Minimum Density Power Divergence Estimator Research Articles

Minimum density power divergence estimation for the generalized exponential distribution

Statistical modeling of rainfall data is an active research area in agro-meteorology. The most common models fitted to such datasets are exponential, gamma, log-normal, and Weibull distributions. As an alternative to some of these models, the generalized exponential (GE) distribution was proposed by Gupta and Kundu (2001a). Rainfall (specifically for short periods) datasets often include outliers, and thus, a proper robust parameter estimation procedure is necessary. Here, we use the popular minimum density power divergence estimation (MDPDE) procedure developed by Basu et al. (1998) for estimating the GE parameters. We derive the analytical expressions for the estimating equations and asymptotic distributions. We analytically compare MDPDE with maximum likelihood estimation in terms of robustness, through an influence function analysis. Besides, we study the asymptotic relative efficiency of MDPDE analytically for different parameter settings. We apply the proposed technique to some simulated datasets and two rainfall datasets from Texas, United States. The results indicate superior performance of MDPDE compared to the other existing estimation techniques in most of the scenarios.

Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

AbstractRecently, a growing interest is evident in modelling dependent competing risks in lifetime prognosis problems. In this work, we propose to model the dependent competing risks by Marshal‐Olkin bivariate exponential distribution. The observable data consists of a number of failures due to different causes across different time intervals. The failure count data is common in instances like one‐shot devices where the state of the subjects is inspected at different inspection times rather than the exact failure times. The point estimation of the lifetime distribution in the presence of competing risk has been studied through a divergence‐based robust estimation method called minimum density power divergence estimation (MDPDE) with and without constraint. The optimal value of the tuning parameter has been obtained. The testing of the hypothesis is performed based on a Wald‐type test statistic. The influence function is derived for the point estimator and the test statistic, reflecting the degree of robustness. Another key contribution of this work is determining the optimal inspection times based on predefined objectives. This article presents the determination of multi‐criteria‐based optimal design. Population‐based heuristic algorithm nondominated sorting‐based multiobjective Genetic algorithm is exploited to solve this optimization problem.

Robust estimation for general integer-valued autoregressive models based on the exponential-polynomial divergence

In this study, we develop a robust estimator for integer-valued one-parameter exponential family autoregressive models, named general integer-valued autoregressive models. This model accommodates a broad class of integer-valued time series models. In particular, we propose a robust estimation method that minimizes the exponential-polynomial divergence (EPD) belonging to the Brègman divergence family. EPD subsumes the density power divergence (DPD), which has been extensively studied by many authors for the past decades. Under regularity conditions, the minimum EPD estimator (MEPDE) is shown to be consistent and asymptotically normal. Comparing the performance of MEPDE with the minimum DPD estimator, we substantiate the validity of MEPDE through a simulation study and real data analysis.

Analysis of one-shot device testing data under logistic-exponential lifetime distribution with an application to SEER gallbladder cancer data

In the literature, the reliability analysis of one-shot devices is found under accelerated life testing in the presence of various stress factors. The application of one-shot devices can be extended to the bio-medical field, where we often evidence that inflicted with a certain disease, survival time would be under different stress factors like environmental stress, co-morbidity, the severity of disease etc. This work is concerned with a one-shot device data analysis and applies it to SEER gallbladder cancer data. The two-parameter logistic-exponential distribution is applied as a lifetime distribution. For robust parameter estimation, weighted minimum density power divergence estimators (WMDPDE) is obtained along with the conventional maximum likelihood estimators (MLE). The asymptotic behaviour of the WMDPDE and the robust test statistic based on the density power divergence measure are also studied. The performances of estimators are evaluated through extensive simulation experiments. Later those developments are applied to SEER gallbladder cancer data. Citing the importance of knowing exactly when to inspect the one-shot devices put to the test, a search for optimum inspection times is performed. This optimization is designed to minimize a defined cost function which strikes a trade-off between the precision of the estimation and experimental cost. The search is accomplished through the population-based heuristic optimization method genetic algorithm.

