Extended Intervened Geometric Distribution

Transcription

1 International Jornal of Statistical Distribtions Applications 6; (): 8- Extended Intervened Geoetric Distribtion C. Satheesh Kar, S. Sreeaari Departent of Statistics, University of Kerala, Trivr, India Eail address: (C. S. Kar), (S. Sreeaari) To cite this article: C. Satheesh Kar, S. Sreeaari. Extended Intervened Geoetric Distribtion. International Jornal of Statistical Distribtions Applications. Vol., No., 6, pp. 8-. Received: March, 6; Accepted: March 9, 6; Pblished: April 5, 6 Abstract: Here we develop an extended version of the odified intervened geoetric distribtion of Kar Sreea (The Aligarh Jornal of Statistics, 4) investigate soe of its iportant statistical properties. Paraeters of the distribtion are estiated by varios ethods of estiation sch as the ethod of factorial oents, the ethod of ixed oents the ethod of axi lielihood. The distribtion has been fitted to a real life data set for illstrating its practical relevance. Keywords: Factorial Moents, Intervened Geoetric Distribtion, Method of Factorial Moents, Method of Mixed Moents, Method of Maxi Lielihood, Probability Generating Fnction, Probability Mass Fnction. Introdction Intervened type distribtions have fond any applications in several areas sch as epideiological stdies, life testing probles etc. In epideiological stdies health agencies taes varios preventive actions. The inforation concerning the effect of sch actions taen by the agencies can statistically analyed by intervened type distribtions. In life testing proble the failed ites dring the observational period are either replaced or repaired. This ind of actions changes the reliability of the syste as only soe of its coponents have longer life. The ipact of sch actions can be stdied by intervened type distribtions. The intervened type distribtions sch as intervened Poisson distribtion (IPD), intervened geoetric distribtion (IGD) odified intervened geoetric distribtion (MIGD) has been stdied by several athors. For exaple see Shangan [, ], Hang Fng [], Scollin [4], Dhanavanthan [5, 6], Kar Shib [7-5], Bartolcci et al [6], Kar Sreea [7] etc. Throgh this paper we consider a new class of intervened geoetric distribtion sitable for ltiple intervention cases naed it as the extended intervened geoetric distribtion (EIGD), which contains the MIGD as its special case. The paper is organied as follows. In Section, we present a odel leading to EIGD obtain expression for its probability ass fnction, ean variance. We also obtain a recrrence relation sefl for the coptation of probabilities of the EIGD. In Section, we consider the estiation of paraeters of the EIGD by the ethod of axi lielihood the distribtion has been fitted to a real life data set for highlighting the seflness of the odel. We need the following series representation in the seqel i= r= i= r= i A(i,r) = A(i r,r) () i B(i,r) = B(i r,r), () i= r= i= r= [a] represents the integer part of a, for any a >. Extended Intervened Geoetric Distribtion Consider a discrete ro variable X having intervened geoetric distribtion with the following probability ass fnction (pf), in which x =,,,..., θ (, ), ρ > sch that ρθ. ( θ)( ρθ) h(x; ρ, θ ) = ρ ( ρ ) θ x x

