Semi Bayes Estimation of the Parameters of Binomial Distribution in the Presence of Outliers

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1 Journal of Statistical Theory and Applications Volume 11, Number 2, 2012, pp ISSN Semi Bayes Estimation of the Parameters of Binomial Distribution in the Presence of Outliers U. J. Dixit Shailaja G. Kelkar Department of Statistics, University of Mumbai, Mumbai-India Abstract Semi Bayes estimation of the Binomial distribution of n based on some prior distribution is studied in the presence of outliers when probability of success is known and unknown. By simulation study it is conjectured that some prior distributions are more useful with respect to the generalized variance of the semi bayes estimator of n and p. Key Words: Binomial distribution, Bayes estimators, outliers. AMS Subject classification:62f10 ulhasdixit@yahoo.co.in kelkar shailaja@yahoo.com

2 U. J. Dixit and S. G. Kelkar Introduction Hamedani and Walter1988) obtained Bayes estimate of the binomial parameter n based on a general prior distribution of n. As special cases such as improper prior and Poisson prior were also considered and formulae for the marginal and posterior distributions were obtained. The standard problem associated with the binomial distribution is that of estimating its probability of success, p. A much less well studied and considerably harder problem is that of estimating the number, n. The estimation of the parameter n is not so straightforward nor well-known and does not appear to have a very large literature. Consider a problem of shooter in Target archery. A shooter needs to aim at the target and hit precisely. To shoot the arrow, the shooter has to adopt the correct stance. He should place the body at a perpendicular angle to the shooting line with the feet apart at shoulder-width distance. The type of stance that the shooter adopts is vital to his performance. There are four types of stances, even, open, close and oblique stance. A shooter practices various stances by marking the exact placement of his feet on the shooting line. Deviation of even by a couple of inches can wreck a shooters aiming and sighting; and it is needless to say that it can begin to plague him with accuracy problems. Let X 1,X 2,...,X m be the number of successful attempts made by the shooter while hitting the target. The shooter has changed his stance on some attempts, say kknown) which results into different probability of success. It implies that out of m attempts, in m k) attempts the probability of success is p and in the remaining k attempts the probability of success is changed to αp, where α > 0, 0 < αp < 1 and α is unknown. Thus, we assume that the random variables X 1,X 2,...,X m are such that k of them are distributed with probability mass function gx, n, p, α), where gx,n,p,α) = Cn,x).αp) x q 1 n x,x = 0,1,...,n, 0 < p < 1, 0 < αp < 1, q 1 = 1 αp, 1) and the remaining m k) random variables are distributed with probability mass function fx,n,p), where fx,n,p) = Cn,x)p x q n x ; x = 0,1,...,n, 0 < p < 1, q = 1 p. 2)

3 Binomial Distribution in the Presence of Outliers 145 In this paper, we have obtained the Bayes estimate of n when the prior distribution of n is i)improper prior ii) Poisson iii)negative binomial, when p is known and unknown. 2 Improper prior Of n We will consider the joint distribution of X 1,X 2,...,X m ) in the presence of k outliers. fx 1,x 2,...x m, n,p) = [Cm,k)] 1 p T q mn T q 1 q )nk) Gx,α,p) where Cm,k) = m! m k)!k!, Gx,α,p) = m k+1 A 1 =1 and T = m x i, see Dixit1989). m k+2 A 2 =A m A k =A k 1 +1 Now we will consider the prior distribution of n as gn), where The joint distribution of X 1,X 2,...,X m ) is fx 1,x 2,...x m,p) = where X = k x A i, and n=x m [Cm,k)] 1 Let n X m) = j n = X m) +j. Consider m n=x m) m k+1 = Cn,x i) )q mn T q 1 q )nk) = A 1 =1 gn) = 1 n. m m k+2 A 2 =A 1 +1 m j=0 m Cn,x i ) 3) αq q 1 ) k x A i, Cn,x i )p T q mn T q 1 q )nk)... m A k =A k αq q 1 ) X gn), 4) Cx m) +j,x i) )q m k q k 1) j q m k)xm T q x m)k 1, 5) = ) Γxm) +j +1) m [ m 1 j=0 [ m 1 Γx m) +1) Γx m) x i) +1) Γx m) x i) +j +1) ] [Γx m) +1)] Γx i) +1)Γx m) x i) +1) q m k q1 k)j, 6) j! ]

