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International Journal of Statistics and Applied Mathematics 2018; 3(3): 111-119
ISSN: 2456-1452
Maths 2018; 3(3): 111-119
© 2018 Stats & Maths
www.mathsjournal.com
Received: 14-03-2018
Accepted: 15-04-2018
Rama Shanker
Department of Statistics College
of Science, Eritrea Institute of
Technology, Asmara, Eritrea
Kamlesh Kumar Shukla
Department of Statistics College
of Science, Eritrea Institute of
Technology, Asmara, Eritrea
Ravi Shanker
Department of Mathematics,
G.L.A. College, N.P University,
Daltonganj, Jharkhand, India
Correspondence
Ravi Shanker
Department of Mathematics,
G.L.A. College, N.P University,
Daltonganj, Jharkhand, India
A quasi Poisson-Akash distribution and its applications
to ecology
Rama Shanker, Kamlesh Kumar Shukla and Ravi Shanker
Abstract
In this paper, a quasi-Poisson-Akash distribution (QPAD) of which Poisson-Akash distribution (PAD) of
Shanker (2017 a) is a particular case, has been proposed by compounding Poisson distribution with quasi
Akash distribution (QAD), introduced by Shanker (2015). The raw moments and central moments have
been obtained. Expressions for coefficient of variation, skewness, kurtosis and index of dispersion have
been given their behaviors have been studied for varying values of the parameters. The QPAD has been
shown to be unimodal and always over-dispersed. The estimation of its parameters has been discussed
using both the method of moments and the method of maximum likelihood. Finally, the goodness of fit
of QPAD has been discussed with two real count datasets from ecology and the fit has been compared
with that of Poisson distribution (PD), Poisson-Lindley distribution (PLD) and Poisson-Akash
distribution (PAD).
Keywords: Poisson-Akash distribution, compounding, moments, log-concavity, over-dispersion,
estimation of parameters, goodness of fit
1. Introduction
The probability mass function (pmf) of Poisson-Akash distribution (PAD) introduced by
Shanker (2017a)
[9]
with parameter
is given by
2 2
3
1
3
2
3 2 3
; ; 0,1,2,..., 0
2
1
x
x x
P x x
(1.1)
Shanker (2017a)
[9]
have studied statistical and mathematical properties of PAD, estimation of
parameter using both the method of moments and the method of maximum likelihood, and
applications to model count data. Shanker (2017b, 2017c)
[10, 11]
have also obtained the size-
biased and zero-truncated forms of PAD and discussed their statistical and mathematical
properties along with the estimation of parameters and applications for modeling count data
which structurally excludes zero counts.
The PAD has been obtained by compounding Poisson distribution with Akash distribution,
introduced by Shanker (2015)
[7]
having probability density function (pdf)
3
2
1
2
; 1 ; 0, 0
2
x
f x x e x
(1.2)
Shanker (2015)
[7]
has studied its statistical and mathematical properties including its shapes
for varying values of parameter, coefficients of variation, skewness, kurtosis, index of
dispersion, hazard rate function, mean residual life function, mean deviations, stochastic
ordering, Renyi entropy measure, order statistics, Bonferroni and Lorenz indices, stress-
strength reliability, estimation of parameter and application for modeling life time data from
engineering and biomedical sciences.
The first four moments about origin and the variance of PAD (1.1) obtained by Shanker
(2017a)
[9]
are given as