Minimum Density Power Divergence Estimator Research Articles

Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator

The recently observed hydrologic extremes are unlike what has been experienced so far. Both the magnitude and frequency of extremes are important indicators that determine the flood safety design criteria. Therefore, how are design criteria updated faced with these extremes? Both a sudden increase of design rainfall by the inclusion of these extremes and complete ignorance are inappropriate. In this study, the changes in extremes were examined and an alternative way to estimate the design rainfall amounts was developed using the data from 60 stations in South Korea. The minimum density power divergence estimator (MDPDE) with the optimal value of a tuning parameter, α, was suggested as an alternative estimator instead of the maximum likelihood estimator (MLE); its performance was evaluated using the Gumbel (GUM) and the generalized extreme value (GEV) distribution. The results revealed an increase in both the frequency and magnitude of extreme events over the last two decades, which imply that the extremes are already occurring. The performance of the MDPDE was evaluated. The results revealed decreased and adjusted values of the design rainfall compared to MLE. On the other hand, the MDPDE of the GEV distribution with a positive shape parameter, ξ, does not show its advantage conditionally because the GEV distribution has a heavier right tail than the GUM distribution (ξ = 0). In contrast, the results showed the high sensitivity of the MLE to the extremes compared to MDPDE.

A Wald-type test statistic for testing linear hypothesis in logistic regression models based on minimum density power divergence estimator

In this paper a robust version of the classical Wald test statistics for linear hypothesis in the logistic regression model is introduced and its properties are explored. We study the problem under the assumption of random covariates although some ideas with non random covariates are also considered. The family of tests considered is based on the minimum density power divergence estimator instead of the maximum likelihood estimator and it is referred to as the Wald-type test statistic in the paper. We obtain the asymptotic distribution and also study the robustness properties of the Wald type test statistic. The robustness of the tests is investigated theoretically through the influence function analysis as well as suitable practical examples. It is theoretically established that the level as well as the power of the Wald-type tests are stable against contamination, while the classical Wald type test breaks down in this scenario. Some classical examples are presented which numerically substantiate the theory developed. Finally a simulation study is included to provide further confirmation of the validity of the theoretical results established in the paper.

Influence analysis of robust Wald-type tests

We consider a robust version of the classical Wald test statistics for testing simple and composite null hypotheses for general parametric models. These test statistics are based on the minimum density power divergence estimators instead of the maximum likelihood estimators. An extensive study of their robustness properties is given though the influence functions as well as the chi-square inflation factors. It is theoretically established that the level and power of these robust tests are stable against outliers, whereas the classical Wald test breaks down. Some numerical examples confirm the validity of the theoretical results.

Generalized Wald-type tests based on minimum density power divergence estimators

In testing of hypothesis, the robustness of the tests is an important concern. Generally, the maximum likelihood-based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper, we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter β. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests is explored through simulations and real data analysis.

The Minimum Density Power Divergence Estimation for the Lognormal Density

In this article, we implement the minimum density power divergence estimation for estimating the parameters of the lognormal density. We compare the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) in terms of robustness and asymptotic distribution. The simulations and an example indicate that the MDPDE is less biased than MLE and is as good as MLE in terms of the mean square error under various distributional situations.

Minimum density power divergence estimator for covariance matrix based on skew $$t$$ t distribution

In this paper, we study the problem of estimating the covariance matrix of stationary multivariate time series based on the minimum density power divergence method that uses a multivariate skew \(t\) distribution family. It is shown that under regularity conditions, the proposed estimator is strongly consistent and asymptotically normal. A simulation study is provided for illustration.

