Abstract
Gamma distribution is a general type of statistical distribution that can be applied in various fields, mainly when the distribution of data is not symmetrical. When predictor variables also affect positive outcome, then gamma regression plays a role. In many cases, the predictor variables give effect to several responses simultaneously. In this article, we develop a multivariate gamma regression (MGR), which is one type of non-linear regression with response variables that follow a multivariate gamma (MG) distribution. This work also provides the parameter estimation procedure, test statistics, and hypothesis testing for the significance of the parameter, partially and simultaneously. The parameter estimators are obtained using the maximum likelihood estimation (MLE) that is optimized by numerical iteration using the Berndt–Hall–Hall–Hausman (BHHH) algorithm. The simultaneous test for the model’s significance is derived using the maximum likelihood ratio test (MLRT), whereas the partial test uses the Wald test. The proposed MGR model is applied to model the three dimensions of the human development index (HDI) with five predictor variables. The unit of observation is regency/municipality in Java, Indonesia, in 2018. The empirical results show that modeling using multiple predictors makes more sense compared to the model when it only employs a single predictor.
Highlights
Gamma distribution is one family of continuous probability distributions and generalizations of exponential distributions [1]
Many researchers study and develop bivariate gamma distribution; among others are Schickedanz and Krause [5], who conducted a study on testing scale parameters from two gamma-distributed data using the generalized likelihood ratio (GLR)
The multivariate gamma regression (MGR) is developed based on multivariate gamma distribution with three parameters
Summary
Gamma distribution is one family of continuous probability distributions and generalizations of exponential distributions [1]. Correa, and Gupta [2] mentioned that the gamma distribution function was first introduced by Swiss mathematician Leonhard Euler (1729) Because this function is considered important, many researchers have studied and developed it. Bhattacharya [3], among others, conducted a study on testing the homogeneity of the parameters (shape and scale) of the gamma distribution. Chen and Kotz [4] conducted a study on the probability density function (pdf) of gamma distribution with three parameters (shape, scale, and location). Many researchers study and develop bivariate gamma distribution; among others are Schickedanz and Krause [5], who conducted a study on testing scale parameters from two gamma-distributed data using the generalized likelihood ratio (GLR). The MGR is developed based on multivariate gamma distribution with three parameters (shape, scale, and location).
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