Estimation in truncated Poisson distributions with concomitant exposure intervals and truncation point

Cohen, A. C.

English

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NASA CONTRACTOR 
REPORT 
NASA CR-61335 
ESTIMATION IN TRUNCATED POISSON DISTRIBUTIONS 
W ITH CONCOMITANT EXPOSURE INTERVALS 
AND TRUNCATION POINTS 
By A. CiiffordCohen 
University of Georgia 
Athens, Georgia 
August 1970 
Final Report 
I 
c 
Prepared for 
N A S A - G E O R G E  C .  M A R S H A L L  S P A C E  F L I G H T  C E N T E R  
Marshall Space Flight Center, Alabama 35812 
https://ntrs.nasa.gov/search.jsp?R=19700030525 2020-03-11T22:46:33+00:00Z
I *  
. 
I 
1. Report No. 
NASA CR-61335 
2. Government Accession No. 3. Recipient's Catalog No. 
4. Title and Subtitle 
ESTIMATION I N  TRUNCATED POISSON DISTRIBUTIONS WITH CONCOMITANT 
EXPOSURE INTERVALS AND TRUNCATION POINTS 
7. Author(s) 
A. Cl i f ford Cohen 
9. F'erfwming Organization Name and AdQe95 
University of Georgia 
Athens, Georgia 
5. Report Date 
6. Performing Organization code 
August 1970 
8. Performing Organization Report No. 
10. Work Unit No. 
11. Contract or Grant No. 
NAS8-11175 
12. Sponsoring Agency Name and Address 
Aero-Astrodynamics Laboratory 
George C. Marshall Space F l i g h t  Center 
Marshall Space F l ight  Center, Alabama 35812 
13. Type of Repon and Period Covered 
CR - F i n a l  Report 
14. Sponsoring Agency code 
15. sP@emnmW Notes Contract Monitor: Lee W. Fal l s ,  Aerospace Environment Division, Aero-Astrodynamics 
Laboratory. 
17. Key Words (Suggested by Author(s)) 
Truncated Poisson Dis t r ibu t ion  
16. Abstract 
e t e r  ti, where t h e  t i  are known pos i t ive  numbers, and where t runcat ion points  ci are specif#ed.ruch 
that x i  2 ci.  
quatior.  f o r  ?, becomes 
Let [x i ) ,  i = 1, ..., n, denote random observations of a truncated Poisson v a r i a b l e  with param- 
Using x i  plus concanitant information provided by ti and c i ,  the  l ikel ihood estimating 
18. Dimibution Statement 
PUBLIC RELEASE: 
where 
19. Security Clanif. (of this repon) 20. Security Clanif. (of this page) 
UNCLASSIFIED UNCUS S IF IED 
Solut ions (est imates)  are r e a d i l y  found using t r i a l  and e r r o r  procedures with l i n e a r  iz te rpola t ion ,  
bu t  any s tandard i t e r a t i v e  procedure would be applicable. The asymptotic variance of h is obtained 
from t h e  second p a r t i a l  of t h e  l ikel ihood function. A n  i l l u s t r a t i v e  example is included. 
21. NO. of pages 22. Rice' 
12 
~ 
'For sale by the Clearinghoure for Federal Scientific and Technical Information 
Springfield. Virginia 22151 
ESTIMATION IN TRUNCATED POISSON DISTRIBUTIONS 
WITH CONCOMITANT EXPOSURE INTERVALS AND TRUNCATION POINTS 
by 
A. Clifford Cohen 
The University of Georgia 
1. INTRODUCTION 
Estimation in the truncated Poisson distribution has previously 
been considered by numerous writers including Bliss [l], David and 
Johnson [4], Moore [6], Plackett 171, Subrahmanian [ 8 ] ,  the writer [2,3], 
and others. Our concern here is with estimation of the Poisson 
parameter when the i- observation of the discrete random variable X 
extends over a specified exposure interval of magnitude t 
the restriction that xi > c 
More specifically, our attention is directed toward maximum likelihood 
estimation based on a random sample consisting of n independent 
observations {x.,t ,c.}, where x. is the number of occurrences of the i i i  1 
event of interest observed during the i- interval of exposure, while 
t. and c. are corresponding concomitant values. Of course, x and c 
1 1 i i 
must be non-negative integers, but ti is permitted to assume any 
positive value. Accident data in which reports are rendered for time 
intervals of varying size with truncation as stipulated fall into the 
category of samples under consideration here. 
defects per unit are reported for units of differing sizes might also 
result in samples of the type considered here. 
th 
subject to i’ 
where c is a-specified truncation point. - i’ i 
th 
Inspection data in which 
With concomitant values ti and c given, the applicable Poisson i 
probability function may be written as 
-At. X .  
