ON INTERVENED NEGATIVE BINOMIAL DISTRIBUTION AND SOME OF ITS PROPERTIES

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1 STATISTICA, anno LII, n. 4, 01 ON INTERVENED NEGATIVE BINOMIAL DISTRIBUTION AND SOME OF ITS PROPERTIES 1. INTRODUCTION Since 1985, intervened tpe distributions have received much attention in the literature. These tpes of distributions provide information on the effectiveness of various preventive actions taken in several areas of scientific research. (Shanmugam, 1985) introduced an intervened Poisson distribution (IPD) as a replacement for positive Poisson distribution where some intervention process changes the mean of rare events. The IPD has been further studied b (Shanmugam, 199), (Huang and Fung, 1989) and (Dhanavanthan, 1998, 000). (Scollnik, 006) developed the intervened generalized Poisson distribution and (Kumar and Shibu, 011) considered a modified version of IPD. (Bartolucci et al., 001) studied an intervened version of geometric distribution with following probabilit mass function (p.m.f.). (1 )(1 ) ( ) 1 1 p P (1 ) 1, (1) in which 1,,3,..., 0 1 and 0. Through this paper we develop an intervened version of negative binomial distribution, which we called as intervened negative binomial distribution (INBD). Negative binomial distribution have found applications in several areas of research such as accidental statistics, birth and death process, medical sciences, pscholog etc. For details see (Johnson et al., 005). In most of these fields there are situations in which observations commence onl when at least one event occurs due to some intervention. For this reason a distribution with support {1,,3,...} is quite relevant and hence we develop a distribution which is suitable for accommodating such intervention mechanisms. The paper is organized as follows. In section we present a model leading to INBD and obtain epression for its probabilit mass function, mean, variance and

2 396 factorial moments. We also obtain a recurrence relation for probabilities of INBD in section. In section 3 we consider the estimation of parameters of INBD b various methods of estimation such as method of factorial moments, method of mied moments and method of maimum likelihood. A real life data is considered for illustrating these procedures.. INTERVENED NEGATIVE BINOMIAL DISTRIBUTION Let Y be a random variable having zero truncated negative binomial distribution with following p.m.f., in which 0 1, r > 0 and = 1,,3,.... r 1 r p( ;, r) P( Y ) (1 ) r 1, () r 1 where [1 (1 ) ]. The characteristic function of Y is r it r Y () t (1 )[(1 e ) 1] (3) Let Z be a random variable following negative binomial distribution with p.m.f. z r 1 r z f( z;,, r) P( Z z) (1 ) ( ), r 1 (4) in which z = 0,1,,..., > 0, r > 0 and 0 such that 0 1. The characteristic function of Z is () (1 )(1 r e it ) r. (5) Z t Assume that Y and Z are statisticall independent. Define = Y + Z. Then the characteristic function of is () t (). t () t where Y Z (,, r)((1 e it ) r 1)(1 e it ) r. (6) (,, r ) [(1 )(1 )] r The distribution of a random variable whose characteristic function is (6) we call the intervened negative binomial distribution or in short the INBD. On replacing e it b s in (6) we get the probabilit generating function (p.g.f.) of as

3 On intervened negative binomial distribution and some of its properties 397 r P ( s) (,, r)[(1 s) 1](1 s) (7) Clearl P (1) 1. r Result.1 Let follows INBD with p.g.f. (7). Then the probabilit mass function g = P(=) of INBD is the following, for 1,,3,..., in which 0< θ <1 and >0. 1 (,, r ) g ( r) ( r) [ ( r )]! 0 (8) Proof From (7) we have the following: P ( s) g s (9) 0 1 (, ) ( s) ( r) ( r) [ ( r )] 0! 0 (10) On equating coefficient of s on right hand side epression of (9) and (10), we get (8). Remark.1 When r =1, (8) reduces to the p.m.f. of intervened geometric distribution as given in (1). Result. For an positive integer k, the k th factorial moment k of INBD is k k k 1 ( k) (1 ) ( r k) (11) [ ( r )] ( r ) 1 in which 1 (1 ) 1, (1 ) and for k 1, k ( k) [ ( k r)][ ( r)]. 0 k 1 Proof The factorial moment generating function H () t of INBD with p.g.f. (7) is

