assume we recognize some likelihood function LS(s;0,3) equivalent to likelihood function LX(xn;0,3). Define a c-finite measure p over (X3,A) by

p(A) = J v(xn )d/(xn), all A eA. (38)

A

Then, from (36), (37), and (8),

P3(A;0) = JLS(s(xn);0,3)dp(xn), all A eA. (39)

A

(iv) Assume we recognize a totally c -finite measure n over (S3, B) such that the measure p s-1 over (S3, B) is absolutely continuous w.r.t. n; i.e., n(B)=0 implis that p s-1(B)=0, where p s-1(B) denotes the p - measure of the inverse image of B.

(v) Assume we recognize a positive B-measurable function (p on S3 such that

J Ls (s; 0,3)p(s)dn(s) = 1 (40)

S3

for all 8g0. Assume further that for any measurable set B of positive n - measure, there exists a 9g0 and a measurable subset B1 of B of positive n - measure over which LS(s;9,3)^>(s) is positive.

From (40), {Ls(s;9,3)^(s):9g0} is a family of densities w.r.t. n For BgB, let

G9(B—) = J Ls (s; 8,3)p(s)dn(s). (41)

B

Thus, (v) provides us with a family of densities, but at this stage we do not know if this recognized family is the family of sampling densities of S.

(vi) Assume we recognize that the family {Ls(s;8,3)^»(s):9g0} is complete, i.e.,

Jp(s) LS (s; 8,3)p(s)dn(s) - 0 for all 8g0 (42)

S3

implies

((s) - 0 (43)

except on a set D with g3(d;9) = 0 for all 8g0.

Theorem 1 (Sampling distribution for truncated law). Under assumptions (i) through (vi), G3 has a density with respect to n and LS(s;9,3)^>(s) is a version of it, i.e.,

Ls(s;8,3Ms) = w (s)[w(9,3)] ng(s; 9) (44)

is the sampling density, g3 (s;9), of the sufficient statistic s(X").

Proof. We show first that (43) and the second part of (v) imply that p (s)-0 (n). For suppose there exists a measurable B with n(B)>0 and p(s)^0 over B. Then BczD, so G3(B;9)=0 for all 8g0. But, from (v), there exists a B1^B for which G3 (B1;9)>0 for some 9, contradicting G3 (B;9)=0 for all 9g0. Now, by a theorem in [10], there exists a non-negative measurable function yon S3 such that

J Q^(s(x"); 9)dp( xn) = J Q^(s; 9)y(s)dn(s) (45)

for every measurable function Q3, in the sense that if either integral exists, then so does the other and the two are equal. In (45), let Q3 (s,8) =xBLS(s;9,3), where xB is the characteristic function of B (BgB). Then there exists a y(s) such that

J Ls (s(xn); 8,3)dp(xn) = J L5 (s; 9,3)y(s)dn(s) (46)

s-1( B) B

for all BgB. Note that the left side of (45) is G»(B;8). In (42), let (p (s) = 1-[ y(s)/p (s)]. From (40) and (46),

1-

¥(s)

LS (s; 8,3)p(s)dn(s) = 0

(47)

3

for all 9g0. Thus, from (43), y(s)=^(s) almost everywhere (n), and, from (47),

Ls(s;0,3)p(s) = W(s)[w(0,3)] ng (s; 0) (48)

is a version of the density of G3 with respect to n. □

X. FINDING RELIABILITY ESTIMATORS FOR TRUNCATED LAWS

VIA THE UNBIASEDNESS EQUIVALENCE PRINCIPLE

Consider a system that is required to operate for a given 'mission time', t. The reliability of this system for the right-truncated distribution of time-to-failure with the probability density function f3 (x;0) may be defined as

3

R(t) = Pr(x > t) = J f3(x;0)dx (49)

Due to the Rao-Blackwell and Lehmann-Scheffe theorem [11] a minimum variance unbiased (MVU) estimator for R may be obtained as

R(t) = J f&( x; s)dx, (50)

where X may be any one of the observations (X1, ..., Xn) from f3 (x;0), S is a complete sufficient statistic for {f3 (x;0)}, and f3 (x;s) is the conditional distribution of Xgiven S=s; f3 (x;s) is obtained as

u x, s) = fix*?) = Wf(x03, (5!)

g3(s;0) w(s)

where

f3( x, s; 0) = Wf (x, 0,3)[w(0,3)]ng (s; 0) (52)

is the joint probability density of X and S, wf (x, 0,3) is an unbiased estimator of

wf(x,0,3) = f3x;0) . (53)

f [ w(0,3)] n

with respect to g(s;0).