Robust estimation for bivariate integer-valued autoregressive models based on minimum density power divergence

This study considers a robust estimation method for bivariate integer-valued autoregressive (BINAR) models using a minimum density power divergence (MDPD) scheme. We first address this problem within a general framework of time series models to verify the strong consistency and asymptotic normality of the MDPD estimator (MDPDE) under regularity conditions. Based on these results, we assert the same asymptotic properties of MDPDE for BINAR models and propose an outlier detection rule based on MDPDE. To assess the performance of the proposed method, we conduct a simulation study. Real data analysis is also carried out for illustration, using the number of monthly earthquakes in the United States.

A hybrid method for density power divergence minimization with application to robust univariate location and scale estimation

We develop a new globally convergent optimization method for solving a constrained minimization problem underlying the minimum density power divergence estimator for univariate Gaussian data in the presence of outliers. Our hybrid procedure combines classical Newton’s method with a gradient descent iteration equipped with a step control mechanism based on Armijo’s rule to ensure global convergence. Extensive simulations comparing the resulting estimation procedure with the more prominent robust competitor, Minimum Covariance Determinant (MCD) estimator, across a wide range of breakdown point values suggest improved efficiency of our method. Application to estimation and inference for a real-world dataset is also given.

Robust estimation of fixed effect parameters and variances of linear mixed models: the minimum density power divergence approach

Many real-life data sets can be analyzed using linear mixed models (LMMs). Since these are ordinarily based on normality assumptions, under small deviations from the model the inference can be highly unstable when the associated parameters are estimated by classical methods. On the other hand, the density power divergence (DPD) family, which measures the discrepancy between two probability density functions, has been successfully used to build robust estimators with high stability associated with minimal loss in efficiency. Here, we develop the minimum DPD estimator (MDPDE) for independent but non-identically distributed observations for LMMs according to the variance components model. We prove that the theoretical properties hold, including consistency and asymptotic normality of the estimators. The influence function and sensitivity measures are computed to explore the robustness properties. As a data-based choice of the MDPDE tuning parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} is very important, we propose two candidates as “optimal” choices, where optimality is in the sense of choosing the strongest downweighting that is necessary for the particular data set. We conduct a simulation study comparing the proposed MDPDE, for different values of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}, with S-estimators, M-estimators and the classical maximum likelihood estimator, considering different levels of contamination. Finally, we illustrate the performance of our proposal on a real-data example.

Robust density power divergence estimates for panel data models

The panel data regression models have become one of the most widely applied statistical approaches in different fields of research, including social, behavioral, environmental sciences, and econometrics. However, traditional least-squares-based techniques frequently used for panel data models are vulnerable to the adverse effects of data contamination or outlying observations that may result in biased and inefficient estimates and misleading statistical inference. In this study, we propose a minimum density power divergence estimation procedure for panel data regression models with random effects to achieve robustness against outliers. The robustness, as well as the asymptotic properties of the proposed estimator, are rigorously established. The finite-sample properties of the proposed method are investigated through an extensive simulation study and an application to climate data in Oman. Our results demonstrate that the proposed estimator exhibits improved performance over some traditional and robust methods in the presence of data contamination.

Robust sure independence screening for nonpolynomial dimensional generalized linear models

AbstractWe consider the problem of variable screening in ultra‐high‐dimensional generalized linear models (GLMs) of nonpolynomial orders. Since the popular SIS approach is extremely unstable in the presence of contamination and noise, we discuss a new robust screening procedure based on the minimum density power divergence estimator (MDPDE) of the marginal regression coefficients. Our proposed screening procedure performs well under pure and contaminated data scenarios. We provide a theoretical motivation for the use of marginal MDPDEs for variable screening from both population as well as sample aspects; in particular, we prove that the marginal MDPDEs are uniformly consistent leading to the sure screening property of our proposed algorithm. Finally, we propose an appropriate MDPDE‐based extension for robust conditional screening in GLMs along with the derivation of its sure screening property. Our proposed methods are illustrated through extensive numerical studies along with an interesting real data application.