4 398 k k0 H () t () t (1) P (1 t ) r (,, r)([1 (1 t) ] 1)[1 (1 t) ] r r r 1t t t (1 ) (1 ) (1 ) (1 ) r kr 1 k (1 ) ( t ). k0 k k k r 1 r 1 ( 1 ) k t k0 0 k 1 (13) On equating coefficient of and (13), we get (11). -1 k (k!) t on the right hand side epressions of (1) Remark. Putting k=1,, 3 in (11) we get the first three factorial moments of INBD as given below. r r 1 1 r ( r 1) [ r( r 1) 1 r 1] r r r r r r 1 r r 1 1 (14) ( 1)( ) [ ( 1)( ) 3 ( 1) ( )] Result.3 The mean E() and variance V() of INBD with p.g.f. (7) are the following: and and E( ) r r (15) V ( ) r (1 ) r (1 ) (1 )( r ) (16) Proof follows from (14) in the light of the relation E ( ) 1

5 On intervened negative binomial distribution and some of its properties 399 V( ) E( ( 1) E( ) [ E( )] [ ] 1 1 Remark.3 When r =1 in (15) and (16) we get the E() and V() of intervened geometric distribution as and E ( ) V( ) (1 ) (1 ) (1 ). Result.4 INBD is over-dispersed or under-dispersed according as ( r 1) 1 r( 1) or ( r 1) 1 r( 1) Proof follows from Result.3. Result.5 The following is a simple recurrence relation for probabilities of INBD, for 1. 1 k1 k1 k (17) k0 ( 1) g r (1 ) g ( ;,, r), in which (, ) 1 ( kr) k ( ;,, r). ( r 1) ( k1) k0 Proof From (10) we have P ( s) s g (18) 0 1 (, ) ( s) ( r) ( r) [ ( r )] 0! 0 (19) Differentiate (18) and (7) with respect to s to obtain the following.

6 400 ' 1 (0) 1 P ( s) ( 1) g s (1 s ) rp ( s ) (, ) r 1s 1 s (1 s ) k k k k ( ) ( ) 0k0 0k0 r s g r s g 0k0 r r 1 k (, ) r ( s) g 0k0 k1 k1 (1 ) k r g s 1 ( k r, ) r ( s ) 0k 0 k k k1 k1 r (1 ) gk [ ;,, r] s (1) 0 k0 Now on equating coefficients of s on the right hand side epressions of (0) and (1), we get (17). 3. ESTIMATION Here we discuss the estimation of parameters of the INBD b method of factorial moments, method of mied moments and the method of maimum likelihood. Method of factorial moment In method of factorial moments, the first three factorial moments of the INBD are equated to the corresponding sample factorial moments m 1, m and m 3 and thus we obtain the following sstem of equations: and r r m () rr ( 1) rr ( 1) r m (3)

7 On intervened negative binomial distribution and some of its properties rr ( 1)( r) rr ( 1)( r) 3 r( r1) ( ) m (4) where 1 and are given in (11). Now, the parameters of the INBD are estimated b solving the non-linear equations (), (3) and (4) using MATHCAD. Method of mied moments In method of mied moments, the parameters are estimated b using the first two sample factorial moments and the first observed frequenc of the distribution. That is, the parameters are estimated b solving the following equation together with () and (3). p r [(1 )(1 )] r 1, (5) N where p 1 is the observed frequenc of the distribution corresponding to the observation =1 and N is the total observed frequenc. Method of maimum likelihood Let a() be the observed frequenc of events, be the highest value of observed. Then the likelihood of the sample is 1 a( ) L [ g( )], which implies log L = 1 a ( ).log[ g ( )] Assume that the parameter r of INBD is known. Let ˆ and ˆ denotes the maimum likelihood estimates of and respectivel. Now, ˆ and ˆ are obtained b solving the normal equation (6) and (7) given below. log L 0 implies a ( ) r a ( )[ 1] 0 (6) 1 1