2 International Jornal of Statistical Distribtions Applications 6; (): 8-9 The probability generating fnction (pgf) of X is the following P (s) = ( θ)( ρ θ)s( s θ) X ( s ρ θ) Let Y be a ro variable having geoetric distribtion with the following pf, in which y =,,,..., θ (, ), ρ > sch that ρθ. The pgf Q Y(s) of Y is f(y; ρ, θ ) = ( ρ θ)( ρ θ ) Q (s) = ( ρ θ)( s ρ θ ) Y Define = X+ Y, in which X Y are assed to be independent is a fixed bt arbitrary positive integer. Then the pgf of is G (s) = P X(s)Q Y(s ) = Λ( ρ, ρ, θ)s( s θ) ( s ρ θ) ( s ρ θ), Λ( ρ, ρ, θ ) = ( θ)( ρ θ) ( ρ θ) The distribtion of a ro variable whose pgf is () is called the extended intervened geoetric distribtion or in short the EIGD. Reslt.. Let follows EIGD with pgf given in eqation nber (). Then the pf g of is the following, for =,,,. g = Λ( ρ, ρ, θ) y ρθ ρθ θ = = ( ) ( ), Λ( ρ, ρ, θ ) is as defined in (), θ (, ), < ρ sch that ρθ, for =,. Proof: We have s g () (4) G (s) = (5) = Λ( ρ, ρ, θ)s( s θ) ( s ρ θ) ( s ρ θ) = Λ( ρ, ρ, θ) ( ρθ) ( θ) ( ρθ) s s s s = = = Λ( ρ, ρ, θ) ρ ρ θ s = = Now by applying the reslts given in eqations () () in eqation (6) we get G (s) = Λ( ρ, ρ, θ) ( ) + ρ ρ θ s = = On eqating co efficient of s in the right h side expressions of (5) (7) we get (4) when ρ ρ = ρ (4) redces to the pf of IGD defined by Bartalcci et.al [8] Reslt. The first three factorial oents µ ( ), µ ( ) µ of the EIGD are ( ) µ = + δ + δ + δ, ( ) µ = δ + ( + ) δ + δ ( + δ ) ( ) + δ ( + δ ) + δ ( δ + δ ) + δ δ ( ) 6 ( ) ( ) ( ) µ = δ + δ + δ + δ + δ + δ + 6 δ ( δ + δ + δ δ ) + 6 δ δ ( + δ + δ ) + δ + δ δ + δ + + δ ( ) ( )( ), in which δ =θ(- θ) -, δ =ρ θ(- ρ θ) - δ =ρ θ(- ρ θ) -. Proof follows fro the fact that in which µ = G (s)/s =, () ( ) µ = G (s)/s = () ( ) µ = G (s)/s = () ( ) r (r) d G (s) = (G r (s)) ds with G (s) as the pgf of. Reslt. The ean variance of EIGD is (6) (7) (8) Mean = +δ + δ + δ (9) Variance = δ(+ δ)+δ(+ δ)+ δ(+δ) () δ for =,, are as defined in Reslt.. Proof: On differentiating the pgf G (s) of given in

3 C. Satheesh Kar S. Sreeaari: Extended Intervened Geoetric Distribtion eqation () with respect to s ptting s=, we get in which () G () = E() = + δ + δ + δ () G () = E(( )) = δ + ( + ) δ + δ ( δ + δ ). + δ ( + δ ) + δ ( + δ ) + δ δ r (r) d G () = (G r (s))/s =. ds Now the ean variance are obtained by sing the following reslts Mean= G () () ( ) () () () Variance ()= G ( ) + G ( ) G ( ) Reslt.4 For, the following is a siple recrrence relation for probabilities g of the EIGD for < for i+ i+ + = + ρ θ i i= g ( ) g Proof: Fro () we have i+ + = ρθ i + i= g ( ) g s g + ( + ρ ) θ g i= i+ i+ i () G (s) = () = Λ( ρ, ρ, θ) s( s θ) ( s ρ θ) ( s ρ θ) () On differentiating () () with respect to s we get the following, in the light of (), ( )g s G (s) G (s) + + = + θ s s θ G (s) G (s)s + ρ θ + ρ θ sρθ s ρθ By applying () () in (4) to obtain (4) which iplies ( )s g+ + = i+ i+ + s g + + ( + ρ ) θ s g i+ i= + ( ρ θ) s g, i= i+ + i+ i+ i+ s g + = ( + ρ ) θ s g i i= + ( ρ θ) s g i= i+ i + (5) On eqating coefficient of s on both sides of the expressions (5), we get ().. Estiation Here we discss the estiation of the paraeters of the EIGD by varios ethods of estiation sch as the ethod of factorial oents, the ethod of ixed oents the ethod of axi lielihood. We asse that is a fixed positive integer the paraeters ρ, ρ θ of the EIGD are estiated for possible vales of. Method of factorial oents In ethod of factorial oents, eqate the first three factorial oents of the EIGD to the corresponding saple / / / factorial oents say,, there by we obtain the following syste of eqations: + δ + δ + δ = (6) / δ + ( + ) δ + δ ( + δ ) + δ ( + δ ) + δ ( δ + δ ) + δ δ = / ( ) ( ) ( ) 6 δ + δ + δ + δ + δ + δ + 6 δ ( δ + δ + δ δ ) + 6 δ δ ( + δ + δ ) + ( ) δ + δ ( δ + δ )( + + δ ) = / (7) (8) δ, δ δ are given in (8). Now the paraeters of EIGD are estiated by solving the non- linear eqations (6), (7) (8). Method of ixed oents In ethod of ixed oents, the paraeters are estiated by sing the first two saple factorial oents the first observed freqency of the distribtion. That is, the paraeters are estiated by solving the following eqation together with (6) (7).