It should be noted that (50) can be obtained by different method as

R(t) = , (54)

W(s)

where wR (t, 0,3) is an unbiased estimator of

wR (t, 0,3) = R(t) (55)

R [w(0,3)]n

with respect to g(s;0).

Example 5.1 (MVU estimator of reliability for the right truncated exponential distribution). Let Xn=(X1, ..., Xn) be a random sample of size n from a population with density (18). Then it follows from (50) (or (54)) that the MVU estimator of R(t) is obtained as

£ (-1)j fn - ([(s - (j3+1))+] n-1 - [(S - (j+1)3)+] n-1)

R(t) = ^--^-• (56)

£ (-1)j . [(S - j3)+ ]

f j J

j=0

I n-1

As a particular case, if 3 ^ ro that is the variable X is assumed unrestricted, the corresponding MVU estimator of reliability reduces to

R(t) = [(1 -1 / S)+ ] n-1. (57)

For instance, suppose that the following failure times, in hours, are available from a given system: 4.2, 9.8, 16, 20 and that the truncation point 3=25 hours and the mission time t=5 hours. Clearly s=50 hours. Substituting these values in (56), the estimate of reliability is obtained as R(t) = 0.824. Had we assumed, however, that the observations are coming from the complete population, the estimate of reliability would have been, from (57), R(t) = 0.729.

Example 5.2 (MVU estimator of reliability for the right-truncated gamma distribution). Let Xn=(Xi, ..., Xn) be a random sample of size n from a population with density

f3(x;0) = w(0,3)-i-a-y-1e-x/CT, 0 < x < 3, a> 0, 0, (58)

where 3 is point of truncation, 0=(ct,£), and w(0,3) is such that

3 1

w(0,3) \——a-5x5-1e -x/adx = 1. (59)

J0

This distribution has found applications in a number of diverse fields, for instance, in fitting of length-of-life data under fatigue. Note that for 5=1, the right-truncated gamma distribution reduces to the right-truncated exponential distribution with parameter a. Although, this distribution is a special case of gamma distribution and gives a good fit to length-of-life data in many situations, it is not suitable since its use carries the implication that at any time future life-length is independent of past history.

To find MVU estimator of R(t) we apply the above technique. If the shape parameter 5 in (58) is assumed to be known, then it is well known that

S = £x (60)

i=1

is a complete sufficient statistic for c. The probability density function of the sampling distribution of S is given by

gs( s; 0) = W(s)[w(0,3)] ng (s; 0) =

[w(0,3)] n rn (S)

-s-<

c r(nS)

-s / c

xT(-1)r " [(s-r3)+ r-1 A

r n >

r=0

S-1

f

V r J

nS-1,

3

(s - r3)

, se(0, n3),

+ J

(61)

where

f

A

nS-1,

3

\

(s - r3)

T

rl^l

r nS- 1Y 3 ^

S-1

+ J {r0, rl, ..., rS-1}:

S-1

T rj=r 1=0 j=0

S-1

n(rj!)T(j !)

1 V m J

(s - r3)

+ J

j=0

(62)

S-1

57=T jri.

j=0

(63)

The joint distribution of X and S is given by

f (x, s; 0) = Wf (x, 0,3)[w(0,3)] n g (s; 0) =

[w(0,3)] nr(n-1)(S) -s/c

cnSr((n - 1)S)

■x e

T (-1)

r n - n

(s - r3- x)(n-1)S-1 A (n - 1)S-1

V r J

3

(s - r3 - x)

+ J

(64)

Thus the conditional distribution of X given S is

f3( x, s; 0) = wf (^ 0,3)

3x;s) = ■

r(nS)

g3(s; 0) W(s) r(S)r((n - 1)S)

n-1

xT (-1)'

r=0

rn - 1^S1 ( 1)S1 r 3 }

x [(s-r3-x)+ ](n-1)S-1 A (n- 1)S-1,--

V r J V (s - r3- x)+J

r n >

T (-1)r (s - r 3)nS-1 A

f

VrJ

nS -1,

3

Y\

(s - r3)

+ J J

(65)

Hence the MVU estimator of R(t) at time t is given by

¡y

R(t) = J f3( x; s)dx

wR (t, 0,3) _ r(nS)

1 rn -D

W(s) r(S)r((n - 1)S)

T (-1)

r=0

VrJ

ffi

r

r=0

-1

x

r=0

X J X5-1 [(5 - r S - x)+ ](n-1)5-1A (n - 1)5 -1,

S

(5 - rS - x)

dx

+ J

( n ^

V r=0

f

(-1)r (5 - rS)n5-1 A

v r j

n5 -1,

3

Y\

(5 - rS)

(66)

+ JJ

It may be remarked that the result (66) though at the first look appears quite unwieldy is not so in practical applications, particularly when the sample size is small.