1 1 e (Ati) 
x.! F(c.) 
, x.=c ,c.+l,..., 
i i i  
1 1 
0 , elsewhere, 
Pr{X=xi It. ,c. 1 = 
1 1  
L 
where 
-Ati X i - m e (Ati) 
F(c.) = C = 1 - F(c.-l) . 
1 1 x =c x '  i i  i' 
2. MAXIMUM LIKELIHOOD ESTIMATION OF X 
The likelihood function for a sample consisting of the n observed 
values { x  ,ti,ci) for which the probability function (1) is applicable, i 
follows as 
-Ati X i 
n e  (Ati) 
L =  n (3) 
i=l x ! F(Ci) i 
2 
I 
i 
On taking logarithms of boRh sides of (3) and differentiating with 
respect to A ,  we obtain 
n n 2 F ( C i )  
-. 1 
n 
- =  alnL - 
a x  1 1 ‘(Ci) a x  
It follows from (2) that 
t 
-Ati c -1 i aF(ci) e (Ati> 
aA (ci-l) ! 
- = f(ci-l) . 
When ( 5 )  is substituted into (4) and the resulting expression is 
equated to zero and simplified, we obtain the estimating equation 
n n n 
1 1 1 
c xi = A [  c ti + c f(ci-l)/F(ci)] , 
in which f(c -1) and F(ci) are defined by (5) and (2) respectively. 
With the aid of an ordinary table of individual and cumulative 
i 
values of the Poisson function such as those of Molina [ S I ,  equation 
( 6 )  can be solved with relative ease for the required estimate A ,  
,. 
using trial and error techniques. 
been found in a sufficiently narrow interval such that 
Once two values Al and A have 2 
n 
1 
G(A1) < xi < G(A2), where G(A) has been written for the right side 
of ( 6 ) ,  the required estimate A can be obtained by interpolating 
(4) 
linearly between A and A 2 .  Of course, standard iterative procedures 1 
3 
.. 
might also be employed in the calculation of A .  
3 .  SOME SPECIAL CASES 
By appropriately restricting the values of t and ci in ( 6 ) ,  we i 
obtain results that apply in various cases of special interest. 
Certain of these specialized results are, of course, quite well known. 
No Truncation - Unequal Intervals 
In this case, c = 0 for all i, 'and equation (6) leads to the i 
estimator 
. . n  n 
h = c x./ c t . 
1 i 1 1 
No Truncation - Eaual Intervals 
(7) 
If in addition to the restriction, c = 0 for all i, we impose the i 
further restriction that ti = t for all i, the applicable estimator 
reduces to 
.. - 
h = x/t , 
n 
1 
where = C x./n. When t = 1, as in the usual case of estimation in the 
Poisson distribution, then (7)  leads to the familiar estimator X = x. 
1 
.. - 
Only Zero Class Truncated In Making Each Observation - Unequal Intervals 
In this case ci = 1 for  each i, and the estimating equation ( 6 )  
reduces to 
n n -Ati 
c xi = [ti/(l-e 11 , 
1 1 
( 9 )  
4 
a r e s u l t  which was obtained independently by one of t h e  writer's 
graduate a s s i s t a n t s ,  Mr. S. S. Srivastava.  
Only Zero Class Truncated In  Making Each Observation - Equal In te rva ls  
When w e  fu r the r  spec ia l i ze  the preceeding r e s u l t  by requir ing tha t  
t i  = t f o r  a l l  i, then (9) becomes 
If w e  spec ia l i ze  s t i l l  fu r the r  by requir ing t h a t  t = 1, then (10) 
reduces t o  
I 
as previously given i n  [3], where a t a b l e  of t he  function on t h e  r i g h t  
s i d e  of (11) i s  provided i n  order t o  fac i l i t a te  so lu t ion  f o r  t h e  required 
estimate A .  
A 
4.. VARIANCE OF ESTIMATES 
L 
The asymptotic variance of X can be expressed as 
I 
I 
for  each of the  cases considered here.  The second der iva t ive  i n  t h e  . 
L 
most general  case under consideration follows from (4) as 
n 
1 c xi n f(ci-2) - f(ci-1) f ( c i - l )  2 (13) 
= -  - [ F(Ci) j + i [ F(Ci ) ]  ' a2 in  L a,* A 2  1 
5 
a result which was previously given in [2]. With the aid of tables 
such as those of Molina [ S I ,  this derivative can easily be evaluated 
once ( 6 )  has been solved for A .  The variance then follows from (12). 
n 
5. AN ILLUSTRATIVE EXAMPLE 
In order to illustrate the practical application of estimators 
derived here, we consider inspection results on the number of surface 
defects per unit found in products of differing surface areas but 
inspections, no truncation takes place. In others, the zeros are 
eliminated, and in still others, both zeros and ones are eliminated. 