4 U. J. Dixit and S. G. Kelkar 146 = m where ) Cx m),x i) ) m F m 1 x m) +1) m ;x m) x 1 +1,...,x m) x m 1) +1;q m k q1) k, 7) x m) +1) m = x m) +1,..m times)...,x m) +1, and mf m 1.) is a hypergeometric function. fx,p) = [Cm,k)] 1 p T q m k)xm) T q x m 1 m)k) αq 1 ) X Cx q m),x i) ) 1 ) m F m 1 x m) +1) m ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1) k 8) En) = x m) + = x m) + n=x m) n x m)) m Cn,x i))p T q mn T q 1 q ) nk αq fx,p) q 1 ) X n=x m) m 1 Cn,x i)) n! x m)!n x m) 1)! pt q mn T q 1 q ) nk αq q 1 ) X Let n X m) 1 = j n = X m) +j +1, = x m) + j=0 m 1 fx,p) Cx m) +j +1,x i) ) x m)+j+1)! x m)!j! fx,p) p T q m k q k 1 )j q m k)x m)+1) T q x m)+1)k 1,,, Consider = x m) + x m) +1) j=0 Cx m) +j +1,x m) +1) fx,p) j=0 m 1 Cx m) +j +1,x i) )p T q m k q k 1) j q m k)x m)+1) T q x m)+1)k 1. m 1 Cx m) +j +1,x m) +1) Cx m) +j +1,x i) )q m k q1) k j = j=0 Γx m) +2) ) Γxm) +j +2) m [ m 1 Γxm) x i) +2) [ m 1 Γx m) x i) +j +2) ) Γx m) +2) ] q m k q1 k)j, Γx i) +1)Γx m) x i) +2) j! ) ] m 1 ) ˆn B = x m) +x m) +1)q m k q1 k xm) +1 x m) x i) +1 m F m 1 xm) +2) m ;x m) x 1) +2,...,x m) x m 1) +2;q m k q1 k ) ). mf m 1 xm) +1) m ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1 k 9)

5 Binomial Distribution in the Presence of Outliers 147 For homogeneous case consider α = 1 in 9) When m = 1 in 10), m 1 ) ˆn B = x m) +x m) +1)q m xm) +1 x m) x i) +1 mf m 1 xm) +2) m ;x m) x 1) +2,...,x m) x m 1) +2;q m) mf m 1 xm) +1) m ;x m) x 1) +1,...,x m) x m 1) +1;qm). 10) ˆn B = x 1 +q. 11) p The expression in 11) is given Hamedani and Walter1988)Page:1832). Case1) p is known and α = 1 We can estimate ˆn B from 10). Case2) p is unknown and α = 1 By using moment estimate of p, where m i = m 1 m j=1 x j i ; i = 1,2 Then we can estimate ˆn B from 10). Case3) p and α are known We can estimate ˆn B from 9). Case4) p and α are unknown From 3) ˆp = 1 m 2 m 1 )2 m, 12) 1 lnlp,α) T lnp+mn T)lnq +nk[lnq 1 lnq]+ln αq q 1 ) X, 13) Let αq q 1 ) X = S, X αq q 1 ) X = S x,and X 2 αq q 1 ) X = S xx. The likelihood Equations are as follows: T p mn T q +nk[ α + 1 α 1S x ]+ q 1 q qq 1 S = 0, 14) nkp + 1 S x q 1 αq 1 S = 0. 15)