8 40 log L 0 implies a ( ). D (, ) 0, (7) 1 in which 1 1. ( r). ( r).. 0 r D(, ) 1 1 ( r). ( r) 0 Here estimator of r is used for obtaining the maimum likelihood estimates ˆ and ˆ of and respectivel. Let r denotes the factorial moment estimator of r and r denotes the mied moment estimate of r. Table 1 gives the data related to the distribution of number of articles of theoretical statistics and probabilit for ears and initial letters N-R of the author s name, for details, see (Kendall, 1961). Here we present the fitting of intervened geometric distribution (IGD), intervened Poisson distributions (IPD), intervened generalized Poisson distributions (IGPD) and intervened negative binomial distributions (INBD). From Table 1, it is obvious that INBD is more suitable for the data compared to IGD, IPD and IGPD. ACKNOWLEDGMENTS The authors would like to epress their sincere thanks to the Editor and the anonmous Referee for carefull reading the paper and for valuable suggestions. Department of Statistics Universit of Kerala, Kariavattom, Trivandrum, India Department of Statistics Government College, Kariavattom, Trivandrum, India C. SATHEESH KUMAR S. SREEJAKUMARI

9 On intervened negative binomial distribution and some of its properties 403 TABLE 1 Distribution of number of articles of theoretical statistics and probabilit for ears from and initial letters N-R of the author s name Observed. freqenc Epected frequenc b method of Factorial moments Mied moments Maimum likelihood IGD IPD IGPD INBD IGD IPD IGPD INBD IGD IPD IGPD INBD INBD using r using r Total Estimates of parameters ˆ 1.16 r ˆ 0.43 r.784 ˆ 0.40 r r.784 ˆ 3.57 ˆ 0.56 ˆ 0.5 ˆ ˆ ˆ 0.68 ˆ 0.97 ˆ ˆ ˆ ˆ 0.08 ˆ 0.54 ˆ 0.51 ˆ 0.11 ˆ 0.47 ˆ ˆ 1.01 ˆ ˆ 0.39 ˆ 0.13 ˆ 1 ˆ 0.58 ˆ ˆ 1.1 ˆ ˆ 0.17 Chi-square value P value based on Chi-square Kolmogrov distance value P value based on Kolmogrov distance

10 404 REFERENCES A. A. BARTOLUCCI, R. SHANMUGAM, K. P. SINGH (001), Developments of the generalized geometric model with application to cardiovascular studies, Sstem Analsis Modeling Simulation, 41, pp P. DHANAVANTHAN (1998), Compound intervened Poisson distribution, Biometrical Journal, 40, pp P. DHANAVANTHAN (000), Estimation of the parameters of compound intervened Poisson distribution, Biometrical Journal, 4, pp M. HUANG, K.Y. FUNG (1989), Intervened truncated Poisson distribution, Sankha, Series B, 51, pp N. L. JOHNSON, A.W. KEMP, S. KOTZ (005), Univariate Discrete Distributions, John Wile and Sons, New York. M.G. KENDAL (1961), Natural law in Science, Journal of Roal Statistical Societ, Series. A, 14, pp C.S. KUMAR, D.S. SHIBU (011), Modified intervened Poisson distribution, Statistica, 71, pp D.P.M. SCOLLNIK (006), On the intervened generalized Poisson distribution, Communication in Statistics-Theor and Methods, 35, pp R. SHANMUGAM (1985), An intervened Poisson distribution and its medical application, Biometrics, 41, pp R. SHANMUGAM (199), An inferential procedure for the Poisson intervention parameter, Biometrics, 48, pp SUMMARY On intervened negative binomial distribution and some of its properties Here we develop a new class of discrete distribution namel intervened negative binomial distribution and derive its probabilit generating function, mean, variance and an epression for its factorial moments. Estimation of the parameters of the distribution is described and the distribution has been fitted to a well known data set.