On Some Properties and Applications of Intervened Gegenbauer Distribution

In this paper, an intervened version of the Gegenbauer distribution is considered and investigated some of its statistical properties. The parameters of the distribution are estimated by the method of maximum likelihood and illustrated using real life data sets. The likelihood ratio test procedure is applied for examining the significance of the intervention parameters and a simulation study is carried out for assessing the performances of the maximum likelihood estimators.


Introduction
Intervened type distributions have received much attention in the literature due to their extensive practical utility. For example these types of distributions have found utility in the study of effectiveness of different types of treatments in connection with various diseases. Also they have been utilized in certain studies on advertisement effectiveness and in quality control for controlling the defective items. (Bartolucci.et.al, 2001) developed an intervened geometric distribution (IGD) as a modification to zero-truncated geometric distribution in connection with a cardiovascular study. Sreeja, 2014, 2016) 32 ISSN-2424-627 IASSL studied some modified versions of the IGD. (Kumar and Sreeja, 2012) have proposed the intervened negative binomial distribution (INBD) as a generalization of the IGD. The INBD is the distribution of the sum of a zero truncated negative binomial random variable and an independent negative binomial random variable. That is, a random variable U is said to follow the INBD if its probability mass function (pmf) fu has the following form, for u = 1, 2, 3, ….
The INBD is suitable for single intervention situations. But there are many practical situations where intervention occurs in multiple form. So in order to model such situations a more general class of the INBD is required. So through this paper we propose a model as a generalization of the INBD which is suitable for tackling multiple intervention situations, named as "the intervened Gegenbauer distribution" or in short "the IGbD" The paper is organized in such a way that in section 2 we develop a model leading to the IGbD and derive expressions for the pmf, moments, mean, variance and recurrence relations of probabilities . In section 3 we discuss the estimation of the parameters of the IGbD by the method of maximum likelihood. A simulation study is carried out in section 4 for IASSL ISBN-1391-4987 33 examining the performance of the estimators of the parameters of the IGbD and to know the effectiveness of intervention parameters. A discussion part is included in section 5.
We need the following in the sequel. For any real valued function A(i; r); we have and for any λ > 0, a > 0 and b > 0 such that a + b < 1; the Gegenbauer polynomials Gn λ (a,b) defined through the generating function where [k] denotes the integer part of k. For details of Gn λ (a,b) see Plunket and Jain(1975).

. Intervened Gegenbauer Distribution
Let Z be a random variable having zero truncated Gegenbauer distribution (ZTGbD) with parameters r, θ1 and θ2. Then the pgf of Z is in which r > 0 and θi > 0 for i = 1, 2 such that θ1 + θ2 ≤1.
Let Y be a random variable following Gegenbauer distribution in which, due to some intervention, the parameters θ1 changes to ρ1θ1and θ2 changes to ρ2θ2 with ρ1 > 0, ρ2 > 0 such that ρ1θ1+ ρ2θ2 ≤ 1 and the parameters ρ1 and ρ2 are called the intervention parameters. Assume that Y and Z are statistically independent. Then the pgf of X = Y + Z is given by in which r > 0, ρi > 0 and θi > 0 for i = 1, 2 such that ρ1θ1+ ρ2θ2 ≤ 1. The distribution of a random variable X with pgf (2.3), we call "the intervened Gegenbauer distribution" or in short "the IGbD". Clearly, when θ2 = 0 the pgf (2. in which r > 0, ρi > 0 and θi > 0 for i = 1, 2 such that ρ1θ1+ ρ2θ2 ≤ 1and Gn λ (a,b) is the Gegenbauer polynomial as defined in (1.3).

Proof:
By definition, the pgf Px(s) of the IGbD is Applying (1.3) in (2.6), we obtain the following.
in the light of (1.2). Now, on equating the coefficients of s x in the right hand side expressions of (2.5) and (2.8), we get (2.4). Next we develop an expression for the k th factorial moment of the IGbD through the following proposition.

Proof
The factorial moment generating function By replacing s by 1 + t in (2.3), we get the in the light of (1.2).
On equating the coefficient of ! k t k on the right hand side expressions of (2.10) and (2.13),we get (2.9).
The mean and variance of the IGbD are respectively  and  are as given in (2.9).
ISSN-2424-627 IASSL Proposition 2.4. The following is a recurrence relation for probabilities of IGbD, for x ≥ 1, On differentiating (2.15) and (2.16) with respect to s , we get the following.

Estimation
Here we discuss the estimation of the parameters of IGbD by the method of maximum likelihood. Let a(x) be the observed frequency of x events, y be the highest value of x observed. Then the likelihood function of the sample is      Note that these likelihood equations do not always have closed form solutions. Therefore, maximum of the likelihood function is attained at the border of domain of the parameters. We obtain the second order partial derivatives of logL with respect to the parameters In order to illustrate the usefulness of the model IGbD, we have considered three sets of real life data and compared with some existing competing intervened type models such as IGD, INBD and MINBD along with ZTNBD and ZTGbD. The numerical results obtained are summarized in Table 1, 2 and 3. From the tables it is clear that the IGbD gives better fit to all those data sets compared to ZTGbD, IGD, ZTNBD,INBD and MINBD. The data given in Table 1 and 2 are related to the number of research publications in the credit of authors . The authors have more number of publications to their credit, that may be due to the influence of some interventions such as measures taken for academic grade, promotion, research incentives etc. The data given in Table 3 is related to the number of fly eggs on flower heads. The data indicates that there are flower heads with more number of eggs which might be due to the effect of certain interventions such as nutritious food, rainfall, temperature, humidity, sunlight etc.
We have computed the bias and mean square errors in each case and presented in Table 4 and 5. From the tables it can be seen that as the sample size increases, the absolute bias and mean square error decreases.
In order to know the effectiveness of the intervention parameter, we have carried out the test H0: ρ1 =0, ρ2 = 0 vs H1 : ρ1 > 0, ρ2 > 0 by using generalized likelihood ratio procedure to the simulated data sets. For the data set generated by the parameter set I, the probability of Type I error is 0.00018 and that of the data set generated by the parameter set II is 0.00026.

Summary and Conclusion
A new class of the intervened type distribution is introduced as a generalization of both the zero truncated negative binomial distribution and the intervened negative distribution through the name intervened Gegenbaur distribution. We obtained several important properties of the distribution and fitted the model to three real life data sets. A brief simulation study is carried out for assessing the performance of the maximum likelihood estimators of the parameters of the distribution and to examine the significance of the intervention parameters.