Download Order Statistics for Negative Binomial Distribution

Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
Order Statistics for Negative Binomial Distribution
Tabark Ayad Ali
Hussain Ahmed Ali
University of Babylon /College of Education for Pure Sciences
[email protected]
Abstract
In this paper, we considers the order statistics from negative binomial distribution .which have moment of
all positive orders.
We take as example for 𝑥3 to find the mean and variance. We define aformula for B . The incomplete beta
function of Negative binomial distribution is 𝐼1−𝐵(𝑥) (𝑖, 𝑀), But The incomplete beta function of binomial
distribution is 𝐼𝐵(𝑥) (𝑖, 𝑀 − 𝑖 + 1).
The purpose of this paper is to study the distribution of 𝑥𝑖 and 𝑅 = 𝑥𝑗 − 𝑥𝑖 (𝑥𝑗 , 𝑥𝑖 , 𝑖 > 𝑗) . The results are
obtained here can be compared to those for a continuous variate.
Keywords :Order Statistics, Negative Binomial, Joint Distribution, Mean, Variance.
‫الخالصة‬
‫ حيث تم وضع صيغ‬. ‫ الذي يمتلك عزوم لكل الرتب الموجبة‬. ‫في هذا البحث ناقشنا االحصاءات المرتبة لتوزيع ثنائي الحدين السالب‬
. ‫ = )𝑁(𝑃 وكذلك وضعت صيغ لحساب دالة بيتا غير الكاملة لتوزيعي ثنائي الحدين السالب وتوزيع ذو الحدين‬1 ‫لحساب الوسط والتباين عندما‬
. R = xj − xi (xj , xi , i > j) ‫ هدف هذا البحث هو دراسة توزيع 𝑖 و‬. ‫ لحساب الوسط والتباين‬x3 ‫اخذنا مثال بالنسبة الى‬
. ‫النتائج التي توصلنا لها يمكن ان تحسب لهذا الموضوع للحالة المستمرة‬
.‫ التباين‬,‫ التوقع‬,‫ التوزيع المشترك‬,‫ ثنائي الحدين السالب‬,‫االحصاءات المرتبة‬:‫الكلمات المفتاحية‬
1. Introduction
Let 𝑥𝑖 (𝑖 = 0,1,2, … , 𝑀) be the number of successes in N independent trials from
anegative binomial distribution with 𝑝 as the probability of success in each trial .
Let 𝑥𝑖 be arranged in order to give
𝑥1 ≤ 𝑥2 ≤ ⋯ ≤ 𝑥𝑖 ≤ ⋯ ≤ 𝑥𝑀
The discrete variable 𝑥𝑖 𝑜𝑟 𝑥(𝑖) take values 1,2, … (𝑖 = 1,2, … , 𝑀).
Table are provided giving the cumulative distribution and the expected value and the
variance of 𝑥(1) and 𝑥𝑀 . joint distribution of 𝑥(𝑖) and 𝑥(𝑗) (𝑖 < 𝑗) is obtained .
The distribution of 𝑥𝑗 − 𝑥𝑖 (𝑗 > 𝑖) is also derived .Given observations, we can see that there
are 𝑖 (𝑀 − 𝑖 + 1) different and mutually exclusive cases of the type 𝑖 – 1 – 𝑘 observations
below 𝑥𝑖 ,
(𝑘 + 𝑚 + 1) observations of 𝑥𝑖 and 𝑛 − 𝑖 − 𝑚 observations above 𝑥𝑖 , with respective
probabilities 𝑃(𝑥𝑖 − 𝑙), 𝑝(𝑥𝑖 ) 𝑎𝑛𝑑 1 − 𝑃(𝑥𝑖 ) ,
(𝑘 = 0,1,2, … 𝑖 − 1 𝑎𝑛𝑑 𝑚 = 0,1,2, … , 𝑀 − 𝑖).
2.Literature Review
The negative binomial distribution (NBD) is one of the most useful probability
distributions and has been successfully employed to model a variety of natural phenomena. It
has been used to construct useful models in many substantive fields: in accident statistics
(Greenwood and Yule(1920) , Arbous and Kerrich(1951), Weber( 1971)) . in birth-death
processes (Furry,1937) ; Kendall 1949) .in psychology (Sichel, 1951).
