Stein‐type shrinkage estimators in gamma regression model with application to prostate cancer data

Abstract

Funding information Natural Sciences and Engineering Research Council (NSERC) of CanadaGamma regression is applied in several areas such as life testing, forecasting cancer incidences, genomics, rainfall prediction, experimental designs, and quality control. Gamma regression models allow for a monotone and no constant hazard in survival models. Owing to the broad applicability of gamma regression, we propose some novel and improved methods to estimate the coefficients of gamma regression model. We combine the unrestricted maximum likelihood (ML) estimators and the estimators that are restricted by linear hypothesis, and we present Stein-type shrinkage estimators (SEs). We then develop an asymptotic theory for SEs and obtain their asymptotic quadratic risks. In addition, we conduct Monte Carlo simulations to study the performance of the estimators in terms of their simulated relative efficiencies. It is evident from our studies that the proposed SEs outperform the usual ML estimators. Furthermore, some tabular and graphical representations are given as proofs of our assertions. This study is finally ended by appraising the performance of our estimators for a real prostate cancer data. KEYWORDS asymptotic quadratic risk, gamma regression, positive-part Stein-type shrinkage estimator, prostate cancer, relative efficiency, Stein-type shrinkage estimator 4310

Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in GaGLM

“…According to (5), the first three order derivative of l n (θ) with respect to θ is continuous and finite for all θ ∈ K θ . This condition ensures the existence of the Taylor expansion, the finite variance of the derivatives of l n (θ).…”

“…Because of the flexibility of the relationship to many other distributions, the Gamma distribution can be a suitable alternative for modelling such kinds of the positive-valued dependent variable. The Gamma distribution-based models have been applied in many areas, such as medical science [4], [5], biology [6], economics [7], [8], forest science [9] and education [10]. Considering the ubiquitous heteroscedasticity of actually applied data, as a member of the well-known GLM, the Gamma distribution based generalized linear model (GaGLM) is more widely used when α and β are both dependent variables.…”

Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in GaGLM

“…According to (5), the first three order derivative of l n (θ θ θ) with respect to θ θ θ is continuous and finite for all θ θ θ ∈ K θ θ θ . This condition ensures the existence of the Taylor expansion, the finite variance of the derivatives of l n (θ θ θ).…”

“…Because of the flexibility of the relationship to many other distributions, the Gamma distribution can be a suitable alternative for modelling such kinds of the positive-valued dependent variable. The Gamma distribution-based models have been applied in many areas, such as medical science [4], [5], biology [6], economics [7], [8], forest science [9] and education [10]. Considering the ubiquitous heteroscedasticity of actually applied data, as a member of the well-known GLM, the Gamma distribution based generalized linear model (GaGLM) is more widely used when α and β (or µ and k ) are both dependent variables.…”

A new class of efficient and debiased two-step shrinkage estimators: method and application

“…Its main advantage over the ridge regression estimator is that it is a linear function of the shrinkage parameter. 2 Lately, shrinkage estimators have been used for different types of datasets such as Mandal et al [22] where the Gamma regression model was considered to analyze the prostate cancer data, Maronna [25] suggesting methods for high dimensional data, and Peterson and Kuhn [30] developed methods to deal with noise variables. Different estimators have been proposed in the literature to improve the original ridge estimator by Hoerl and Kennard [9].…”