6 U. J. Dixit and S. G. Kelkar 148 From 14) and 15) we get one more equation m 1 = npbα+ b), 16) write 15) as nkpα S x S = 0. 17) Let t 1 p,α) = x npbα+ b), 18) and t 2 p,α) = S x S nkpα. 19) Expanding the equations 18) and 19) by Taylor series around p 0,α 0 ), t 1 p,α) = t 1 p 0,α 0 )+ pt 1pp 0,α 0 )+ αt 1αp 0,α 0 )+om 1 ), 20) and t 2 p,α) = t 2 p 0,α 0 )+ pt 2pp 0,α 0 )+ αt 2αp 0,α 0 )+om 1 ), 21) where α = α α 0, p = p p 0, t 1p p,α) and t 1α p,α) are the first order derivative of t 1 p,α) with respect to p and α respectively. Similarly t 2p p,α) and t 2α p,α) are the first order derivative of t 2 p,α) with respect to p and α respectively. t 1p = nbα+ b), and Write 18) and 19) as follows t 1α = npb, t 2p = α 1 qq 1 [ S xx S S x S )2 ] nkα, t 2α = 1 αq 1 [ S xx S S x S )2 ] nkp. t 1 p,α) = c 1 +g p+h α = 0, 22)

7 Binomial Distribution in the Presence of Outliers 149 and t 2 p,α) = c 2 +e p+f α = 0. 23) Solving 22) and 23) and α = c 2g +c 1 e fg eh, p = c 2h+c 1 f eh fg. The corrections p and α are obtained such that we may expect p = p+p 0, α = α+α 0. Selectthe initial solutions p 0 and α 0 then from 9) calculate ˆn B and call it as n 0. After iterations we will get p and α. Kale1962) had considered the multiparameter case and shown that this method is justified. From that we will get ˆn B. For details, see Dixit and Kelkar2011). 3 Prior of n is Poissonλ) Let gn) = e λ λ n, n = 0,1,2,..., λ > 0. 24) n! The joint probability distribution of X 1,X 2,...,X m ) is fx 1,x 2,...x m,n,p) = [Cm,k)] 1 m Cn,x i )p T q mn T q 1 q )nk) αq ) X e λ λ n, 25) q 1 n! fx 1,x 2,...x m,p) = n=x m) [Cm,k)] 1 m Cn,x i )p T q mn T q 1 q )nk) αq ) X e λ λ n, 26) q 1 n!

8 U. J. Dixit and S. G. Kelkar 150 After simplification fx,p) = [Cm,k)] 1 p q )T e λ q mx m) q m 1 1 αq q )x m)k ) X Cx q m),x i) )[x m)!] 1 1 ) m 1 F m 1 x m) +1) m 1 ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1λ) k. 27) Bayes estimate of n in squared error loss function is m 1 ) ˆn B = x m) +q m k) q1 k xm) +1 x m) x i) +1 m 1 F m 1 xm) +2) m 1 ;x m) x 1) +2,...,x m) x m 1) +2;q m k q1 k λ) m 1F m 1 xm) +1) m 1 ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1 k 28) λ). For homogeneous case α = 1, ˆn B = x m) +q m λ When m = 1, m 1 xm) +1 x m) x i) +1 ) m 1 F m 1 xm) +2) m 1 ;x m) x 1) +2,...,x m) x m 1) +2;q m λ ) ). m 1F m 1 xm) +1) m 1 ;x m) x 1) +1,...,x m) x m 1) +1;q m 29) λ ˆn B = x 1 +qλ. 30) The expression in 30) is given Hamedani and Walter1988)Page:1834). Case1) p, λ are known and α = 1 We can estimate ˆn B from 29). Case2) p is unknown, λ is known and α = 1 By using moment estimate of p from 12) and then estimate ˆn B from 29). Case3) p, α and λ are known We can estimate ˆn B from 28). Case4) p and α are unknown, λ is known By using method of mle expressions from 13) to 23), we can estimate ˆn B. 4 Prior of n is Negative Binomialθ, r) gn) = Γr +n) Γr)Γn+1) 1 θ)r θ n, n = 0,1,2,...,r = 1,2,3..., θ > 0. 31)