4 International Jornal of Statistical Distribtions Applications 6; (): 8- (,, ) p Λ ( ρ, ρ, θ ) = (9) N Λ ρ ρ θ is as defined in (), p is the observed freqency of the distribtion corresponding to the observation x = N is the total freqency. Method of axi lielihood Let a() be the observed freqency of events be the highest vale of observed. Then the lielihood fnction of the saple is which iplies L = (g ) = a() logl a()log (g ) Let ρˆ ˆ, ρ, θ denotes the axi lielihood estiators of ρ ρ, θ respectively. Now, ρˆ ˆ ˆ, ρ, θ are obtained by ˆ solving the lielihood eqations (), () () as given below. iplies ρ = = ρ E (, ρ, ρ, θ ) = logl = ρ ρθ Ψ4(, ρ, ρ, θ) a()( + ) + a() =, ρ θ Ψ (, ρ, ρ, θ) () ρ 4(,,, ) = = ρ Ψ ρ ρ θ = Ψ(, ρ, ρ, θ ) is as defined in (). iplies logl = ρ ρθ Ψ5(, ρ, ρ, θ) a() + a() = ρ θ Ψ (, ρ, ρ, θ) () iplies logl = θ Ψ (, ρ, ρ, θ) a() (,, ) a() a() Ψ ρ ρ θ + = Ψ(, ρ, ρ, θ) () θ ρ θ Ψ ρ ρ θ = ρ θ (,, ), θ ρθ ρθ ρ (,,, ) ( ) = = ρ Ψ ρ ρ θ = ρ 5(,,, ) = = ρ in which Ψ ρ ρ θ = Ψ(, ρ, ρ, θ ) is as defined in (). We present the fitting of the intervened geoetric distribtion (IGD) the extended intervened geoetric distribtion (EIGD) for particlar vales of for the data set on the cont of the nber of Eropean red ites on apple leaves taen fro Jani Shah [8]. We estiate the paraeters by the ethod of factorial oents, the ethod of ixed oents the ethod of axi lielihood. We have copted the vales of χ statistics in the case of each odel the nerical reslts are saried in Table,Table Table. Fro the tables it is obvios that the EIGD with = gives a better fit copared to IGD as well as for the case =, =(MIGD) =4. Table. (Observed freqencies O i Expected freqencies E i calclated by the ethod of factorial oents). x Oi Ei Expected freqencies of EGD for different vales of (IGD) = = = = Total Estiated vales of paraeters ρ ˆ =.7 θ ˆ =.596 ρ ˆ =. ρ ˆ =.8 θ ˆ =.6 ρ ˆ =. ρ ˆ =.54 θ ˆ =.46 ρ ˆ =.47 ρ ˆ =.49 θ ˆ =.6 Chi- sqare vale p-vale ρ ˆ =.5 ρ ˆ =.87 θ ˆ =.6