정규-지수분포에 대한 최소밀도함수승간격 추정법

최소밀도함수승간격 추정법은 Baus 등 (1998)에 의해 처음 소개된 이후 많은 관심의 대상이 되었다. 최소밀도함수승간격 추정량은 우수한 로버스트 성질을 갖고 효율성도 최우추정량에 필적한 것으로 알려져 있다. 본 논문에서는 생물정보학에서 사용되는 노말-지수 분포에 근거한 추정량을 최소밀도함수승간격 추정법을 사용하여 구하는 방법을 다루고자 한다. 그런데 그 과정에서 간격을 적분을 통해 구하는 것이 매우 어려움으로 인해 직접적인 적분 대신 라플라스 근사를 시도할 것을 제안한다. 그 결과 추정량이 다소 효율성이 줄어들지만 로버스트 성질을 갖고 있음을 수학적 방법과 모의실험을 통하여 보였다. The minimum density power divergence estimation has been a popular topic in the field of robust estimation for since Basu et al. (1988). The minimum density power divergence estimator has strong robustness properties with the little loss in asymptotic efficiency relative to the maximum likelihood estimator under model conditions. However, a limitation in applying this estimation method is the algebraic difficulty on an integral involved in an estimation function. This paper considers a minimum density power divergence estimation method with approximated divergence avoiding such difficulty. As an example, we consider the normal-exponential convolution model introduced by Bolstad (2004). The estimated divergence in this case is too complicated; consequently, a Laplace approximation is employed to obtain a manageable form. Simulations and an empirical study show that the minimum density power divergence estimators based on an approximated estimated divergence for the normal-exponential model perform adequately in terms of bias and efficiency.

Minimum density power divergence estimator for Poisson autoregressive models

The robust estimation for Poisson autoregressive models is studied. As a robust estimator, a minimum density power divergence estimator (MDPDE) is considered. It is shown that under regularity conditions, the MDPDE is strongly consistent and asymptotically normal. Simulation results are provided for illustration. A real data analysis is implemented for the polio incidence data.

Local robust and asymptotically unbiased estimation of conditional Pareto-type tails

We introduce a non-parametric robust and asymptotically unbiased estimator for the tail index of a conditional Pareto-type response distribution in presence of random covariates. The estimator is obtained from local fits of the extended Pareto distribution to the relative excesses over a high threshold using an adjusted minimum density power divergence estimation technique. We derive the asymptotic properties of the proposed estimator under some mild regularity conditions, and also investigate its finite sample performance with a small simulation experiment. The practical applicability of the methodology is illustrated on a dataset of calcium content measurements of soil samples.

A Robust Estimation for the Composite Lognormal-Pareto Model

Cooray and Ananda (2005) proposed a composite lognormal-Pareto model to analyze loss payment data in the actuarial and insurance industries. Their model is based on a lognormal density up to an unknown threshold value and a two-parameter Pareto density. In this paper, we implement the minimum density power divergence estimation for the composite lognormal-Pareto density. We compare the performances of the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) by simulations and an example. The minimum density power divergence estimator performs reasonably well against various violations in the distribution. The minimum density power divergence estimator better fits small observations and better resists against extraordinary large observations than the maximum likelihood estimator.

Robust estimation for copula Parameter in SCOMDY models

In this study, we study the robust estimation for the copula parameter in semiparametric copula‐based multivariate dynamic (SCOMDY) models proposed by Chen and Fan (2006). To this end, instead of the pseudo maximum likelihood estimator in Chen and Fan (2006), we use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998). It is shown that the MDPDE is consistent and asymptotically normal under regularity conditions. We compare the performance between the two estimators when outliers exist through a simulation study.

Minimum density power divergence estimator for diffusion processes

In this paper, we consider the robust estimation for a certain class of diffusion processes including the Ornstein–Uhlenbeck process based on discrete observations. As a robust estimator, we consider the minimum density power divergence estimator (MDPDE) proposed by Basu et al. (Biometrika 85:549–559, 1998). It is shown that the MDPDE is consistent and asymptotically normal. A simulation study demonstrates the strong robustness of the MDPDE.