As a particular case, if S ^ ro that is the random variable X is assumed unrestricted, the distribution of the sufficient statistics from equation (61) reduces to

£(5; 0) =

1

r(nS)G

n5

5n5-1e-/ff,

5e(0,ro)

(67)

and the corresponding MVU estimator of reliability at time t is given by

S

X

R(t)=105) £ j-'0 -t' , (68)

ja(S-1 - j)![(n-1)5 + j]!

which corresponds to Basu's [12] equation (9).

Y. CONCLUSIONS

The authors hope that this work will stimulate further investigation using the approach on specific applications to see whether obtained results with it are feasible for realistic applications.

It will be noted that the similar approach also can be used to find the sampling distribution for truncated law when some or all of its truncation parameters are left unspecified.

For instance, consider Example 3.3, where it is assumed that the truncation parameter S=(Si,S2) is unknown. It is known that the statistic (X(i), X(n), S), where

X(1) = min X,., (69)

( ) 1<i<n

X(n) = mux X,, (70)

1<i < n

and

S= £ X,., (71)

i=2

is a complete sufficient statistic for a set of parameters (31,32,0). In this case, the likelihood function of a sample is determined as

n-1

LX (x(1), x(n), x2,..., xn-10, 3) = n(n - 1)fa (x(1);Q)U (x(n);0)n f»(x' ;0)

i=2

n-1 1 -X X,-/0

n-2

e

i=2

= (X(1), X(n)

; 0)[(0, X(i), X(n) )]n-2 П / ( X, ; 0) =h, ( X(i) X( n) ; 0)[(0, xm , X( n))]n

i=2 0

(72)

where

h, ( X(1), X(n)0) = n(n -1)[ F, ( X( n);0) - F, ( X( n)0)]n - 2 f ( х(И;0)./в ( xw;0),

n-1

x(1) < x(n), x(1), x(n)6 [31,32], (73)

is the joint probability density function of the order statistics xw and x(„)), Fg.(-) is the probability distribution function. It is well known that

S = T X, s e [0,»), (74)

is a complete sufficient statistic for the family {fx;0)}. It follows from (2) and (72) that

g3 (s;0) = w (s)[w(0 , x(1), x(n) )]n - 2 g (s;0)

e s/ 0 r n -

Г(п - 2)0n-2[e~X(1)/0- e X(2)/0]n-2 j=0 ^ j )

(-1)n-2-j [(5 - (n - 2)X(1) - (n - 2 - j)(X(2) - X(1)))+ ] n-3

e~sш n-2fn - 2^

s/0 _ -

.....(-1)n-2-j [(s - (n - 2)X(1) - (n - 2 - j)(X(2) - X(1)))+ ] n-3

Г(п - 2)0n-VX(1)/0- e X(2)/0]n-2 j=11 j )

se[(n-2)x(1),(n-2)x(n)], n > 3, (75)

where

sn-3

g ( s;0) =---- e"s/0, s e [0, œ) , (76)

Г(и - 2)0

x(1), x(n))]n-2 = X(1) — x(2)/g]n-2 , (77)

[e - e ]

1 ^ r n - 2 ^ 2 . 3

W(s) = —r£| . (-1)n-2-j[(5- (n -2)x(1) - (n -2- j)(x(2) - x(1)))+ ] n-3, (78)

.,n-3 ^

j=0 v j J

E{w(5)}= Jw(5)g(5;0)ds = [e~x(1)/0 -e x(2)/—]n-2. (79) 0

Thus, the sampling distribution of the sufficient statistic (X(1), X(n), S) for is given

by

g— (^ x( ^ 5;0) = hS(X(1), x( n)0)gS (5;0). (80)

In other words, we have the following results.

In a singly truncated case, when a truncation point on the left, S1, is unknown, a sampling distribution of the sufficient statistic (X(1), S) for is given by

gSi (x(-), 5;0) = hS(x(-);0) g—(5;— (81)

where

X ~ S (x ;0) = w—, 3) f(x ;0) = -

- F (S-;0)'

X,. ~ fS-(x. ;0) = w(d,S-)f (x, —) = f(x, ;0), x, > S-, i = 1, ..., n, (82)

^ 1 - F (3;—)

h— (xO);0) = n[1 - F— (xO);0)]n-1 f— (x^;—) (83)

is the probability density function of the order statistic X(1),

g— (50) = w(5)[w(0, x(-))] n-1 g (5; 0), (84)

5=5(X2, ..., Xn).