Based on the total sample which results from combining the several 
sub-samples involved, we proceed to estimate the Poisson parameter X 
which characterizes the underlying common production process, and to 
determine the asymptotic variance of this estimate. The total sample 
available for this purpose consists of inspection results on 422 
separate manufactured uhits. These data are presented in Table 1, with 
the total sample broken down into eight sub-samples having values of 
(t,c) as follows: (1,0), (1.5~) ,(2,1), (2.5,1), (3,1), (1,2), (2,2), and 
(3,2). The value t = 1 indicates a surface area of one unit, while 
t = 2 indicates a surface area of two units, etc. For the total 
number of defects found in the 422 units inspected, we have 
422 422 
C x = 1597. 
1 
For the sum of the t's, we have C ti = lOO(1) 
1 i 
+ 94(1.5) + 50(2) + 32(2.5) +28(3) + 30(1) + 34(2)  + 54(3) = 765- 
I 
. 
These values are substituted into ( 6 )  which must then be solved for 
6 
No. Defect 
X 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Column 
Totals 
Cxnx 
TABLE 1 
NUMBER OF SURFACE DEFECTS FOUND IN 
FINAL INSPECTION OF 422 MANUFACTURED UNITS 
Frequencies n 
X 
c=o 
t=l 
13 
28 
27 
17 
10 
3 
2 
100 
200 
t=l.5 -
d 
16 
21 
22 
16 
11 
6 
1 
1 
94 
294 
C -
t=2 
==. 
-
- 
5 
6 
9 
11 
9 
4 
3 
2 
1 
- 
50 
- 
203 
t=2.5 -
< -
1 
3 
5 
6 
5 
5 
4 
2 
1 
32 
- 
154 
t=3 
7 
0 
1 
3 
5 
4 
5 
4 
3 
1 
2 
28 
162 
- 
t=l 
7 
L 
14 
9 
5 
1 
1 
30 
86 
c=2 -
t=2  - 
>< - 
6 
5 
7 
6 
5 
3 
1 
1 
- 
34 
- 
153 
- 
- 
7 
b 
3 
3 
7 
7 
9 
8 
6 
5 
3 
2 
1 
54 
145 
Row 
Totals 
13 
50 
81 
73 
67 
4 6  
37 
23 
15 
9 
5 
2 
1 
422 
- 
1597 
7 
,. 
the required estimate A .  
which comes from the sub-sample of size 100 with t = 1 and c = 0. In 
As a first approximation, we try X = 2.0000, 1 
general, satisfactory first approximations might be obtained from any 
one or a combination of the available sub-samples. In some circum- 
stances, a judicious guess might even suffice. 
at the final estimate h = 2.027, by linear interpolation as follows 
We subsequently arrive 
4 A 
2.1’00 1652.91 
422 
C x = 1597.00 2.027 
1 
2.000 1576.75 
i 
where G ( X )  has been written for the right side of the estimating 
equation (6). 
and (13) to calculate V ( h )  = 0.0027. 
For the variance of the above estimate, we employ (12) 
A 
8 
REFERENCES 
[l] Bliss, C. I., (1948). Estimation of the mean and its error from 
incomplete Poisson distributions, Connecticut Agricultural 
Experiment Station Bulletin 513, 12 p. 
[2] Cohen, A. C., Jr. (1954). Estimation of the Poisson parameter 
from truncated samples and from censored samples, J. Am. Stat. 
Assn. 49, 158-68. 
[3] Cohen, A. C., Jr. (1960). Estimating the parameter in a conditional 
Poisson distribution, Biometrics 16, 203-11. 
[4] David, F .  N., and Johnson, N. L . ,  (1952). The truncated Poisson, 
Biometrics 8 .  275-85. 
[SI Molina, E. C., (1942). Poisson's Exponential Binomial Limit, D. 
Van Nostrand Co., Inc. 
[6] Moore, P. G., (1952). The estimation of the Poisson parameter 
from a truncated distribution, Biometrika, 39, 247-51. 
[7] Plackett, R. L . ,  (1953). The truncated Poisson distribution, 
Biometrics 9. 485-88. 
[8] Rider, Paul R., (1953). Truncated Poisson distributions, J .  Am. 
Stat. Assn. 48, 826-30. 
[9] Subrahmanian, K., (1965). A note on estimation in the truncated 
Poisson, Biometrika 52, 279-82. 
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