The first work in actuarial literature that has come to my attention involving the negative
binomial was by Keffer in 1929 in connection with a group life experience rating plan. He
developed the theory in relationship to the relative dispersion of loss ratios about their true
mean.
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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
A. L. Bailey first utilized the negative binomial in the Proceedings of the Casualty
Actuarial Society in 1950. The distribution resulting from
repeated trials of this experiment will be in thc form of a negative binomial distribution . This
model is suggested by the mathematical development of Feller (1957) .
Creel and Loomis (1990, 1991) estimated truncated and untruncated Poisson and negative
binomial models for California deer hunting and derived welfare measures from these models
see (Miaou, 1994; ; Hadi et al., 1995; Shankar et al.(1995) Poch and Mannering, (1996).
In addition, zero-inflated Poisson and zeroinflated negative binomial models were also
employed to analyze accident frequencies by Shankar et al. (1997) . also employed to analyze
accident frequencies by Lee and Mannering (2002) and Lee et al. (2002) .
In biology (Anscombe(1950); Bliss and Fisher (1953); Anderson (1965); Boswell and
Patil ( 1970); Elliot (1977) . In ecology (Martin and Katti (1965); Binn (1986); White and
Bennetle ( 1996)). in entomology (Wilson and Room( 1983) . In epidemiology (Byers et al.
(2003) . in information sciences (Bookstein( 1997) . In meterology (Sakamato ( 1972) .
In addition, many other physical and biological applications have been described by
( Biggeri (1998) and Eke et al. (2001) .
3. Moment of order statistics
Gupta ,S.Shanti .and S. Panchspakesan(1967) "Department of Statistics Division of
Mathematical Sciences Mimeograph Series No. 120"
𝑁−1 𝑥
) 𝑝 (1 − 𝑝)𝑁−𝑥 ,
𝑥−1
𝑁 = 𝑥, 𝑥 + 1, … , 𝑥 > 0
Where 𝑁𝑏(𝑥) probability density function of Negative binomial distribution.
𝑁𝑏(𝑥) = 𝑃(𝑥) ≡ 𝑃(𝑁; 𝑝, 𝑥) = (
𝑠
𝐵(𝑠) = ∑ 𝑃(𝑥) .
𝑥=𝑟
𝐵(𝑝, 𝑞) = (𝛤(𝑝)𝛤(𝑞))/(𝛤(𝑝 + 𝑞) )
𝑝
1
𝐼𝑝 (𝑎, 𝑏) =
∫ 𝑢𝑎−1 (1 − 𝑢)𝑏−1 𝑑𝑢 .
𝐵(𝑎, 𝑏)
0
Let 𝑝𝑖 (𝑥) be the probability that the ith order statistic 𝑥(𝑖) is equal to
𝑥 and let 𝑃𝑖 (𝑥) = 𝑝{𝑥(𝑖) ≤ 𝑥} be the 𝑐. 𝑑. 𝑓. 𝑜𝑓 𝑥(𝑖) .
𝑖−1 𝑀−𝑖
𝑝𝑖 (𝑥) = ∑ ∑
𝑘=0 𝑚=0
𝑀! {𝐵(𝑥 − 1)}𝑖−1−𝑘 {𝑝(𝑥)}𝑘+𝑚+1 {1 − 𝐵(𝑥)}𝑀−𝑖−𝑚
(𝑖 − 1 − 𝑘)! (𝑘 + 𝑚 + 1)! (𝑀 − 𝑖 − 𝑚)!
Where 𝐵(𝑥 − 1) = 0 𝑓𝑜𝑟 𝑥 = 0 . This can be rewritten as
𝐵(𝑥)
𝑀
𝑝𝑖 (𝑥) = 𝑖 ( )
𝑖
∫ 𝜔𝑖−1 (1 − 𝜔)𝑀−1 𝑑𝜔
𝐵(𝑥−1)
= 𝐼1−𝐵(𝑥) (𝑖, 𝑀) − 𝐼1−𝐵(𝑥−1) (𝑖, 𝑀).