In a singly truncated case, when a truncation point on the right, S2, is unknown, a sampling distribution of the sufficient statistic (X(n), S) for (S2,0) is given by

g—,(x( n^ 5;0) = h—(x( n);0) g—, (5;— (85)

where

X, ~ fs2(x, ;0) = w(d,S2)f(x—) = -——f(xi ;0), x, < —, / = -, n, (86) 2 F (S2—)

h— (x(n)— = n[F— (x(n))rlf— (x(n)— (87)

is the probability density function of the order statistic X(n),

gS, (5;—) = W(5)[w(0, x(n))] n-1 g(5—), (88)

s=s(Xi, ..., X„.i).

In a doubly truncated case, when a lower truncation point, and an upper truncation point, 32, are unknown, a sampling distribution of the sufficient statistic (X(1), X(n), S) for (31,32,0) is given by

ga( xoo> s;0) = h( x»;0) g, (s;0).

(89)

where

X ~ /• (x;0) = w(0,3,3) f (x;0) =

1

F(32;0) - F(3;0)

f (x;0), 3i <x < 32, - = 1, ..., n,

h, ( x(i), xw;0) = n[ F, (x ( n)0) - F, (xO);0)]n-2 f ( x( ^;0)

(90)

(91)

is the joint probability density function of the order statistic X(1) and X(

(n)>

g3 (s;0) = w(s)[w(0, x( n) )] n-2 g ( s;0),

(92)

s=s(X2, Xn-1)-

If, say, we deal with a left-truncated exponential distribution,

where

fAx;0) = w(0,3)f(x;0), 3 < x <«>,

w(0,3i) = ■

-3/0

(93)

(94)

and a truncation point on the left, 31, is unknown, then it follows immediately from (81) that the sampling distribution of the sufficient statistic (X(1), S), S = T"= 2X, for 3,0) is given by

g,( x^ s;0) = h3( Xd;0) g, (s;0) =

f -xm/e\

e ()

n

-3i/e

K~ J

i i

x(i)/e

[s - (n -1)x(i)]n-2 i

- s/0

n-2

[e~ x(1)/0]n-1 r(n -1)0

n-i

n -n(x(1)-3)/0 [s (n 1)X(i)] -[s_(„_1)x(1)]/0

= — e 0

r(n -1)0

n-i

which corresponds to the well-known result [11].

(95)

1

e

n-i

n-2

s

x

e

ACKNOWLEDGMENTS

This research was supported in part by Grant No. 06.1936 and Grant No. 07.2036 from the Latvian Council of Science and the National Institute of Mathematics and Informatics of Latvia.

REFERENCES

[1] Chapman, D.G. Estimating the Parameters of a Truncated Gamma Distribution. Ann. Math. Stati5t, Vol. 27, 1956, pp. 498-506.

[2] Kenyon, K.W., Scheffer, V.B. and Chapman, D.G. A Population Study of the Alaska Fur Seal Herd U.S. WUdUfe, Vol. 12, 1954, pp. 1-77.

[3] Deemer, W.L. and Votaw, D.F.Jr. Estimation of Parameters of Truncated or Censored Exponential Distributions. Ann. Math. Stat,5t, Vol. 25, 1955, pp. 498-504.

[4] Bain, L.J. and Weeks, D.L. A Note on the Truncated Exponential Distribution. Ann. Math. Stati5t., Vol. 35, 1964, pp. 1366-1367.

[5] Charalambides, C.A. Minimum Variance Unbiased Estimation for a Class of Left-Truncated Discrete Distributions. Sankhya, Vol. 36, 1974, pp. 397-418.

[6] Cacoullos, T. A Combinatorial Derivation of the Distribution of the Truncated Poisson Sufficient Statistic. Ann. Math. Stati5t, Vol. 32, 1961, pp. 904-905.

[7] Tukey, J.W. Sufficiency, Truncation and Selection. Ann. Math. Stat,5t, Vol. 20, 1949, pp. 309-311.

[8] Jordan, C. Calculu5 of FmUe Difference5. New York: Chelsea, 1950.

[9] Tate, R.F. and Goen, R.L. Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution. Ann. Math. Stati5t, Vol. 29, 1958, pp. 755-765.

[10] Halmos, P.R. Mea5ure Theory. New York: Van Nostrand, Inc., 1950.

[11] Zacks, S. The Theory of Stati5tical Inference. New York: John Wiley & Sons, Inc.,

1971.

[12] Basu, A.P. Estimation of Reliability for Some Distributions Useful in Life Testing. Technometric5, Vol. 6, 1964, pp. 215-219.