Extended Intervened Geometric Distribution

Here we develop an extended version of the modified intervened geometric distribution of Kumar and Sreeja (The Aligarh Journal of Statistics, 2014) and investigate some of its important statistical properties. Parameters of the distribution are estimated by various methods of estimation such as the method of factorial moments, the method of mixed moments and the method of maximum likelihood. The distribution has been fitted to a real life data set for illustrating its practical relevance.


Introduction
Intervened type distributions have found many applications in several areas such as epidemiological studies, life testing problems etc. In epidemiological studies health agencies takes various preventive actions. The information concerning the effect of such actions taken by the agencies can statistically analyzed by intervened type distributions. In life testing problem the failed items during the observational period are either replaced or repaired. This kind of actions changes the reliability of the system as only some of its components have longer life. The impact of such actions can be studied by intervened type distributions. The intervened type distributions such as intervened Poisson distribution (IPD), intervened geometric distribution (IGD) and modified intervened geometric distribution (MIGD) has been studied by several authors. For example see Shanmugan [1,2], Huang and Fung [3], Scollink [4], Dhanavanthan [5,6], Kumar and Shibu [7][8][9][10][11][12][13][14][15], Bartolucci et al [16], Kumar and Sreeja [17] etc.
Through this paper we consider a new class of intervened geometric distribution suitable for multiple intervention cases and named it as the extended intervened geometric distribution (EIGD), which contains the MIGD as its special case. The paper is organized as follows. In Section 2, we present a model leading to EIGD and obtain expression for its probability mass function, mean and variance. We also obtain a recurrence relation useful for the computation of probabilities of the EIGD. In Section 3, we consider the estimation of parameters of the EIGD by the method of maximum likelihood and the distribution has been fitted to a real life data set for highlighting the usefulness of the model.
We need the following series representation in the sequel where [a] represents the integer part of "a", for any a > 0

Estimation
Here we discuss the estimation of the parameters of the EIGD by various methods of estimation such as the method of factorial moments, the method of mixed moments and the method of maximum likelihood. We assume that m is a fixed positive integer and the parameters ρ 1 , ρ 2 and θ of the EIGD are estimated for possible values of m.

Method of factorial moments
In method of factorial moments, equate the first three factorial moments of the EIGD to the corresponding sample factorial moments say / 1 m , / 2 m , and / 3 m and there by we obtain the following system of equations: where 1 δ , 2 δ and 3 δ are given in (8). Now the parameters of EIGD are estimated by solving the non-linear equations (16), (17) and (18).

Method of mixed moments
In method of mixed moments, the parameters are estimated by using the first two sample factorial moments and the first observed frequency of the distribution. That is, the parameters are estimated by solving the following equation together with (16) and (17).
where ( ) 1 2, , Λ ρ ρ θ is as defined in (3)   We present the fitting of the intervened geometric distribution (IGD) and the extended intervened geometric distribution (EIGD) for particular values of m for the data set on the count of the number of European red mites on apple leaves taken from Jani and Shah [18]. We estimate the parameters by the method of factorial moments, the method of mixed moments and the method of maximum likelihood. We have computed the values of χ 2 statistics in the case of each model and the numerical results are summarized in Table 1, Table 2 and Table 3. From the tables it is obvious that the EIGD with m=3 gives a better fit compared to IGD as well as for the case m=1, m=2(MIGD) and m=4.  (IGD)  m=1  m=2  m=3  m=4  1  38  27  28  26  40  29  2  17  22  23  17  16  19  3  10  14  14  15  6  10  4  9  8  7  9  10  5  5 +  6  9  8  13  8  17  Total  80  80  80