Further
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(3.2)
(3.1)
Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
𝐵(𝑥)
𝑀
𝑝𝑖 (𝑥) = 𝑖 ( ) ∫ 𝜔𝑖−1 (1 − 𝜔)𝑀−1 𝑑𝜔 = 𝐼1−𝐵(𝑥) (𝑖, 𝑀) (3.3)
𝑖
0
𝑁−1
𝐸(𝑥(𝑖) ) = ∑[1 − 𝑝𝑖 (𝑥)]
(3.4)
𝑥=0
𝑁−1
𝐸(𝑥
2
(𝑖)
𝑁−1
) = 2 ∑ 𝑥[1 − 𝑝𝑖 (𝑥)] + ∑[1 − 𝑝𝑖 (𝑥)]
𝑥=0
(3.5)
𝑥=0
𝑣𝑎𝑟(𝑥𝑖 ) = 𝐸(𝑥 2 (𝑖) ) − [𝐸(𝑥(𝑖) )]
2
(3.6)
Example
1
let ( 𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 ) be order statistic , i=3 , 𝑝 = 𝑞 = 2 , 𝑥3 = 3 , find the mean and
veriance of 𝑥3 ? when 𝑀 = 3 , 𝑁 = 4 , 𝑍 = 2 .
where 𝜔 = 𝑍𝑃(𝑥𝑖 ) + (𝑥𝑖 − 1)
2 1 1 2 1
𝑃(𝑥3 = 3) = ( ) ( ) ( ) =
0 2 2
8
1 1 1
1
𝑃(𝑥3 − 1) = 𝑃(2) = ( ) ( ) ( ) =
0 2 2
4
1 1 1
+ =
8 4 2
𝜔 = 2∙
𝐵(𝑥)
𝑀
𝑃(𝑥) = 𝑖 ( ) ∫ 𝜔𝑖 (1 − 𝜔)𝑀−𝑖 𝑑𝜔
𝑖
𝑥
0
𝐵(𝑥) = ∑ 𝑝𝑖 (1 − 𝑝)𝑖−𝑟
, 𝑖 = 0,1,2,3
𝑟=𝑖
1
5
13
𝐵(0) = 0 , 𝐵(1) = , 𝐵(2) = , 𝐵(3) =
2
8
8
𝑃3 (0) = 0
1
2
3
1 3 1 0
3
𝑃3 (1) = 3 ( ) ∫ ( ) ( ) 𝑑𝜔 =
3
2
2
16
0
5
8
3
1 3 1 0
15
(2)
𝑃3
= 3 ( ) ∫ ( ) ( ) 𝑑𝜔 =
3
2
2
64
0
13
8
3
1 3 1 0
39
𝑃3 (3) = 3 ( ) ∫ ( ) ( ) 𝑑𝜔 =
3
2
2
64
0
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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
4−1
𝐸(𝑥3 ) = ∑
[1 − 𝑃3 (𝑥)].
𝑥=0
= [1 − 𝑃3 (0)] + [1 − 𝑃3 (1)] + [1 − 𝑃3 (2)] + [1 − 𝑃3 (3)] .
3
15
39
13
49
25
= 1 + (1 − ) + (1 − ) + (1 − ) = 1 + ( ) + ( ) + ( )
16
64
64
16
64
64
64 + 52 + 74 190
=
64
64
𝑁−1
𝐸( 𝑥
2
(3) )
𝑁−1
= 2 ∑ 𝑥 [1 − 𝑃3 (𝑥)] + ∑[1 − 𝑃3 (𝑥)]
𝑥=0
3
∑ 𝑥 [1 − 𝑃3 (𝑥)] = 0. [1] + 1. [
𝑥=0
𝑥=0
13 49 75
52 + 98 + 75 225
+
+ ]=
=
64 32 64
64
64
𝑬 (𝒙𝟐 (𝟑)) = 2.
