Journal of Probability and Statistics

Volume 2019, Article ID 8575424, 13 pages

https://doi.org/10.1155/2019/8575424

## The New Odd Log-Logistic Generalized Inverse Gaussian Regression Model

^{1}ESALQ, Universidade de São Paulo, Piracicaba, Brazil^{2}DEINFO, Universidade Federal de Pernambuco, Recife, Brazil^{3}UFMS, Universidade Federal de Mato Grosso do Sul, Paranaíba, Brazil

Correspondence should be addressed to Edwin M. M. Ortega; rb.psu@niwde

Received 7 August 2018; Accepted 10 December 2018; Published 10 January 2019

Academic Editor: Aera Thavaneswaran

Copyright © 2019 Julio Cezar Souza Vasconcelos 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 define a new four-parameter model called the odd log-logistic generalized inverse Gaussian distribution which extends the generalized inverse Gaussian and inverse Gaussian distributions. We obtain some structural properties of the new distribution. We construct an extended regression model based on this distribution with two systematic structures, which can provide more realistic fits to real data than other special regression models. We adopt the method of maximum likelihood to estimate the model parameters. In addition, various simulations are performed for different parameter settings and sample sizes to check the accuracy of the maximum likelihood estimators. We provide a diagnostics analysis based on case-deletion and quantile residuals. Finally, the potentiality of the new regression model to predict price of urban property is illustrated by means of real data.

#### 1. Introduction

The inverse Gaussian (IG) distribution is widely used in several research areas, such as life-time analysis, reliability, meteorology and hydrology, engineering, and medicine. Some extensions of the IG distribution have appeared in the literature. For example, the* generalized inverse Gaussian* (GIG) distribution with positive support is introduced by Good [1] in a study of population frequencies. Several papers have investigated the structural properties of the GIG distribution. Sichel [2] used this distribution to construct mixtures of Poisson distributions. Statistical properties and distributional behavior of the GIG distribution were discussed by Jørgensen [3] and Atkinson [4]. Dagpunar [5] provided algorithms for simulating this distribution. Nguyen et al. [6] showed that it has positive skewness. More recently, Madan et al. [7] proved that the Black-Scholes formula in finance can be expressed in terms of the GIG distribution function. Koudou [8] presented a survey about its characterizations and Lemonte and Cordeiro [9] obtained some mathematical properties of the* exponentiated generalized inverse Gaussian* (EGIG) distribution.

In this paper, we study a new four-parameter model named the* odd log-logistic generalized inverse Gaussian* (OLLGIG) distribution which contains as special cases the GIG and IG distributions, among others. Its major advantage is the flexibility in accommodating several forms of the density function, for instance, bimodal and unimodal shapes. It is also suitable for testing goodness-of-fit of some submodels.

Our main objective is to study a new regression model with two systematic structures based on the OLLGIG distribution. We obtain some mathematical properties and discuss maximum likelihood estimation of the parameters. For these models, we presented some ways to perform global influence (case-deletion) and, additionally, we developed residual analysis based on the quantile residual. For different parameter settings and sample sizes, various simulation studies were performed and the empirical distribution of quantile residual was displayed and compared with the standard normal distribution. These studies suggest that the empirical distribution of the quantile residual for the OLLGIG regression model with two regression structures a high agreement with the standard normal distribution.

This paper is organized as follows. In Section 2, we define the OLLGIG distribution. In Section 3, we obtain some of its structural properties. We define the OLLGIG regression model in Section 4 and evaluate the performance of the maximum likelihood estimators (MLEs) of the model parameters by means of a simulation study. In Section 5, we adopt the case-deletion diagnostic measure and define quantile residuals for the fitted model. Further, we perform various simulations for these residuals. In Section 6, we provide two applications to real data to illustrate the flexibility of the OLLGIG regression model. Finally, some concluding remarks are offered in Section 7.

#### 2. The OLLGIG Distribution

The GIG distribution [3] has been applied in several areas of statistical research. The cumulative distribution function (cdf) and probability density function (pdf) of the GIG distribution are given by (for )andwhere is the location parameter, is the scale parameter, is the shape parameter, is the modified Bessel function of the third kind and index , , and .

We denote by a random variable having density function (2). The mean and variance of arerespectively.

The moment generating function (mgf) of reduces toWe use the reparameterized GIG distribution according to GAMLSS in software R. For example, we have Other properties of the GIG distribution are investigated by Jørgensen [3].

The statistical literature is filled with hundreds of continuous univariate distributions. Recently, several methods of introducing one or more parameters to generate new distributions have been proposed. Based on the* odd log-logistic generator* (OLL-G) [10], we define the OLLGIG cdf, say , by integrating the log-logistic density function as follows:where , is a position parameter, is a scale parameter, and and are shape parameters. Clearly, is a special case of (5) when .

Henceforth, we write to simplify the notation. The OLLGIG density function can be expressed as

The main motivations for the OLLGIG distribution are to make its skewness and kurtosis more flexible (compared to the GIG model) and also allow bi-modality. We have , where and . Thus, the parameter represents the quotient of the log odds ratio for the new and baseline distributions. Note that the pdf and cdf of the OLLGIG distribution depend on integrals, which are calculated numerically in the same way as those of the Birnbaum-Saunders distribution.

Hereafter, we assume that the random variable follows the OLLGIG cdf (5) with parameters , say . The OLLGIG distribution contains as special cases the GIG distribution when and the IG distribution when and .

Some plots of the OLLGIG density for selected parameter values are displayed in Figure 1. It is evident that the proposed distribution is much more flexible, especially in relation to bi-modality (for ), than the GIG and IG distributions.