On distance-type Gaussian estimation
Maximum likelihood estimation method is a popular and simple method to estimate the parameters of a model. However, there are cases where this method cannot be applied or is not recommended to be applied. For instance, this method cannot be applied when the model, which describes the data, is intractable or when it contains intractable normalizing constant which is a function of the unknown population parameters. Moreover, since maximum likelihood estimation is highly affected by the presence of outliers in the data, it is not recommended to be applied in such cases. To overcome the first problem, \textit{T. Zhang} [J. Multivariate Anal. 169, 234--247 (2019; Zbl 1433.62080)] proposed, under the assumption that the computation of the population mean vector \(\mu\) and the population covariance matrix \(\Sigma\) is not so arduous, the replacement of the initial intractable model by the normal model. This approach is called general Gaussian estimation. On the other hand, minimum distance estimation methods can be used in order to tackle the presence of outliers, since these methods are characterized by good efficiency and robustness properties. Among the existing minimum distance estimation methods, the Minimum density power divergence (DPD) estimation method, introduced by \textit{A. Basu} et al. [Biometrika 85, No. 3, 549--559 (1998; Zbl 0926.62021)], has a prominent position. In this paper, which consists of eight sections, a new procedure for estimating the parameters of a model is proposed by combining Zhang's Gaussian estimation approach [\textit{T. Zhang}, J. Multivariate Anal. 169, 234--247 (2019; Zbl 1433.62080)] and the minimum density power divergence estimators of \textit{A. Basu} et al. [Biometrika 85, No. 3, 549--559 (1998; Zbl 0926.62021)]. In this context, after an excellent introduction which gives clearly the motivation of the current work (Section 1), the Minimum DPD Gaussian estimation (MDPDGE) method is introduced in Section 2 of the paper. The consistency and asymptotic normality of the estimators obtained by this method are proved in Section 3. In Section 4, the behavior of the MDPDGE in the univariate case is illustrated via some elementary examples by considering the Poisson, Exponential and Binomial distribution. Section 5 concentrates on the estimation of the parameters of the multivariate elliptic family of distributions by using both Zhang's Gaussian approach and the MDPDGE method. The general results obtained are exemplified for the following specific members of the elliptic family of distributions: multivariate normal, multivariate t with v degrees of freedom and Kotz type symmetric distribution. In Section 6, the robustness of the proposed MDPDGE is evaluated numerically by considering the Poisson and multivariate t distribution. Section 7 summarizes the contribution of the paper and gives some possible topics for a future research. Finally, some technical details are given in the last section (Section 8). Summarizing, the paper, which is included in a special issue devoted to the celebrations of the 50-year anniversary of the Journal of Multivariate Analysis, is very well written, and the interest reader can read it pleasantly. The main contribution of the paper is that it proposes a method of estimation which meets both the case of intractable model and the presence of outliers in the data, a case which was not previously considered by Zhang's Gaussian approach.
minimum density power divergence Gaussian estimation
scientific article; zbMATH DE number 7451352
scientific article