9 Binomial Distribution in the Presence of Outliers 151 The joint probability distribution of X 1,X 2,...,X m ) is fx 1,x 2,...x m,n,p) = [Cm,k)] 1 m fx 1,x 2,...x m,p) = After simplification Cn,x i )p T q mn T q 1 q )nk) αq q 1 ) X Γr +n) Γr)Γn+1) 1 θ)r θ n, 32) n=x m) [Cm,k)] 1 m Cn,x i )p T q mn T q 1 q )nk) αq q 1 ) X Γr +n) Γr)Γn+1) 1 θ)r θ n, 33) m 1 fx,p) = [Cm,k)] 1 Cr +x m) 1,x m) ) Cx m),x i ) p q )T 1 θ) r q m k) q1θ) k x αq m) ) X q 1 ) m F m 1 x m) +1) m 1,r +x m) ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1θ) k.34) Bayes estimate of n in squared error loss function is m 1 ) xm) ) ˆn B = x m) +q m k) q1θ k x m) +1 +r x m) x i) +1 r m F m 1 xm) +2) m 1,r+x m) +1;x m) x 1) +2,...,x m) x m 1) +2;q m k q1 k θ) mf m 1 xm) +1) m 1,r+x m) ;x m) x 1) +1,...,x m) x m 1) +1;q m k q1 k.35) θ) For homogeneous case, α = 1 m 1 ) xm) ) ˆn B = x m) +q m x m) +1 +r θ x m) x i) +1 r m F m 1 xm) +2) m 1,r+x m) +1;x m) x 1) +2,...,x m) x m 1) +2;q m θ ) mf m 1 xm) +1) m 1,r +x m) ;x m) x 1) +1,...,x m) x m 1) +1;q m θ ). 36) Case1) p, θ, r are known and α = 1 We can estimate ˆn B from 36). Case2) p is unknown, θ, r are known and α = 1 By using moment estimate of p from 12) and then estimate ˆn B from 36). Case3) p, α, θ and r are known We can estimate ˆn B from 35). Case4) p and α are unknown, θ and r are known By using method of mle expressions from 13) to 23), we can estimate ˆn B.

10 U. J. Dixit and S. G. Kelkar Numerical Study Our all results of estimation of n are same as Draper and Guttman1971) and Hamedani and Walter1988) in case of improper and poisson prior λ=13) of n. i.e. In homogeneous case when improper and poisson priors are used the estimator of n is 13 for m=2, x 1 =10, x 2 =12 and p=0.8. Moreover we also have used Negative Binomial prior for n and the estimate of n is 12 where m=2, x 1 =10, x 2 =12, p=0.8, θ=0.6 and r=3. In order to get the idea of selecting prior of n, we have generated one thousand samples of size m = 1020)50 from the Binomial distribution with n = 10, p=.4,.5,.6 and.8, α=.4,.5,.6 and.8 and k=0,1,2,3 for the following three prior distributions of n, i)g 1 n) = 1 for all n ii)g 2 n) = e λ λ n n! n = 0,1,2,..., λ = 2 and 3 iii)g 3 n) = Γr+n) ΓrΓn+1) 1 θ)r θ n n = 0,1,2,...,r = 1,2,3, θ =.4 and.6 We would like to comment on bias of ˆn, ˆp and ˆα. A)Homogeneous case k = 0): 1.Bias of ˆn, ˆp need not be positive for m = 1010)50. 2.Bias of ˆp is more for m=10, 20 for an improper prior distribution of n as compared to the other prior distributions of n. 3.Bias of ˆp is almost same for m=30 for all prior distributions of n. 4.Bias of ˆp is less for m=40, 50 for an improper prior distribution of n as compared to the other prior distributions of n. 5.We can not draw any such conclusions in case of bias of ˆn for all prior distributions of n. 6.For m > 50 the bias of ˆn is less than 1 and bias of ˆp tends to zero. B)Outliers case k > 0): We have plotted graphs for various values of p,α) for n = Bias of ˆp is always positive for all prior distributions of n. 2.Bias of ˆn and ˆα need not be positive for all prior distributions of n. 3.Bias of ˆn, ˆp and ˆα is more in case of improper prior distribution of n as compared to the Poison and Negative Binomial prior distributions of n.