Modeling and inference for multivariate time series of counts based on the INGARCH scheme

Modeling multivariate time series of counts using the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) scheme is proposed. The key idea is to model each component of the time series with a univariate INGARCH model, where the conditional distribution is modeled with a one-parameter exponential family distribution, and to use a (nonlinear) parametric function of all components to recursively produce the conditional means. It is shown that the proposed multivariate INGARCH (MINGARCH) model is strictly stationary and ergodic. For inference, the quasi-maximum likelihood estimator (QMLE) and the minimum density power divergence estimator (MDPDE) for robust estimation are adopted, and their consistency and asymptotic normality are verified. As an application, the change point test based on the QMLE and MDPDE is illustrated. The Monte Carlo simulation study and real data analysis using the number of weekly syphilis cases in the United States are conducted to confirm the validity of the proposed method.

Robust Interval Estimation Using Density Power Divergence

The maximum likelihood estimator (MLE) is known to be asymptotically efficient but it is not robust with respect to outliers and model misspecification. Basu et al. (1998) suggest a family of divergence measures called density power divergences , which is based on density. Each measure in this family is indexed by a single tuning parameter a. This parameter controls the trade-off between asymptotic efficiency and robustness of the estimators. The L₂-distance and the Kullback-Leibler divergence belong to this family. With a appropriately chosen tuning parameter, one can get a minimum density power divergence estimator (MDPDE). For 0 < a < 1, the estimator is in between MLE and minimum L₂-distance estimator L₂E. Note that MLE is efficient-but-nonrobust while L₂E is robust-butinefficient. Hong, Kim (2001) suggest an automatic selection of a. In this paper we will suggest a confidence interval using MDPDE when the data set is contaminated. Bootstrap resampling will be used to obtain the confidence interval. It is expected that the resulting confidence intervals (called MDPD bootstrap confidence intervals) are robust with respect to the outliers. The performances of the MDPDE bootstrap confidence intervals are investigated via simulation study.

Robust density power divergence based tests in multivariate analysis: A comparative overview of different approaches

Hypothesis testing is one of the fundamental paradigms of statistical inference. The three canonical hypothesis testing procedures available in the statistical literature are the likelihood ratio (LR) test, the Wald test and the Rao (score) test. All of them have good optimality properties and past research has not identified any of these three procedures to be a clear winner over the other two. However, the classical versions of these tests are based on the maximum likelihood estimator (MLE), which, although the most optimal estimator asymptotically, is known for its lack of robustness under outliers and model misspecification. In the present paper we provide an overview of the analogues of these tests based on the minimum density power divergence estimator (MDPDE), which presents us with an alternative option that is strongly robust and highly efficient. Since these tests have, so far, been mostly studied for univariate responses, here we primarily focus on their performances for several important hypothesis testing problems in the multivariate context under the multivariate normal model family.

On distance-type Gaussian estimation

In this paper, we develop a new procedure for estimating the parameters of a model by combining Zhang’s (2019) recent Gaussian estimator and the minimum density power divergence estimators of Basu et al. (1998). The proposed estimator is called the Minimum Density Power Divergence Gaussian Estimator (MDPDGE). The consistency and asymptotic normality of the MDPDGE are proved. The MDPDGE is applied to some classical univariate distributions and it is also investigated for the family of elliptically contoured distributions. A numerical study illustrates the robustness of the proposed estimator.

Robust parametric inference for finite Markov chains

We consider the problem of statistical inference in a parametric finite Markov chain model and develop a robust estimator of the parameters defining the transition probabilities via minimization of a suitable (empirical) version of the popular density power divergence. Based on a long sequence of observations from a first-order stationary Markov chain, we have defined the minimum density power divergence estimator (MDPDE) of the underlying parameter and rigorously derived its asymptotic and robustness properties under appropriate conditions. Performance of the MDPDEs is illustrated theoretically as well as empirically for some common examples of finite Markov chain models. Its applications in robust testing of statistical hypotheses are also discussed along with (parametric) comparison of two Markov chain sequences. Several directions for extending the MDPDE and related inference are also briefly discussed for multiple sequences of Markov chains, higher order Markov chains and non-stationary Markov chains with time-dependent transition probabilities. Finally, our proposal is applied to analyze corporate credit rating migration data of three international markets.