225 190 640
+
=
64
64
64
𝒗𝒂𝒓 (𝒙𝟑 ) = 𝐸(𝑥 2 (3)) − [𝐸(𝑥3 )]2
=
640 36100 4860
−
=
64
4096
4096
4 . Joint distribution of 𝒙(𝒊) 𝒂𝒏𝒅 𝒙(𝒋) , 𝒊 < 𝒋
BIGGERI, A. (1998 ), FISHER,R.A. (1941), Gupta ,S.Shanti .and S. Panchspakesan(1967)
Let 𝑝𝑖,𝑗 (𝑥, 𝑦)(𝑖 < 𝑗) be the probability that x(i) is equal to 𝑥 and 𝑥(𝑗) is equal to y and let
𝑝𝑖,𝑗 (𝑥, 𝑦) = 𝑝(𝑥(𝑖) ≤ 𝑥, 𝑥(𝑗) ≤ 𝑦). If 𝑥 ≥ 𝑦 ,
𝑝𝑖,𝑗 (𝑥, 𝑦) = 𝑝(𝑥(𝑗) ≤ 𝑦)
𝐵(𝑦)
𝑀
= 𝑗 ( ) ∫ 𝑢𝑖−1 (1 − 𝑢)𝑀−1 𝑑𝑢 )
𝑗
(4.1)
0
By repeated application of the results
𝑝
𝑛
𝑛+𝑠−1 𝑛
1
∑(
) 𝑝 (𝑘 − 𝑝)𝑠 =
∫ 𝑢𝑎−1 (1 − 𝑢)𝑛−1 𝑑𝑢
𝑠
𝐵(𝑎, 𝑛)
𝑠=𝑎
0
and
𝑝
𝑏
𝑛+𝑠−1 𝑛
1
∑(
) 𝑝 (𝑘 − 𝑝)𝑠 =
∫ 𝑢𝑏 (1 − 𝑢)𝑛−1 𝑑𝑢
𝑠
𝐵(𝑏 + 1, 𝑛)
𝑠=0
0
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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
Where 0 ≤ 𝑝 < 1 and 𝑝 < 𝑘 ; we obtain
𝒑𝒊,𝒋 (𝒙, 𝒚)
1
𝐵(𝑥)
𝐵(𝑥)
𝑀!
𝑗−𝑖−1
𝑀
𝑢𝑖−1 (1 − 𝑢)𝑀−1 𝑑𝑢 −
∫ 𝑑𝑣 ∫ 𝜔𝑖−1 ( 𝑣 − 𝜔)
= 𝑖( ) ∫
(𝑖 − 1)! (𝑗 − 𝑖 − 1)! (𝑀 − 1)!
𝑖
𝐵(𝑦)
0
0
(1 − 𝑣)𝑀−1 𝑑𝜔
(𝟒. 𝟐)
𝜔 = 𝑧𝑃(𝑥𝑖 ) + 𝑃(𝑥𝑖 − 1)
,
𝑣 = 𝑃(𝑥𝑗 ) − 𝑦 𝑃(𝑥𝑗 )
we get the relation
1
𝐵(𝑥)
𝑀!
