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On Size-Biased Logarithmic Series Distribution and Its Applications

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On Size-Biased Logarithmic Series Distribution and Its Applications 71 The Open Statistics and Probability Journal, 2009, 1, 71-75 Open Access On Size-Biased Logarithmic Series Distribution and Its Applications Khurshid Ahmad Mir* Department of Statistics, Govt. College (Boys), Baramulla, Kashmir, India Abstract: In this paper, a size-biased logarithmic series distribution (SBLSD), a particular case of the weighted logarith- mic series distribution, taking the weights as the variate values is defined. The moments and recurrence relation of (SBLSD) are obtained. Negative moments and inverse ascending factorial moments of the size-biased logarithmic series distribution have been derived in terms of hyper-geometric function. Recurrence relations for these moments have also been derived using properties of hyper-geometric functions. Different estimation methods for the parameter of the model are discussed. R- Software has been used for making a comparison among the three different estimation methods and with the logarithmic series distribution. Key Words: Size-biased logarithmic series distribution, Negative moments, Inverse ascending factorial moments, Bayes’ esti- mator, Beta distribution, R-Software. 1. INTRODUCTION The variance of LSD is given as The logarithmic series distribution (LSD) characterized 3 2 μ2 = ()1[12]+ (6) by a parameter is given by x In this paper, we have made an attempt to study proposed 1 PX()= x = ; x= 1, 2….. (1) size-biased LSD, its moments and recurrence relations. log(1 )x Negative moments and inverse ascending factorial moments of the size-biased logarithmic series distribution have been The model (1) is a limiting form of zero-truncated nega- derived in terms of hyper-geometric function. Recurrence tive binomial distribution. Negative moments of discrete relations for these moments have also been derived using distributions, mainly the binomial, Poisson and negative bi- nomial have been investigated by various authors [Stephan properties of hyper-geometric functions. In order to make a [1], Grab and Savage [2], Mendenhall and Lehman [3], Go- comparative analysis among the three estimation methods for vindarajulu [4,5], Tiku [6], Stancu [7], Chao and Strawder- the parameter of the size-biased logarithmic series distribu- man [8], Gupta [9,10], Cressie et al. [11], Cressie and Bork- tion (SBLSD), one of the standard software packages R- ent [12] and Roohi[13]. Inverse ascending factorial moments Software is used which is meant for data analysis and graph- have only been dealt with by Lepage [14] and Jones [15]. ics. Best et al. [16] discussed the test of fit for the model (1). 2. SIZE-BIASED LOGARITHMIC SERIES DISTRI- Sadinle [17] linked the negative binomial distribution with BUTION (SBLSD) the logarithmic series and Shanumugam [18] studied the characterization of model (1). A brief list of authors and their A size-biased logarithmic distribution (SBLSD) is ob- works can be seen in Johnson, Kotz and Kemp [19]. tained by taking the weight of the LSD (1) as x. The first four moments of LSD are given as We have from (1) and (2) . 1 1 μ1 = (1) , where (2) xPX ()== x , = x=1 1 log(1 ) 3 μ2 = (1 ) (1 ) (3) x = 52 x=1 1 μ3 = (1 ) (1 )(1 ) (4) This gives the size-biased logarithmic series distribution μ = (1)(1)27432+ 6++ 21 (5) 4 () (SBLSD) as 21x 1 PX11[]== x {[1,;;]} F AA; x=1, 2… (7) *Address correspondence to this author at the Department of Statistics, 21 Govt. College (Boys), Baramulla, Kashmir, India; Tel: 91-1942426584; Where 01<< and {[1,;;]}(1)FAA1 = Fax: 91-19424265; E-mail: [email protected] 1876-5270/09 2009 Bentham Open On Size-Biased Logarithmic Series Distribution The Open Statistics and Probability Journal, 2009, Volume 1 72 2.1. Moments 1 μμ10()s==, where () s 1 th (1 ) The r moment μr ()s of SBLSD (7) about origin is ob- tained as The second moment of (7) about origin can also be ob- tained by using the relation (13) sEXrr xPXx μr ()==() 1 [ = ] ; r=1,2… 4. NEGATIVE MOMENTS AND INVERSE ASCEND- x=1 ING FACTORIAL MOMENTS xrx1 Theorem I: Suppose the random variable X has a size- μr ()s = (1 ) (8) x=1 biased logarithmic series distribution with parameter , then the relation obviously μ0 ()s = 1 and for r 1 EX(+=+ A )12 {( A 1) F[1, AA ; ; ] } -12 F[1, A ++ 1; A 2; ] 1 11 sxrx μr ()= holds. x=1 Proof: Since X has a size-biased logarithmic series distribu- 1 rx tion, then = x x x =1 1 EX A12 F AA 1 ()+= {[1,;;]}1 1 r +1 xA == xPXx[] x=1 ()+ x=1 212 =+{(AFAAFAA 1) 11[1, ; ; ]} [1, ++ 1; 2; ] (14) 1 μμrr()s = +1 (9) This completes the proof. th where μr+1 is the (r + 1) moments about origin of LSD (1). Theorem II: Suppose the random variable X has a size- biased logarithmic series distribution with parameter , then The moments of SBLSD can be obtained by using relations the kth inverse ascending factorial moment is given as (2) to (4) in (9) μ =+{(k 1)212 F [1, A ; A ; ]} F[1,1; k + 2; ], k = 1,2... and 0<<1. 1 []k 1 1 s k μ1 () = (10) 1 ()1 Proof: E Here μ[]k = [ i=1 Xi+ Using relation (4) in (9), we get x1 2 = {[1,;;]}21FAA ()1 ()1 1 s x=1 (xx++ 1)( 2).......( xk + ) μ2 ()= 4 (11) ()1 1.22 =+++ {21FAA [1, ; ; ]} [ ....] Which gives the variance of SBLSD (7) as 1 (k+1)! (kk++ 2)! ( 3)! 1 s On simplification, we get μ2 ()= 2 (12) ()1 212 μ[]k =+{(kFAAFk 1) 1[1, ; ; ]} 1[1,1; + 2; ] (15) The higher moments of SBLSD (7) about origin can also be obtained by using (9) if so desired. 5. RECURRENCE RELATION FOR NEGATIVE AS- CENDING FACTORIAL MOMENTS 3. RECURRENCE RELATION OF MOMENTS ABOUT ORIGIN OF SIZE-BIASED LSD Theorem III: Suppose the random variable X has a size- biased logarithmic series distribution with parameter and The recurrence relation can be obtained by differentiating (8) μ[]k is the kth inverse ascending factorial moment of X, as then the relation μ s 2 r () rx 1 (k +1) μ = [ k +1 1 + k]μ + = x {}(1 ) [k+1] ()() [k] x=1 1 μ , k = 2,3...and 0<<1 ()[k1] 11 1 = μμrr+1 ()ss() μ r () s (1 ) holds. Proof: we know that μ s r () 1 1 2 1 2 μrr+1 ()ss=+ μ () (13) μ =+{(kFAAFk 1) [1, ; ; ]} [1,1; + 2; ] (1 ) []k 1 1 1 2 1 2 μ[1]k + =+{(kFAAFk 2) 1 [1, ; ; ]} 1[1,1; + 3; ] For r = 0, we get Using the identity (see Rainville [20], page 71) 73 The Open Statistics and Probability Journal, 2009, Volume 1 Khurshid Ahmad Mir a c +1 2 F [a,b;c; z] = a 2 F [a +1, b;c; z] (c 1) 2 F [a,b;c 1; z], ()1 1 1 (16) for a=1,b=1,c=k+3,z=, we get k + 2 1 2 F k 2 F k 2 F k 1[1,1; + 3; ] = 1[1,1; + 2;] 1[2,1; + 3; ] (17) k +1 k +1 Now again using the identity given by (Rainville [20], page 71) c b z 2 F a b c z = a 2 F a b c z z 2 F a b c + z for ()1 1[ , ; ; ] 1[ 1, ; ; ] ( ) 1[ , ; 1; ], a=2,b=1,c=k+2,z= , we get c k + 2 ()()k + 21 22Fk[1,2;3;]+= Fk[1,1;2;] + 2 Fk[2,1;2;]+ (18) 11()kk++11() 1 Substituting (18) in (17) we get k + 2 1 k + 2 1 2 F k + = 2 F k + + ()()2 F k + 1[1,1; 3; ] 1- 1[1,1; 2; ] 2 1[2,1; 2; ] k +1 ()k+1 k +1 () Again using (5.1), for a=1, b=1, c=k+2 and z=, we have k + 2 1 k 1 k + 2 1 2 F k () 2 F k ()()2 F k 1[2,1; + 2;] = 1- 1[1,1; + 2;]+ 1[1,1; +1; ] k +1 ()k+1 ()k +1 ()k +1 k + 21kk()121()()+ k1 2 FAA 1 2 Fk 2 Fk μ[1]k + =+{( 2) 1 [1, ; ; ]} [ 1- 11[1,1;++ 2; ] [1,1; + 1; ]] kk++1k+11 () () () k + 1 1 This gives ˆ = 1 (20) x 21kk+ ()1 μμμ[1]kkk+ =+[]22 [] [1] ()kk++11() 6.2. Method of Maximum Likelihood 2 (k +1) μ = [ k +1 1 + k]μ + 1 μ , k = 2,3... (19) Let XX, ,..... Xn be a random sample from size-biased [k+1] ()() [k] ()[k1] 12 logarithmic series distribution, then corresponding likelihood 22 Where Fk11[][1,1;+=+ 2; ]() k 1 ! FAA[1, ; ; ]μ k and function is given as 22 Fk11[1][1,1;+= 1; ]() kFAA ! [1, ; ; ]μ k n xni L = ()1 ; x=1, 2… (21) 6. ESTIMATION METHODS L n yn where y x In this section, we discuss the various estimation methods = ()1 , = i (22) for size-biased logarithmic series distribution and verify their efficiencies. The log likelihood function can be written as logLn= log(1 )+ ( yn ) log and the likelihood equa- 6.1. Method of Moments tion as In the method of moments replacing the population mean log Ln()yn 1 =+=0 μ1 = by the corresponding sample mean ()1 ()1 n On solving, we get the maximum likelihood estimate as 1 xx= i , we get 1 n ˆ = 1 , which coincides with the moment estimate. i=1 x On Size-Biased Logarithmic Series Distribution The Open Statistics and Probability Journal, 2009, Volume 1 74 6.3. Bayesian Method of Estimation 7. COMPUTER SIMULATION AND CONCLUSIONS Since 01<< , therefore we assume that prior informa- It is very difficult to compare the theoretical perform- tion about is from beta distribution.
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