5 C. Satheesh Kar S. Sreeaari: Extended Intervened Geoetric Distribtion Table. (Observed freqencies O i Expected freqencies E i calclated by the ethod of ixed oents). x Oi Ei Expected freqencies of EGD for different vales of (IGD) = = = = Total Estiated vales of paraeters ρ ˆ =.56 θ ˆ =.5 ρ ˆ =.5 ρ ˆ =.87 θ ˆ =.56 ρ ˆ =.5 ρ ˆ =.87 θ ˆ =.59 ρ ˆ =.46 ρ ˆ =.587 θ ˆ =.9 Chi- sqare vale p-vale Table. (Observed freqencies O i Expected freqencies E i calclated by the ethod of axi lielihood). x Oi Ei Expected freqencies of EGD for different vales of (IGD) = = = = Total Estiated vales of paraeters ρ ˆ =.75 θ ˆ =.46 ρ ˆ =.894 ρ ˆ =.7 θ ˆ =.84 ρ ˆ =.594 ρ ˆ =.47 θ ˆ =.64 ρ ˆ =.84 ρ ˆ =.448 θ ˆ =.84 ρ ˆ =.57 ρ ˆ =.87 θ ˆ =.48 Chi- sqare vale p-vale ρ ˆ =.47 ρ ˆ =.7 θ ˆ =.75 Acnowledgeents Athors wish to express their sincere gratitde to the Editor-in Chief anonyos referees for fritfl coents. References [] Shanga, R. (985) An intervened Poisson distribtion its edical application, Bioetrics 4, 5-9. [] Shanga, R. (99) An inferential procedre for the Poisson intervention paraeter, Bioetrics 48, [] Hang, M., Fn, K. Y. (989) Intervened trncated Poisson distribtions, Sanhya Series, 5, -. [4] Scollni, D. P. M. (6) On intervened generali-ed Poisson distribtions, Conication in Statistics - Theory Methods, 5, [5] Dhanavanthan, P. (998) Copond intervened Poisson distribtions, Bioetrical Jornal, 4, [6] Dhanavanthan, P () Estiation of paraeters of copond intervened Poisson distribtions, Bioetrical Jornal, 4, 5-. [7] Kar, C. S., Shib, D. S. () Modified intervened Poisson distribtions, Statistica, 7, [8] Kar, C. S., Shib, D. S. () Estiation of the paraeters of a finite ixtre of intervened Poisson distribtion. Statistical Methods in Interdisciplinary Stdies, Dept. of Statistics, Maharaas College, Cochin, 9-. [9] Kar, C. S., Shib, D. S. (a). Soe finite ixtres of intervened Poisson distribtion. The Aligarh Jornal of Statistics,, [] Kar, C. S., Shib, D. S. (b). Generalied intervened Poisson distribtion. Jornal of Applied Statistical Sciences, 9, -4. [] Kar, C. S., Shib, D. S. (c). An alternative to trncated intervened Poisson distribtion. Jornal of Statistics Applications, 5, -4. [] Kar, C. S., Shib, D. S. () On soe aspects of intervened generalied Herite distribtion, Metron, 7, 9-9. [] Kar, C. S., Shib, D. S. () Finite ixtres of extended intervened Poisson distribtion, In Collection of Recent Statistical Methods Applications, Departent of Statistics, University of Kerala Pblishers, Trivr, [4] Kar, C. S., Shib, D. S. () An extended version of intervened Poisson distribtion, Research Jornal of Fatia Mata National College Kolla, 4, 9-4. [5] Kar, C. S., Shib, D. S. (4) On finite ixtres of odified intervened Poisson distribtion its applications, Jornal of Statistical Theory Applications, (4),

6 International Jornal of Statistical Distribtions Applications 6; (): 8- [6] Bartolcci, A. A., Shanga, R., Singh, K. P. () Developent of the generalied geoetric odel with application to cardiovasclar stdies, Syste Analysis, Modelling silation, 4, [8] Jani, P. N., Shah, S. M. (979). On fitting of the generalied logarithic series distribtion. Jornal of the Indian Society for Agricltral Statistics., -. [7] Kar, C. S., Sreeaari. S. (4) Modified intervened geoetric distribtion, The Aligarh Jornal of Statistics, 4, -.