∫ 𝑑𝑣 ∫0 𝜔𝑖−1 (𝑣
(𝑖−1)!(𝑗−𝑖−1)!(𝑀−𝑗)! 𝐵(𝑥)
− 𝜔)𝑗−𝑖−1 (1 − 𝑣)𝑀−1 𝑑𝜔 (4.3)
𝐵(𝑥)
𝐵(𝑥)
𝑀
𝑀
𝑖−1
𝑀−1
= 𝑖( ) ∫
𝑢 (1 − 𝑢)
𝑑𝑢 − 𝑖 ( ) ∫
𝑢 𝑗−1 (1 − 𝑢)𝑀−1 𝑑𝑢
𝑖
𝑖
0
0
= 𝐼1−𝐵(𝑥) (𝑖, 𝑀) − 𝐼1−𝐵(𝑥) (𝑗, 𝑀)
𝑁
(4.4)
𝑁
𝐸(𝑥(𝑖) 𝑥(𝑗) ) = ∑ ∑ 𝑥𝑦 𝑝𝑖,𝑗 (𝑥, 𝑦)
𝑥=0 𝑦=0
𝑁−1 𝑁
𝑁
= ∑ 𝑥 2 𝑝𝑖,𝑗 (𝑥, 𝑥) + ∑ ∑ 𝑥𝑦 𝑝𝑖,𝑗 (𝑥, 𝑦)
𝑥=0
(4.5)
𝑥=0 𝑦=𝑥+1
so
𝑁
𝑁−1
𝑁
𝑐𝑜𝑣(𝑥(𝑖) , 𝑥(𝑗) ) = ∑ 𝑥 2 𝑝𝑖,𝑗 (𝑥, 𝑥) + ∑ ∑ 𝑥𝑦 𝑝𝑖,𝑗 (𝑥, 𝑦)
𝑥=0
𝑁−1
𝑥=0 𝑦=𝑥+1
𝑁−1
−{∑[1 − 𝑝𝑖 (𝑥)]} {∑[1 − 𝑝𝑗 (𝑥)]} (𝟒. 𝟔)
𝑥=0
𝑥=0
5 . Distribution of 𝒀𝒊,𝒋 = 𝑿(𝒋) − 𝑿(𝒊) (𝒋 > 𝒊)
BIGGERI, A. (1998 ), FISHER,R.A. (1941), Gupta ,S.Shanti .and S. Panchspakesan(1967)
𝑌𝑖,𝑗 represents a generalized range and can take values 0,1, … . . , 𝑁 .for 𝑟 ≥ 0 ,
𝑃(𝑌𝑖,𝑗 = 𝑟) =
𝑀!
𝑖−1 (𝑉
∑𝑛−𝑟
− 𝜔)𝑗−𝑖−1 (1 − 𝑉)𝑀−1 𝑑𝑣𝑑𝜔 (5.1)
𝑘=0 (𝑖−1)!(𝑗−𝑖−1)!(𝑀−1)! ∫ ∫𝐴 𝜔
Where A is the region given by
𝑣≥𝜔
{ 𝐵(𝑘) ≥ 𝜔 ≥ 𝐵(𝑘 − 1)
𝐵(𝑘 + 𝑟) ≥ 𝑣 ≥ 𝐵(𝑘 + 𝑟 − 1)
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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(23): 2015
This can be rewritten as
𝑝{𝑌𝑖,𝑗 = 𝑟}
𝐵(𝑘)
𝑁
𝑀!
∑
(𝑖 − 1)! (𝑗 − 𝑖 − 1)! (𝑀 − 1)!
𝑘=0
=
𝑁−𝑟
𝐵(𝑘)
∫ 𝑑𝜔 ∫ 𝜔𝑖−1 (𝑉 − 𝜔)𝑗−𝑖−1 (1 − 𝑉)𝑀−1 𝑑𝑣 , 𝑟 = 0 (𝟓. 𝟐)
𝐵(𝑘−1)
𝐵(𝑘)
𝑀!
∑
(𝑖 − 1)! (𝑗 − 𝑖 − 1)! (𝑀 − 1)!
𝑘=0
𝜔
∫ 𝑑𝜔
𝐵(𝑘−1)
𝐵(𝑘+𝑟)
𝜔𝑖−1 (𝑉 − 𝜔)𝑗−𝑖−1 (1 − 𝑉)𝑀−1 𝑑𝑣 , 𝑟 > 0 )
∫
𝐵(𝑘+𝑟−1)
(𝟓. 𝟑)
{
so
𝑁
𝑁−𝑟
𝐸(𝑌𝑖,𝑗 ) = ∑ 𝑟 ∑
𝑟=1 𝑘=0
𝐵(𝑘+𝑟)
𝐵(𝑘)
∫ 𝑑𝜔
𝐵(𝑘−1)
𝑀!
(𝑖 − 1)! (𝑗 − 𝑖 − 1)! (𝑀 − 1)!
