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Akash Distribution, Compounding, Moments, Skewness, Kurtosis, Estimation of Parameter, Statistical Properties, Goodness of Fit

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Akash Distribution, Compounding, Moments, Skewness, Kurtosis, Estimation of Parameter, Statistical Properties, Goodness of Fit International Journal of Probability and Statistics 2017, 6(1): 1-10 DOI: 10.5923/j.ijps.20170601.01 The Discrete Poisson-Akash Distribution Rama Shanker Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea Abstract A Poisson-Akash distribution has been obtained by compounding Poisson distribution with Akash distribution introduced by Shanker (2015). A general expression for the r th factorial moment has been derived and hence the first four moments about origin and the moments about mean has been obtained. The expressions for its coefficient of variation, skewness and kurtosis have been obtained. Its statistical properties including generating function, increasing hazard rate and unimodality and over-dispersion have been discussed. The maximum likelihood estimation and the method of moments for estimating its parameter have been discussed. The goodness of fit of the proposed distribution using maximum likelihood estimation has been given for some count data-sets and the fit is compared with that obtained by other distributions. Keywords Akash distribution, Compounding, Moments, Skewness, Kurtosis, Estimation of parameter, Statistical properties, Goodness of fit lifetime data from engineering and medical science and 1. Introduction concluded that Akash distribution has some advantage over Lindley and exponential distributions, Lindley distribution A lifetime distribution named, “Akash distribution” has some advantage over Akash and exponential having probability density function distributions and exponential distribution has some advantage over Akash and Lindley distributions due to their θ 3 over-dispersion, equi-dispersion, and under-dispersion for fx( ,θθ) =( 1 +x2 ) e−θ x ; x >> 0, 0 (1.1) θ 2 + 2 various values of their parameters. Recently, Shanker and Shukla (2016) has introduced a two-parameter weighted has been proposed by Shanker (2015) for modeling Akash distribution (WAD) andstudied its various lifetime data from biomedical science and engineering and mathematical and statistical properties, estimation of shown that the proposed distribution is a two-component parameter and application for modeling lifetime data. mixture of exponential distribution having scale parameter Shanker (2016) has also proposed a two-parameter quasi θ and a gamma distribution having shape parameter 3 and Akash distribution and studied its statistical and θ 2 mathematical properties, estimation of parameters using scale parameter θ with their mixing proportions 2 maximum likelihood estimation and method of moments θ + 2 along with its application for modeling lifetime data from 2 engineering and biomedical sciences. and respectively. Various mathematical and θ 2 + 2 In the present paper, a Poisson mixture of Akash statistical properties of Akash distribution including its distribution introduced by Shanker (2015) named, shape, moments, skewness, kurtosis, hazard rate function, “Poisson-Akash distribution (PAD) has been proposed. Its mean residual life function, stochastic ordering, mean various mathematical and statistical properties including its deviations, distribution of order statistics, Bonferroni and shape, moments, coefficient of variation, skewness, and Lorenz curves, Renyi entropy measure and stress-strength kurtosis have been discussed. The estimation of its parameter reliability have been discussed by Shanker (2015). It has has been discussed using maximum likelihood estimation been shown by Shanker (2015) that Akash distribution and method of moments. The goodness of fit of PAD along provides much closer fit than Lindley and exponential with Poisson distribution and Poisson-Lindley distribution distributions for modeling lifetime data from medical science (PLD), a Poisson mixture of Lindley (1958) distribution and and engineering. Further, Shanker et al (2016) have done a introduced by Sankaran (1970), has been given with some detailed comparative study on Akash, Lindley and count data-sets. exponential distributions for modeling different types of * Corresponding author: 2. Poisson-Akash Distribution [email protected] (Rama Shanker) Published online at http://journal.sapub.org/ijps Suppose the parameter λ of Poisson distribution follows Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved Akash distribution (1.1). Then the Poisson mixture of Akash 2 Rama Shanker: The Discrete Poisson-Akash Distribution distribution (1.1) can be obtained as by compounding Poisson distribution with Lindley ∞ −λ x 3 distribution, introduced by Lindley (1958) having e λθ −θλ PX==⋅+ x 1 λλ2 e d (2.1) probability density function (p.d.f) ( ) ∫ 2 ( ) x! θ + 2 2 0 θ θ f( x,θθ) =( 1 +xe) x ; x >> 0, 0 (2.4) 3 ∞ θ +1 θ −+(θλ1) + = edλλxx+ 2 λ 2 ∫ Ghitany et al (2008) have discussed various interesting (θ + 2!) x 0 properties of Lindley distribution, estimation of parameter and application for modeling waiting time data from a bank. 22 θ 3 xx++3(θθ ++ 23) Shanker et al (2015) have detailed study on modeling of =⋅=;x 0,1,2,...,θ > 0 .