Robust Post-Hoc Multiple Comparisons: Skew t Distributed Error Terms

ABSTRACTCensoring can be occurred in many statistical analyses in the framework of experimental design. In this study, we estimate the model parameters in one-way ANOVA under Type II censoring. We assume that the distribution of the error terms is Azzalini's skew normal. We use Tiku's modified maximum likelihood (MML) methodology which is a modified version of the well-known maximum likelihood (ML) in the estimation procedure. Unlike ML methodology, MML methodology is non-iterative and gives explicit estimators of the model parameters. We also propose new test statistics based on the proposed estimators. The performances of the proposed estimators and the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. A real life data is analysed to show the implementation of the methodology presented in this paper at the end of the study.

The goal of the present work is to estimate the nonlinear correlation between two random variables when the sample is drawn from a Farlie-Gumbel-Morgenstern (FGM)-type bivariate gamma distribution.In the context of estimating the dependence parameter, a maximum likelihood (ML) methodology is used.Thus, the present work offers ML estimators based on simple random sampling (SRS) and ranked set sampling (RSS).Additionally, we consider generalized modified RSS (GMRSS), which only requires a single rank to obtain a sample.Using GMRSS, we aim to observe the effect of the rth order statistic and its concomitant on the ML estimator.According to the Monte Carlo simulation, it is clearly seen that RSS provides an ML estimator as efficient as the ML estimator based on SRS.On the other hand, it appears that the ML estimator based on GMRSS (with minimum or maximum ranked pairs) is the best option among the studied ML estimators.Moreover, these findings are made even more meaningful by the fact that GMRSS is easier to obtain than SRS and RSS.

This paper studies estimation of the parameters of the generalized Gompertz distribution based on ranked-set sample (RSS). Maximum likelihood (ML) and Bayesian approaches are considered. Approximate confidence intervals for the unknown parameters are constructed using both the normal approximation to the asymptotic distribution of the ML estimators and bootstrapping methods. Bayes estimates and credible intervals of the unknown parameters are obtained using differential evolution Markov chain Monte Carlo and Lindley’s methods. The proposed methods are compared via Monte Carlo simulations studies and an example employing real data. The performance of both ML and Bayes estimates is improved under RSS compared with simple random sample (SRS) regardless of the sample size. Bayes estimates outperform the ML estimates for small samples, while it is the other way around for moderate and large samples.

The goal of this PhD Thesis is the definition of new robust estimators, thereby extending the available theory and exploring new directions for applications in finance. The Thesis contains three papers, which analyse three different types of estimators: M-, Minimum Distance- and R-estimators. The focus is manly of their infinitesimal robustness, but global robustness properties are also considered. The first paper (“Higher-order infinitesimal robustness”) studies M-estimators and it is a joint work with Elvezio Ronchetti and Fabio Trojani. Using the higher-order von Mises expansion, we go beyond the Influence Function and we extend Hampel's paradigm of robust ness, introducing higher-order infinitesimally robust M-estimators. We show that a bounded estimating function having also bounded gradient with respect to the parameter ensures, at the same time, the stability of the: (i) second-order approximated bias (B-robustness); (ii) asymptotic variance (V-robustness) and (iii) saddle point density approximation. An application in finance (risk management) concludes the paper. The second paper (“On robust estimation via pseudo-additive information measures”) is jointly written with Davide Ferrari and it studies a new class of Minimum Divergence (in the following, MD) estimators. The theoretical contribution of the paper is to show that robustness is dual to information theory. Information theory plays a crucial role in statistical inference : Maximum Likelihood estimators are related to it through the minimization of Shannon entropy (namely, minimization of the Kullback-Leibler divergence). The fundamental axiom characterizing Shannon entropy is additivity. Relaxing this assumption, we obtain a generalized entropy (called q-entropy) which exploits the link between information theory and infinitesimal robustness. Minimizing the q-entropy, we define a new class of MD robust re-descending estimators, featuring B-, V-robustness and that have also good global robustness properties in terms of high-breakdown. The third paper (“Semi-parametric rank-based tests and estimators for Markov processes”) contains the preliminary results of a working paper that I have started in Princeton, working with Marc Hallin. The paper deals with R-estimators and rank-based tests. Precisely, combining the flexibility of the semi-parametric approach with the distribution-freeness of rank statistics, we define R-estimators and tests for stationary Markov processes. An application for inference and testing in stochastic volatility (SV) models concludes the paper.