Robust estimation for the covariance matrix of multivariate time series based on normal mixtures

In this paper, we study the robust estimation for the covariance matrix of stationary multivariate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) designed by Basu et al. (1998). To supplement the result of Kim and Lee (2011), we employ a multivariate normal mixture family instead of a multivariate normal family. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration. A real data analysis applied to the portfolio selection problem is also considered.

The Minimum Density Power Divergence Approach in Building Robust Regression Models

It is well known that in situations involving the study of large datasets where influential observations or outliers maybe present, regression models based on the Maximum Likelihood criterion are likely to be unstable. In this paper we investigate the use of the Minimum Density Power Divergence criterion as a practical tool for parametric regression models building. More precisely, we suggest a procedure relying on an index of similarity between estimated regression models and on a Monte Carlo Significance test of hypothesis that allows to check the existence of outliers in the data and therefore to choose the best tuning constant for the Minimum Density Power Divergence estimators. Theory is outlined, numerical examples featuring several experimental scenarios are provided and main results of a simulation study aiming to verify the goodness of the procedure are supplied.

Robust estimation for the covariance matrix of multi-variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi-variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998). Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.

Robust estimation for order of hidden Markov models based on density power divergences

In this paper, we study the robust estimation for the order of hidden Markov model (HMM) based on a penalized minimum density power divergence estimator, which is obtained by utilizing the finite mixture marginal distribution of HMM. For this task, we adopt the locally conic parametrization method used in [D. Dacunha-Castelle and E. Gassiate, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. (1997), pp. 285–317; D. Dacunha-Castelle and E. Gassiate, Testing the order of a model using locally conic parametrization: population mixtures and stationary arma processes, Ann. Statist. 27 (1999), pp. 1178–1209; T. Lee and S. Lee, Robust and consistent estimation of the order of finite mixture models based on the minimizing a density power divergence estimator, Metrika 68 (2008), pp. 365–390] to avoid the difficulties that arise in handling mixture marginal models, such as the non-identifiability of the parameter space and the singularity problem with the asymptotic variance. We verify that the estimated order is consistent and simulation results are provided for illustration.

Estimation of a tail index based on minimum density power divergence

In this paper, we consider the minimum density power divergence estimator for the tail index of heavy tailed distributions in strong mixing processes. It is shown that the estimator is consistent and asymptotically normal under regularity conditions. The simulation results demonstrate that the estimator is robust in the presence of outliers.

Minimum density power divergence estimator for GARCH models

In this paper, we study the robust estimation for the generalized autoregressive conditional heteroscedastic (GARCH) models with Gaussian errors. As a robust estimator, we consider a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (Biometrika 85:549–559, 1998). It is shown that the MDPDE is strongly consistent and asymptotically normal. Our simulation study demonstrates that the MDPDE has robust properties in contrast to the maximum likelihood estimator. A real data analysis is performed for illustration.

Robust and Efficient Estimation for the Generalized Pareto Distribution

In this article we implement the minimum density power divergence estimator (MDPDE) for the shape and scale parameters of the generalized Pareto distribution (GPD). The MDPDE is indexed by a constant α ≥ 0 that controls the trade-off between robustness and efficiency. As α increases, robustness increases and efficiency decreases. For α = 0 the MDPDE is equivalent to the maximum likelihood estimator (MLE). We show that for α > 0 the MDPDE for the GPD has a bounded influence function. For α < 0.2 the MDPDE maintains good asymptotic relative efficiencies, usually above 90%. The results from a Monte Carlo study agree with these asymptotic calculations. The MDPDE is asymptotically normally distributed if the shape parameter is less than (1 + α)/(2 + α), and estimators for standard errors are readily computed under this restriction. We compare the MDPDE, MLE, Dupuis’ optimally-biased robust estimator (OBRE), and Peng and Welsh’s Medians estimator for the parameters. The simulations indicate that the MLE has the highest efficiency under uncontaminated GPDs. However, for the GPD contaminated with gross errors OBRE and MDPDE are more efficient than the MLE. For all the simulated models that we studied the Medians estimator had poor performance.