(2.2) lifetime data using exponential and Lindley distributions. θ 23+ x+ 2 (θ +1) Further, Shanker et al (2016) have discussed the comparative We name this distribution “Poisson-Akash distribution applications of Akash, Lindley and exponential distributions (PAD)”. Shanker et al (2016) have detailed study on for modeling lifetime data from biomedical sciences and applications of PAD to model count data from different engineering. Shanker and Hagos (2015) have detailed and fields of knowledge. critical study on applications of PLD to model count data It would be recalled that Sankaran (1970) obtained from biological sciences. Poisson-Lindley distribution (PLD) having probability mass The graphs of the pmf of Poisson-Akash distribution function (p.m.f) (PAD) and Poisson-Lindley distribution (PLD) for different values of their parameter are shown in the figure 1. θθ2 ( x ++2) = = =θ > (2.3) PX( x) x+3 ;x 0,1,2,..., 0 (θ +1) Poisson-Akash Distribution Poisson-Lindley Distribution θ=1 θ=1.5 θ=2 θ=2.5 θ=1 θ=1.5 θ=2 θ=2.5 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 P(x) P(x) 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 2 4 6 8 10 0 2 4 6 8 10 x x Poisson-Akash Distribution Poisson-Lindley Distribution θ=3 θ=3.5 θ=4 θ=4.5 θ=3 θ=3.5 θ=4 θ=4.5 1 1 0.8 0.8 0.6 0.6 P(x) P(x) 0.4 0.4 0.2 0.2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 x x Figure 1. Graphs of probability mass function of PAD and PLD for different values of the parameter θ International Journal of Probability and Statistics 2017, 6(1): 1-10 3 3. Moments and Associated Measures Using gamma integral and a little algebraic simplification, we get finally, a general expression for the r th factorial The r th factorial moment about origin of Poisson-Akash moment of PAD (2.2) as distribution (PAD) (2.2) can be obtained as r!θ 2 ++( rr 12)( +) (r) µλ′ = EEX| , where µ ′ = ;r = 1,2,3,.... (3.1) (r) ( ) (r) r 2 θθ( + 2) (r) X= XX( −1)( X − 2) ...( X −+ r 1) Substituting r =1, 2, 3, and 4 in (3.1), the first four factorial moments about origin can be obtained and using the Using (2.1), the r th factorial moment about origin of relationship between factorial moments about origin and PAD (2.2) can be obtained as moments about origin, the first four moment about origin of the PAD (2.2) are obtained as µλ′ = (r) (r) EEX( | ) 2 ′ θ + 6 µ1 = 3 ∞ ∞ −λ x θθ2 + 2 θλ(r) e −θλ ( ) = x 1+ λλ2 ed 2 ∫ ∑ ( ) x! 32 θ + 2 0 x=0 θθθ+2 ++ 6 24 µ ′ = 2 22 3 ∞ ∞ −−λ xr θθ+ 2 θλe −θλ ( ) = λr 1+ λλ2 ed 2 ∫ ∑ ( ) 43 2 θ + 2 xr= ( xr− )! θθ++6 12 θ ++ 72 θ 120 0 µ ′ = 3 32 Taking xr+ in place of x within the bracket, we get θθ+ 2 ( ) ∞ θλ3 ∞ e−λ x 54 3 2 µ′ = λr + λλ2 −θλ ′ θθ+++14 42 θ 192 θ ++ 720 θ 720 (r) ∑ (1 )ed µ4 = θ 2 + ∫ x! 42 2 0 x=0 θθ( + 2) The expression within the bracket is clearly unity and Using the relationship between moments about mean and hence we have the moments about origin, the moments about mean of the θ 3 ∞ PAD (2.2) are obtained as µ′ = λλr 1+ 2 ed−θλ λ (r) 2 ∫ ( ) θ + 2 0 θθ54++8 θ 3 + 16 θ 2 + 12 θ + 12 µσ=2 = 2 2 θθ22+ 2 ( ) θθ87654++3 12 θ + 54 θ + 88 θ + 132 θ 32 + 96 θ ++ 72 θ 48 µ = 3 3 θθ32( + 2) θθ11+10 10 ++ 30 θ 9 197 θ 8 + 576 θ 7 + 1208 θ 6 + 2144 θ 5 + 2584 θ 4 +2928θθθ32 + 2496 ++ 1440 720 µ = 4 4 θθ42( + 2) β The coefficient of variation (CV. ) , coefficient of skewness ( β1 ) , coefficient of kurtosis ( 2 ) , and index of dispersion (γ ) of the PAD (2.2) are thus given by σ θθ54++8 θ 3 + 16 θ 2 + 12 θ + 12 = = CV. 2 µ1′ θ + 6 87654 32 µ3 θθ++3 12 θ + 54 θ + 88 θ + 132 θ + 96 θ ++ 72 θ 48 β1 = = µ 32 54 3 2 32 2 (θθ++8 θ + 16 θ + 12 θ + 12) 4 Rama Shanker: The Discrete Poisson-Akash Distribution θθ11+10 10 ++ 30 θ 9 197 θ 8 + 576 θ 7 + 1208 θ 6 + 2144 θ 5 432 µ4 +2584θθθθ + 2928 + 2496 ++ 1440 720 β2 = = µ 2254 3 2 2 (θθ++8 θ + 16 θ + 12 θ + 12) σ2 θθ 54++8 θ 3 + 16 θ 2 + 12 θ + 12 γ = = µ′ 22 1 θθ( ++26)( θ ) ′ The expressions for µµ1, 2 , C.V, ββ 12 , and γ of PLD (2.3) obtained by Sankaran (1970) and Ghitany and Al-Mutairi (2009) are given by θ + 2 θθθ32+4 ++ 62 µ ′ = , µ = 1 2 2 θθ( +1) θθ2 ( +1) σθθθ32+4 ++ 62 CV.
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