In this paper, we consider a mixture of two uniform distributions and derive L-moment estimators of its parameters. Three possible ways of mixing two uniforms, namely with neither overlap nor gap, with overlap, and with gap, are studied. The performance of these L-moment estimators in terms of bias and efficiency is compared to that obtained by means of the conventional method of moments (MM), modified maximum likelihood (MML) method and the usual maximum likelihood (ML) method. These intensive simulations reveal that MML estimators are the best in most of the cases, and the L-moment estimators are less subject to bias in estimation for some mixtures and more efficient in most of the cases than the conventional MM estimators. The L-moment estimators are, in some cases, more efficient than the ML and MML estimators.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

EWMA control charts based on robust estimators: A powerful tool for monitoring a process with a non-normal distribution

An algorithm is presented to initiate tracks of a ballistic missile in the initial exoatmospheric phase, using line of sight (LOS) measurements from one or more moving platforms (typically satellites). The major feature of this problem is the poor target motion observability which results in a very ill-conditioned estimation problem. The Gauss-Newton iterative least squares minimization algorithm for estimating the state of a nonlinear deterministic system with nonlinear noisy measurements has been previously applied to the problem of angles-only orbit determination using more than three observations. A major shortcoming of this approach is that convergence of the algorithm depends strongly on the initial guess. By using the more sophisticated Levenberg-Marquardt method in place of the simpler Gauss-Newton algorithm and by developing robust new methods for obtaining the initial guess in both single and multiple satellite scenarios, the above mentioned difficulties have been overcome. In addition, an expression for the Cramer-Rao lower bound (CRLB) on the error covariance matrix of the estimate is derived. We also incorporate additional partial information as an extra pseudomeasurement and determine a modified maximum likelihood (ML) estimate of the target state and the associated bound on the covariance matrix. In most practical situations, probabilistic models of the target altitude and/or speed at the initial point constitute the most useful additional information. Monte Carlo simulation studies on some typical scenarios were performed, and the results indicate that the estimation errors are commensurate with the theoretical lower bounds, thus illustrating that the proposed estimators are efficient.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.

Estimating the midpoints of interval-valued data poses significant challenges, particularly when uncertainty and variability are inherent. This paper advances midpoint estimation by leveraging both Maximum Likelihood (ML) and Bayesian methodologies, incorporating various forms of additional information through parameter constraints. In practical applications, such information often naturally arises, enabling the imposition of constraints such as order, rectangular, spherical, and bounded-order conditions on the parameter space. By integrating these constraints, the estimators can exploit structural relationships among parameters to achieve improved precision and accuracy. Through comprehensive simulation studies, we assess the performance of ML and Bayesian estimators under these constrained settings, demonstrating that Bayesian estimators consistently outperform ML estimators, particularly when employing noninformative priors. The constrained Bayesian approach not only results in lower mean squared errors (MSEs) but also yields more precise credible intervals compared to the confidence intervals obtained via ML, showcasing its robustness and efficiency. Furthermore, a real-world climatological case study validates our theoretical and simulation findings, demonstrating the practical utility of the proposed methods for interval-valued temperature data. This example illustrates how integrating additional information–often readily available–enhances the accuracy and reliability of midpoint estimation in multivariate contexts. Our work underscores the potential of structural constraints in symbolic data analysis, providing actionable insights across diverse scientific domains.

There is a wide range of outliers in spatial data, and these potential outliers will have a great impact on parameter estimation and corresponding statistical inference. Relying on the framework of maximum likelihood estimation (MLE), we investigate the asymptotic distribution of robust ML estimator under the mixed spatial autoregressive models with outliers and compare it with that of the ML estimator. Furthermore, based on the asymptotic theoretical result, we conduct the confidence interval of robust MLE and MLE. Similar to the results of MLE, we construct the second-order-corrected robust confidence interval using the parametric and semi-parametric bootstrap method. Simulation studies using Monte Carlo show that the robust estimator with the Huber loss function is more accurate and outperforms the MLE in most sample settings when data is contaminated by outliers. Then the use of the method is demonstrated in the analysis of the Neighborhood Crimes Data and the Boston Housing Price Data. The results further support the eligibility of the robust method in practical situations.

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

This paper proposes a joint maximum likelihood and Bayesian methodology for estimating Gaussian mixture models. In Bayesian inference, the distributions of parameters are modeled, characterized by hyperparameters. In the case of Gaussian mixtures, the distributions of parameters are considered as Gaussian for the mean, Wishart for the covariance, and Dirichlet for the mixing probability. The learning task consists of estimating the hyperparameters characterizing these distributions. The integration in the parameter space is decoupled using an unsupervised variational methodology entitled variational expectation-maximization (VEM). This paper introduces a hyperparameter initialization procedure for the training algorithm. In the first stage, distributions of parameters resulting from successive runs of the expectation-maximization algorithm are formed. Afterward, maximum-likelihood estimators are applied to find appropriate initial values for the hyperparameters. The proposed initialization provides faster convergence, more accurate hyperparameter estimates, and better generalization for the VEM training algorithm. The proposed methodology is applied in blind signal detection and in color image segmentation.

Powerful mathematical tools have been developed for trading in stocks and bonds, but other markets that are equally important for the globalized world have to some extent been neglected. We decided to study the shipping market as an new area of development in mathematical finance. The market in shipping derivatives (FFA and FOSVA) has only been developed after 2000 and now exhibits impressive growth. Financial actors have entered the field, but it is still largely undiscovered by institutional investors. The first part of the work was to identify the characteristics of the market in shipping, i.e. the segmentation and the volatility. Because the shipping business is old-fashioned, even the leading actors on the world stage (ship owners and banks) are using macro-economic models to forecast the rates. If the macro-economic models are logical and make sense, they fail to predict. For example, the factor port congestion has been much cited during the last few years, but it is clearly very difficult to control and is simply an indicator of traffic. From our own experience it appears that most ship owners are in fact market driven and rather bad at anticipating trends. Due to their ability to capture large moves, we chose to consider Levy processes for the underlying price process. Compared with the macro-economic approach, the main advantage is the uniform and systematic structure this imposed on the models. We get in each case a favorable result for our technology and a gain in forecasting accuracy of around 10% depending on the maturity. The global distribution is more effectively modelled and the tails of the distribution are particularly well represented. This model can be used to forecast the market but also to evaluate the risk, for example, by computing the VaR. An important limitation is the non-robustness in the estimation of the Levy processes. The use of robust estimators reinforces the information obtained from the observed data. Because maximum likelihood estimation is not easy to compute with complex processes, we only consider some very general robust score functions to manage the technical problems. Two new class of robust estimators are suggested. These are based on the work of F. Hampel ([29]) and P. Huber ([30]) using influence functions. The main idea is to bound the maximum likelihood score function. By doing this a bias is created in the parameters estimation, which can be corrected by using a modification of the following type and as proposed by F. Hampel. The procedure for finding a robust estimating equation is thus decomposed into two consecutive steps : Subtract the bias correction and then Bound the score function. In the case of complex Levy processes, the bias correction is difficult to compute and generally unknown. We have developed a pragmatic solution by inverting the Hampel's procedure. Bound the score function and then Correct for the bias. The price is a loss of the theoretical properties of our estimators, besides the procedure converges to maximum likelihood estimate. A second solution to for achieving robust estimation is presented. It considers the limiting case when the upper and lower bounds tend to zero and leads to B-robust estimators. Because of the complexity of the Levy distributions, this leads to identification problems.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.