(5.4)
𝜔𝑖−1 (𝑉 − 𝜔)𝑗−𝑖−1 (1 − 𝑉)𝑀−1 𝑑𝑣
∫
(5.5)
𝐵(𝑘+𝑟−1)
But we also know that
𝐸(𝑌𝑖,𝑗 ) = 𝐸(𝑋(𝑗) ) − 𝐸(𝑋(𝑖) )
𝐵(𝑥)
𝑁−1
𝑁−1
𝐵(𝑥)
𝑥=0
0
𝑀
𝑀
= ∑ 𝑖 ( ) ∫ 𝜔𝑖−1 (1 − 𝜔)𝑀−1 𝑑𝜔 − ∑ 𝑗 ( ) ∫ 𝜔 𝑗−1 (1 − 𝜔)𝑀−1 𝑑𝜔
𝑖
𝑗
𝑥=0
𝑁−1
0
= ∑[𝐼1−𝐵(𝑥) (𝑖, 𝑀) − [𝐼1−𝐵(𝑥) (𝑗, 𝑀)]
(5.6)
𝑥=0
(5.5) and (5.6) lead to the relation
𝑁
𝑛−𝑟
∑ ∑𝑟
𝑟=1
𝐵(𝑘)
∫ 𝑑𝜔
𝑀!
∙
(𝑖 − 1)! (𝑗 − 𝑖 − 1)! (𝑀 − 1)!
𝑘=0
𝐵(𝑘+𝑟)
∫
𝜔𝑖−1 (𝑉 − 𝜔)𝑗−𝑖−1 (1 − 𝑉)𝑀−1 𝑑𝑣
𝐵(𝑘−1)
𝐵(𝑘+𝑟−1)
𝑁−1
= ∑[𝐼1−𝐵(𝑥) (𝑖, 𝑀) − [𝐼1−𝐵(𝑥) (𝑗, 𝑀)]
(5.7)
𝑥=0
6. Conclusions
A . Since 𝑃(𝑁) = 1 ,
𝑁
𝑁−1
𝐸(𝑥𝑖 ) = ∑ 𝑥𝑖 𝑝 ∗ (𝑥𝑖 ) = 𝑁 − ∑ 𝑃 ∗ (𝑥𝑖 )
𝑥𝑖 =0
𝑥𝑖 =0
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where 𝑃 ∗ (𝑥𝑖 ) is defined by
𝐵(𝑥)
𝑝𝑖 (𝑥) = 𝑖(𝑀𝑖) ∫0 𝜔𝑖−1 (1 − 𝜔)𝑀−1 𝑑𝜔 = 𝐼1−𝐵(𝑥) (𝑖, 𝑀) .
Hence, we have
𝑁−1
𝐸(𝑥𝑖 ) = ∑ {1 − 𝑝 ∗ (𝑥𝑖 )}
𝑥𝑖 =0
For the variance of 𝑥𝑖 , we have
𝑁
𝑁−1
𝐸(𝑥𝑖 2 ) = ∑ 𝑥𝑖 2 𝑝 ∗ (𝑥𝑖 ) = 𝑁 2 − ∑ (2𝑥𝑖 + 1)𝑃 ∗ (𝑥𝑖 )
𝑥𝑖=0
𝑥𝑖 =0
That is,
𝑁−1
𝐸(𝑥𝑖
2)
= 2 ∑ 𝑥𝑖 {1 − 𝑃 ∗ (𝑥𝑖 )} + 𝐸(𝑥𝑖 )
𝑥𝑖=0
Hence, the variance of 𝑥𝑖 , is
𝑁−1
𝑉(𝑥𝑖 ) = 2 ∑ 𝑥𝑖 {1 − 𝑃 ∗ (𝑥𝑖 )} + 𝐸(𝑥𝑖 ){1 − 𝐸(𝑥𝑖 )}
𝑥𝑖=0
B . The incomplete beta function of Negative binomial distribution is
𝑰𝟏−𝑩(𝒙) (𝒊, 𝑴)
But
The incomplete beta function of binomial distribution is
𝐼𝐵(𝑥) (𝑖, 𝑀 − 𝑖 + 1)
7. References
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BINNS, M. R. 1986: “Behavioral dynamics and the negative binomial distribution”, Oikos,
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BLISS, C. I. and R.A. FISHER 1953: “Fitting the negative binomial distribution to biological
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