Computational and Mathematical Methods in Medicine

Volume 2016 (2016), Article ID 8727951, 12 pages

http://dx.doi.org/10.1155/2016/8727951

## Different Estimation Procedures for the Parameters of the Extended Exponential Geometric Distribution for Medical Data

^{1}Statistics Department, Institute of Mathematical and Computer Sciences (ICMC), São Paulo University (USP), 13560-970 São Carlos, SP, Brazil^{2}Department of Social Medicine, Ribeirão Preto School of Medicine (FMRP), São Paulo University (USP), 14049-900 Ribeirão Preto, SP, Brazil

Received 19 May 2016; Accepted 3 July 2016

Academic Editor: Ezequiel López-Rubio

Copyright © 2016 Francisco Louzada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have considered different estimation procedures for the unknown parameters of the extended exponential geometric distribution. We introduce different types of estimators such as the maximum likelihood, method of moments, modified moments,* L*-moments, ordinary and weighted least squares, percentile, maximum product of spacings, and minimum distance estimators. The different estimators are compared by using extensive numerical simulations. We discovered that the maximum product of spacings estimator has the smallest mean square errors and mean relative estimates, nearest to one, for both parameters, proving to be the most efficient method compared to other methods. Combining these results with the good properties of the method such as consistency, asymptotic efficiency, normality, and invariance we conclude that the maximum product of spacings estimator is the best one for estimating the parameters of the extended exponential geometric distribution in comparison with its competitors. For the sake of illustration, we apply our proposed methodology in two important data sets, demonstrating that the EEG distribution is a simple alternative to be used for lifetime data.

#### 1. Introduction

Many researches are interested in search distributions which can be used to describe real data sets. Generalizations of the standard exponential distribution have been introduced in the literature for this purpose, such as Gamma, Weibull, and Generalized Exponential distribution [1]. Another useful generalization is known as extended exponential geometric distribution. Initially, the development of such distribution was made by Adamidis and Loukas [2] proposing exponential geometric distribution with two parameters, in which the hazard function could be decreasing. In a further paper, Adamidis et al. [3] explored extended exponential geometric (EEG) distribution. Let be a random variable representing a lifetime data, with extended exponential geometric (EEG) distribution; its probability density function (PDF) is given byfor all , , and . One of its peculiarities is that its hazard function can be increasing or decreasing, depending on the values of its parameters, giving great flexibility of fit for real applications.

This model arises naturally in competing risks scenarios. Let , where* M* is a random variable with geometrical distribution and are independent of* M* and are assumed to be independent and identically distributed according to exponential distribution; then the random variable has EEG distribution with , also known as exponential geometric (EG) distribution [2]. Considering the same assumptions and , the random variable has EEG distribution with , also known as Complementary Exponential Geometric distribution [4]. Due to its importance, some generalizations of the EEG distribution have been proposed, such as the Beta exponential geometric distribution [5], Exponentiated Exponential-Geometric distribution [6], Complementary Exponentiated Exponential Geometric distribution [7], and Generalized Exponential Geometric distribution [8].

Despite the fact that EEG distribution has good flexibility, a few estimation procedures have been proposed in the literature. Adamidis et al. [3] derived the maximum likelihood estimators (MLE) for the unknown parameters of the EEG distribution. Ramos et al. [9] developed a Bayesian analysis under noninformative priors. However, considering the frequentist approach, it is well known that, usually, for small samples, the MLE does not perform well. In this paper, we proposed nine new estimators for the parameters of the EEG distribution, which are given considering the following estimation procedures: the method of moments, modified moments, ordinary least squares, weighted least squares,* L*-moments, percentile, maximum product of spacings, Cramer-von Mises type minimum distance, and Anderson-Darling estimator.

The main aim of this paper is twofold. First, it aims to develop a guideline for choosing the most efficient estimators among ten different estimation procedures for the EEG distribution, which would be of interest to applied statisticians. Second, it aims to demonstrate that the EEG distribution is a simple alternative to be used in applications in medicine.

The originality of this study comes from the fact that, for the EEG distribution and considering the frequentist approach, only the MLE has been presented in the literature. The performances of the different estimation methods are compared using extensive numerical simulations. Additionally, these results are analogous for the exponential geometric distribution and the Complementary Exponential Geometric distribution. Related studies for other distributions can be found in Gupta and Kundu [10], Mazucheli et al. [11], Teimouri et al. [12], and Dey et al. [13].

The paper is organized as follows. In Section 2, we discuss some properties of the EEG distribution. In Section 3, we present ten estimation procedures for the parameters of our proposed model. In Section 4, a simulation study is presented in order to identify the most efficient estimators. In Section 5, we apply our proposed methodology in two real data sets. Some final comments are presented in Section 6.

#### 2. Extended Exponential Geometric Distribution

Let be a random variable with density function (1); the distribution function is given by

The survival and hazard functions of distribution is given, respectively, byThe hazard function (3) is decreasing for , is constant for , and is monotonically increasing when . Figure 1 presents different forms for the density and hazard functions for the EEG distribution considering different values of and .