Parametric Point Estimation of the Geeta Distribution

Geeta distribution is a new discrete random variable distribution defined over all the positive integers with two parameters. This distribution belongs to the family of Location-parameter (LDPD) system and is of the form L – shaped model. Pareto and Yule distributions belong to the same family but these distributions have a disadvantage of having a single parameter which makes them not versatile to meet the needs of modern complex data sets. Geeta distribution is found to be very versatile and flexible to fit observed count data sets and can be used efficiently to model different types of sets. This paper investigates the characteristics of Geeta distribution, such as the existence of the mean, variance, moment generating function, probability generating function and that the sum of probabilities for all values of X for Geeta Distribution model is unity. It is well known that the sample mean is the estimator of a population mean from a given population of interest as a point estimator which assume a single number that is obtained by taking a random sample of a specified size from the entire population, depending on whether the population mean and variance is known or unknown These point estimators were obtained by employing the method of Moments, Maximum Likelihood (MLE) and Bayesian estimator. Further the estimators were subjected to the conditions like unbiasedness, efficiency, sufficiency and completeness which are properties of a good estimator. For the first aspect, the results of the mean, variance, moments and generating functions were achieved that proves the distribution is a probability density function (pdf). The methods of moments and the maximum likelihood and their properties were applied and yielded the desired and expected results for any given probability distribution. The best estimator obtained is best linear unbiased estimator (BLUE).


Introduction
The estimation of parameters of a probability distribution is a problem of basic importance in modern mathematical statistics [1]. However, several theories of estimation are available which apply under general conditions and are categorized into two broad classes, the parametric interval estimation and parametric point estimation [2]. This paper focuses on the parametric point estimation and any given distribution expressed in the mathematical form either in its discrete or continuous state, the techniques of obtaining the estimators shall apply.
Given a population with a probability density function f(x i , θ 1 , θ 2 , …, θ n ) which would completely be determined if the values of the parameters θ 1 , θ 2 , …, θ n were known. Let X 1 , X 2 ..., X n be a random sample of observations X 1 , X 2 ,..., X n from this population, it is required to determine some model, it belongs to a family of Modified Power Series distribution(MPSD), the langrangian series distributions and location parameter distribution. The Yule distribution and Pareto which belongs to the same family(MPSD) have a single parameter but fails the test of handling large data sets when it comes to applications in modern technologies [3]. Geeta distribution model is very versatile in meeting the needs of modern complex data sets and this is attributed to the presence of the two unknown constants when compared to the distribution of the same class. The unknown constants can be estimated using the estimation techniques and it is believed that these constants contain a lot of information since estimation is accompanied by statistical inference.
Geeta distribution is defined as a discrete random variable, X, over the set of all positive integers, with the probability mass function [4] given by ( ) The family of Geeta probability model belongs to the classes of the modified power series distribution (MPSD) and the Langrangian series distribution. It can also be expressed as a location parameter probability distribution [3] given below ( ) where µ is the mean and β > 1. Note that this form does not have an upper limit on β. Consul (1990) has shown that the Geeta distribution (4) can be characterized by its variance: ( ) ( )( ) the variance will be less than the mean, µ and will have the range: σ will become larger than µ.

Bar Diagrams for Geeta Distribution Model
The successive probabilities for various values of X can be easily computed from the values: The probabilities for the Geeta distribution (4) were computed for µ = 1. [1]. It is clear from these graphs that ( ) Pr 1 X = reduces as µ increases and the probabilities for all other values of x increase but the model always remains L-shaped. Thus the tail becomes more and more heavy and longer with the increase in the value of µ . There is a similar effect when the value of µ is kept fixed and the value of β is slowly increased. The value of ( )

Bar -Diagram for Geeta
decreases and the probabilities for other values of x increase as β increases. However, these changes for β are at a much slower pace than the changes for µ with the result that the Geeta probability model becomes more suitable and versatile than some other models for abundance data sets.

Methodology
The parametric point estimation is twofold; finding point estimators and then determining good, or optimum estimator. In the preliminaries stages a statistic is obtained by taking a sample of size, n from the entire population. A statistic can be expressed a function of sample observation and there are many statistics from same population that can be obtained. Assume that some characteristics of the elements in a population can be represented by a random variable X whose density is , where the form of the density is assumed known except that it contains an unknown parameter θ.
The methods stated here like the least squares estimators (LSE) is commonly used in estimating parameters in linear regression models assuming a normal distribution of errors of observation and the condition of linear combination of the values with error variance being minimum. The maximum likelihood estimates (MLE) which yields an unbiased estimator, the method of moments, Bayesian estimators, minimum variance estimators [6,7] and Cramer Rao lower bound [8] that is used to obtain the Uniformly Minimum variance unbiased estimator (UMVUE) are some of the techniques employed when estimating the unknown parameters.

Methods of Finding Estimators
Definition 1: Any statistic whose function of observable random variables that is itself (a random variable) whose values are used to estimate τ (θ) or θ where τ (.) is some function of the parameter θ, is defined to be an estimator of τ (θ) or θ.

Method of Moments
One of the simplest methods of estimation is by moments  [11,12]. With the Geeta distribution given as in (1), the th r moment of the population is defined as ( ) r r E X µ′ = (10) and the th j sample moment j M ′ is defined by Now from (11) we have: and from (10) we have: We see that (12) and (13)

Maximum Likelihood Estimator
A maximum likelihood estimator, if it exists is the most efficient estimator; it happens sometimes that the method of maximum likelihood leads to complicated equations as we shall see in obtaining the estimate of parameter in Geeta distribution [9].
Let , , … , be the values of a random sample 1 2 , , , n X X X ⋯ from The maximum likelihood estimator of θ is obtained by taking the logs of the likelihood function, then differentiating partially with respect to θ and equate to zero,; that is;

Properties of Estimators
We have seen that many estimates will be available to us in any given situation. It is therefore desired to investigate some properties of point estimates. This will help in choosing the best estimates [7].

Unbiasedness
An An estimate that is not unbiased is called biased.

Consistency
Let , , … , be a sequence of independent and identically distributed random variables with common density function

( )
; ; ∈ Ω . A sequence of point estimates ( 1 , 2 , … , ) = will be called consistent for It is important to remember that consistency is essentially a large sample property.
If n T is a sequence of estimates such that ( ) Var T → as → ∞ then T n is consistent for θ.

Efficiency
Let T1, T2 be two unbiased estimators for a parameter θ.
 Clearly the efficiency of the most efficient estimates is 1, and the efficiency of any unbiased estimator 1 T is > 1.

Sufficiency
Let , , … , be a sequence of independent and identically distributed random variables with common density function

( )
; ; ∈ Ω ., where θ may be vector. A statistic = ( 1 , 2 , … , ) is defined to be a sufficient statistic if and only if the conditional distribution of ( , , … , ) given T t = does not depend on for any value of t . Example Poisson sufficient statistic. Let X 1 , X 2 be independent Poisson variables with common expectation λ, so that their joint distribution is Then, the conditional distribution of X 1 given X 1 + X 2 = t is given by Since this is independent of λ, so is the conditional distribution given t of (X 1 ,., X 2 = t − X 1 ), and hence T = X 1 +X 2 is a sufficient statistics for λ. To see how to reconstruct (X 1 ,, X 2 ) from T, note that so that, that is, the conditional distribution of X 1 given t is the binomial distribution b(1/2, t) corresponding to t trials with success probability 1 /2. Let . and . = − . be respectively the number of heads and the number of tails in t tosses with a fair coin. Then, the joint conditional distribution of ( . , . ) given t is the same as that of (X 1 , X 2 ) given t.

Results of Point Estimators
The following results were obtained. Theorem 1: The point estimator of β and θ using the method of moments is given by: Theorem 2: The maximum likelihood estimator of β is given by the intersection of the two graphs The maximum likelihood estimator of µ is given by ˆX µ = . Proof: The Geeta distribution is as given in (1), and its likelihood function is given by Taking the logs of the likelihood function, and partially differentiating with respect to 0 and equating to zero we have ˆX µ = Differentiating the log-likelihood function (20) partially with respect to and equating the results to zero, we have which further simplifies to 1 log ( 1) log 1 which can be re-written as

Properties of the Point Estimators
We have seen that many estimates will be available to us in any given situation. It is therefore desired to investigate some properties of point estimates. This will help in choosing the best estimates. The following results of the properties of the various point estimators were obtained.
Theorem 3 is an unbiased estimator of θ .