Robust estimation of Pareto-type tail index through an exponential regression model

In this paper, we introduce a robust estimator of the tail index of a Pareto-type distribution. The estimator is obtained through the use of the minimum density power divergence with an exponential regression model for log-spacings of top order statistics. The proposed estimator is compared to existing minimum density power divergence estimators of the tail index based on fitting an extended Pareto distribution and exponential regression model on log-ratio of spacings of order statistics. We derive the influence function and gross error sensitivity of the proposed estimator of the tail index to study its robustness properties. In addition, a simulation study is conducted to assess the performance of the estimators under different contaminated samples from different distributions. The results show that our proposed estimator of the tail index has better mean square errors and is less sensitive to an increase in the number of top order statistics. In addition, the estimation of the exponential regression model yields estimates of second-order parameters that can be used for estimation of extreme events such as quantiles and exceedance probabilities. The proposed estimator is illustrated with practical datasets on insurance claims and calcium content in soil samples.

Robust Estimation for Bivariate Poisson INGARCH Models

In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the minimum density power divergence estimator. We demonstrate the proposed estimator is consistent and asymptotically normal under certain regularity conditions. Monte Carlo simulations are conducted to evaluate the performance of the estimator in the presence of outliers. Finally, a real data analysis using monthly count series of crimes in New South Wales and an artificial data example are provided as an illustration.

A Robust Generalization of the Rao Test

This article presents new families of Rao-type test statistics based on the minimum density power divergence estimators which provide robust generalizations for testing simple and composite null hypotheses. The asymptotic null distributions of the proposed tests are obtained and their robustness properties are also theoretically studied. Numerical illustrations are provided to substantiate the theory developed. On the whole, the proposed tests are seen to be excellent alternatives to the classical Rao test as well as other well-known tests.

Improving the Efficiency of Robust Estimators for the Generalized Linear Model

The distance constrained maximum likelihood procedure (DCML) optimally combines a robust estimator with the maximum likelihood estimator with the purpose of improving its small sample efficiency while preserving a good robustness level. It has been published for the linear model and is now extended to the GLM. Monte Carlo experiments are used to explore the performance of this extension in the Poisson regression case. Several published robust candidates for the DCML are compared; the modified conditional maximum likelihood estimator starting with a very robust minimum density power divergence estimator is selected as the best candidate. It is shown empirically that the DCML remarkably improves its small sample efficiency without loss of robustness. An example using real hospital length of stay data fitted by the negative binomial regression model is discussed.

Robust Wald-type tests under random censoring.

Randomly censored survival data are frequently encountered in biomedical or reliability applications and clinical trial analyses. Testing the significance of statistical hypotheses is crucial in such analyses to get conclusive inference but the existing likelihood-based tests, under a fully parametric model, are extremely nonrobust against outliers in the data. Although there exists a few robust estimators given randomly censored data, there is hardly any robust testing procedure available in the literature in this context. One of the major difficulties here is the construction of a suitable consistent estimator of the asymptotic variance of robust estimators, since the latter is a function of the unknown censoring distribution. In this article, we take the first step in this direction by proposing a consistent estimator of asymptotic variance of the M-estimators based on randomly censored data without any assumption on the censoring scheme. We then describe and study a class of robust Wald-type tests for parametric statistical hypothesis, both simple as well as composite, under such a set-up. Robust tests for comparing two independent randomly censored samples and robust tests against one sided alternatives are also discussed. Their advantages and usefulness are demonstrated for the tests based on the minimum density power divergence estimators and illustrated with clinical trials and other medical data.

Robustness of Minimum Density Power Divergence Estimators and Wald-type test statistics in loglinear models with multinomial sampling

In this paper we propose a new family of estimators, Minimum Density Power Divergence Estimators (MDPDE), as a robust generalization of maximum likelihood estimators (MLE) for the loglinear model with multinomial sampling by using the Density Power Divergence (DPD) measure introduced by Basu et al. (1998). Based on these estimators, we further develop two types of confidence intervals (asymptotic and bootstrap ones), as well as a new robust family of Wald-type test statistics for testing a nested sequence of loglinear models. Furthermore, we study theoretically the robust properties of both the MDPDE as well as Wald-type tests through the classical influence function analysis. Finally, a simulation study provides further confirmation of the validity of the theoretical results established in the paper.