11 Binomial Distribution in the Presence of Outliers Interestingly for k=3, bias of ˆα is almost same for m=1010)50. 5.For m > 50 bias of ˆn is less than 1, bias of ˆp and bias of ˆα tends to zero. In case of multiparametric case we have to compare the efficiencies of estimators on the basis of the determinant of variance-covariance matrix. Here we have calculated the determinant of variance-covariance matrix of ˆp, ˆα,ˆn) under g 1, g 2 and g 3. In homogeneous case i.e.k = 0 and α = 1, we have presented determinant in Tables 1-4 for m = 1020)50 with n = 10, p=.4,.5,.6 and.8. Determinant of variance-covariance matrix under g 1 is more as compared to under g 2 and g 3.We can conjecture that estimates of n and p under g 2 and g 3 are better than under g 1. But there is no significant difference in the determinant under g 2 and g 3. For outlier case i.e.k=1 and 3, we have plotted the generalized variance-covariance matrix. Determinant under g 1 is significantly less than the determinant under g 2 and g 3. In this case estimates of p, α and n are better under g 1 than the estimates of p, α and n under g 2 and g 3. Here also determinant does not change significantly under g 2 and g 3.

12 U. J. Dixit and S. G. Kelkar 154 Appendix Determinant of ˆp and ˆn Table 1.λ = 2 θ = 0.4 r=1 m Prior p = 0.4) p = 0.5) p = 0.6) p = 0.8) 10 Improper Poisson NegBin Improper Poisson NegBin Improper Poisson NegBin Table 2.λ = 2 θ = 0.4 r=3 m Prior p = 0.4) p = 0.5) p = 0.6) p = 0.8) 10 Improper Poisson NegBin Improper Poisson NegBin Improper Poisson NegBin

13 Binomial Distribution in the Presence of Outliers 155 Determinant of ˆp and ˆn Table 3.λ = 3 θ = 0.6 r=1 m Prior p = 0.4) p = 0.5) p = 0.6) p = 0.8) 10 Improper Poisson NegBin Improper Poisson NegBin Improper Poisson NegBin Table 4.λ = 3 θ = 0.6 r=3 m Prior p = 0.4) p = 0.5) p = 0.6) p = 0.8) 10 Improper Poisson NegBin Improper Poisson NegBin Improper Poisson NegBin

14 U. J. Dixit and S. G. Kelkar 156 Bias of nˆ n=10 k=1!" #!$%&' r=3 n=10 k=3!"'' #!$%&''' r=3

15 Binomial Distribution in the Presence of Outliers 157 Bias of pˆ n=10 k=1!" #!$%&' r=3 n=10 k=3!"'' #!$%&''' r=3

16 U. J. Dixit and S. G. Kelkar 158 Bias of ˆ n=10 k=1!" #!$%6 r=3 n=10 k=3!"& #!$%6 r=3

17 Binomial Distribution in the Presence of Outliers 159 Generalized variance ofpˆ,ˆ and nˆ n=10 k=1 =2 r= 1 =0.4 n=10 k=1 =2 r= 3 =0.4

18 U. J. Dixit and S. G. Kelkar 160 Generalized variance ofpˆ, ˆ and nˆ n=10 k=1!=3 r= 1 "=0.6 n=10 k=1!=3 r= 3 "=0.6

19 Binomial Distribution in the Presence of Outliers 161 Generalized variance ofpˆ,ˆ and nˆ n=10 k=3 =2 r= 1 =0.4 n=10 k=3 =2 r=3 =0.4

20 U. J. Dixit and S. G. Kelkar 162 Generalized variance ofpˆ,ˆ and nˆ n=10 k=3 =3 r= 1 =0.6 n=10 k=3 =3 r= 3 =0.6

21 Binomial Distribution in the Presence of Outliers 163 Acknowledgment The authors are thankful to the editors and the referees for their valuable comments. References [1] Dixit, U.J.1989).Estimation of parameters of the gamma distribution in the presence of outliers. Commun.Statist Theor. Meth., 18: [2] Dixit, U.J. and Kelkar, S.G.2011).Estimation of the Parameter of Binomial Distribution in the Presence of Outliers.Accepted for publication in Journal of Applied Statistical Sciences. [3] Draper, N. and Guttman, I.1971).Bayesian estimation of the Binomial parametertechnometrics, 13: [4] Hamedani, G.G. and Walter, G.G.1988).Bayes estimation of the binomial parameter. ncommun.statist Theor. Meth., 176): [5] Kale, B.K.1962).On the solution of the likelihood equation by iteration process-the multiparametric case.biometrika,49: