arXiv subjects

Diaa Al Mohamad

Publications and source records attributed to Diaa Al Mohamad.

9 recordsLinked to original sources

arXiv2018

Simultaneous Confidence Intervals for Ranks With Application to Ranking Institutions

When a ranking of institutions such as medical centers or universities is based on an indicator provided with a standard error, confidence intervals should be calculated to assess the quality of these ranks. We consider the problem of constructing simultaneous confidence intervals for the ranks of means based on an observed sample. For this aim, the only available method from the literature uses Monte-Carlo simulations and is highly anticonservative especially when the means are close to each other or have ties. We present a novel method based on Tukey's honest significant difference test (HSD). Our new method is on the contrary conservative when there are no ties. By properly rescaling these two methods to the nominal confidence level, they surprisingly perform very similarly. The Monte-Carlo method is however unscalable when the number of institutions is large than 30 to 50 and stays thus anticonservative. We provide extensive simulations to support our claims and the two methods are compared in terms of their simultaneous coverage and their efficiency. We provide a data analysis for 64 hospitals in the Netherlands and compare both methods. Software for our new methods is available online in package ICRanks downloadable from CRAN. Supplementary materials include supplementary R code for the simulations and proofs of the propositions presented in this paper.

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arXiv2018

An improvement of Tukey's HSD with application to ranking institutions

When a ranking of institutions such as medical centers or universities is based on an indicator provided with a standard error, confidence intervals should be calculated to assess the quality of these ranks. We consider the problem of constructing simultaneous confidence intervals (CIs) for the ranks of centers based on an observed sample. We present a novel method based on Tukey's honest significant difference test (HSD) which is the first method to produce valid simultaneous CIs for ranks. Moreover, we introduce a new variant of Tukey's HSD based on the sequential rejection principle. The new algorithm ensures familywise error control, and produces simultaneous confidence intervals for the ranks uniformly shorter than those provided by Tukey's HSD for the same level of significance. We illustrate the method through both simulations and real data analysis from 64 hospitals in the Netherlands. Software for our new methods is available online in package \texttt{ICRanks} downloadable from CRAN. Supplementary materials include supplementary R code for the simulations and proofs of the propositions presented in this paper.

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arXiv2018

Adaptive Critical Value for Constrained Likelihood Ratio Testing

We present a new way of testing ordered hypotheses against all alternatives which overpowers the classical approach both in simplicity and statistical power. Our new method tests the constrained likelihood ratio statistic against the quantile of one and only one chi-squared random variable with a data-dependent degrees of freedom instead of a mixture of chi-squares. Our new test is proved to have a valid finite-sample significance level $α$ and provides more power especially for sparse alternatives (those with a few or moderate number of null constraints violations) in comparison to the classical approach. Our method is also easier to use than the classical approach which requires to calculate or simulate a set of complicated weights. Two special cases are considered with more details, namely the case of testing orthants $μ_1<0, \cdots, μ_n<0$ and the isotonic case of testing $μ_1<μ_2<μ_3$ against all alternatives. Contours of the difference in power are shown for these examples showing the interest of our new approach.

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arXiv2017

Semiparametric two-component mixture models under linear constraints

We propose a structure of a semiparametric two-component mixture model when one component is parametric and the other is defined through linear constraints on its distribution function. Estimation of a two-component mixture model with an unknown component is very difficult when no particular assumption is made on the structure of the unknown component. A symmetry assumption was used in the literature to simplify the estimation. Such method has the advantage of producing consistent and asymptotically normal estimators, and identifiability of the semiparametric mixture model becomes tractable. Still, existing methods which estimate a semiparametric mixture model have their limits when the parametric component has unknown parameters or the proportion of the parametric part is either very high or very low. We propose in this paper a method to incorporate a prior linear information about the distribution of the unknown component in order to better estimate the model when existing estimation methods fail. The new method is based on $φ-$divergences and has an original form since the minimization is carried over both arguments of the divergence. The resulting estimators are proved to be consistent and asymptotically normal under standard assumptions. We show that using the Pearson's $χ^2$ divergence our algorithm has a linear complexity when the constraints are moment-type. Simulations on univariate and multivariate mixtures demonstrate the viability and the interest of our novel approach.

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arXiv2017

Simultaneous confidence sets for ranks using the partitioning principle - Technical report

Ranking institutions such as medical centers or universities is based on an indicator accompanied with an uncertainty measure such as a standard deviation, and confidence intervals should be calculated to assess the quality of these ranks. We consider the problem of constructing simultaneous confidence intervals for the ranks of centers based on an observed sample. We present in this paper a novel method based on multiple testing which uses the partitioning principle and employs the likelihood ratio (LR) test on the partitions. The complexity of the algorithm is super exponential. We present several ways and shortcuts to reduce this complexity. We provide also a polynomial algorithm which produces a very good bracketing for the multiple testing by linearizing the critical value of the LR test. We show that Tukey's Honest Significant Difference (HSD) test can be written as a partitioning procedure. The new methodology has promising properties in the sens that it opens the door in a simple and easy way to construct new methods which may trade the exponential complexity with power of the test or vice versa. In comparison to Tukey's HSD test, the LR test seems to give better results when the centers are close to each others or the uncertainty in the data is high which is confirmed during a simulation study.

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arXiv2016

Estimation of parametric and semiparametric mixture models using phi-divergences

The study of mixture models constitutes a large domain of research in statistics. In the first part of this work, we present phi-divergences and the existing methods which produce robust estimators. We are more particularly interested in the so-called dual formula of phi-divergences. We build a new robust estimator based on this formula. We study its asymptotic properties and give a numerical comparison with existing methods on simulated data. We also introduce a proximal-point algorithm whose aim is to calculate divergence-based estimators. We give some of the convergence properties of this algorithm and illustrate them on theoretical and simulated examples. In the second part of this thesis, we build a new structure for two-component mixture models where one component is unknown. The new approach permits to incorporate a prior linear information about the unknown component such as moment-type and L-moments constraints. We study the asymptotic properties of the proposed estimators. Several experimental results on simulated data are illustrated showing the advantage of the novel approach and the gain from using the prior information in comparison to existing methods which do not incorporate any prior information except for a symmetry assumption over the unknown component.

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arXiv2016

Two Iterative Proximal-Point Algorithms for the Calculus of Divergence-based Estimators with Application to Mixture Models

Estimators derived from an EM algorithm are not robust since they are based on the maximization of the likelihood function. We propose a proximal-point algorithm based on the EM algorithm which aim to minimize a divergence criterion. Resulting estimators are generally robust against outliers and misspecification. An EM-type proximal-point algorithm is also introduced in order to produce robust estimators for mixture models. Convergence properties of the two algorithms are treated. We relax an identifiability condition imposed on the proximal term in the literature; a condition which is generally not fulfilled by mixture models. The convergence of the introduced algorithms is discussed on a two-component Weibull mixture and a two-component Gaussian mixture entailing a condition on the initialization of the EM algorithm in order for the later to converge. Simulations on mixture models using different statistical divergences are provided to confirm the validity of our work and the robustness of the resulting estimators against outliers in comparison to the EM algorithm.

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arXiv2016

A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models

Estimators derived from a divergence criterion such as $φ-$divergences are generally more robust than the maximum likelihood ones. We are interested in particular in the so-called MD$φ$DE, an estimator built using a dual representation of $φ$--divergences. We present in this paper an iterative proximal point algorithm which permits to calculate such estimator. This algorithm contains by its construction the well-known EM algorithm. Our work is based on the paper of \citep{Tseng} on the likelihood function. We provide several convergence properties of the sequence generated by the algorithm, and improve the existing results by relaxing the identifiability condition on the proximal term, a condition which is not verified for most mixture models and hard to be verified for non mixture ones. Since convergence analysis uses regularity conditions (continuity and differentiability) of the objective function, which has a supremal form, we find it useful to present some analytical approaches for studying such functions. Convergence of the EM algorithm is discussed here again in a Gaussian and Weibull mixtures in the spirit of our approach. Simulations are provided to confirm the validity of our work and the robustness of the resulting estimators against outliers.

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arXiv2015

Towards a better understanding of the dual representation of phi divergences

The aim of this paper is to study different estimation procedures based on $φ-$divergences. The dual representation of $φ-$divergences based on the Fenchel-Legendre duality is the main interest of this study. It provides a way to estimate $φ-$divergences by a simple plug-in of the empirical distribution without any smoothing technique. Resulting estimators are thoroughly studied theoretically and with simulations showing that the so called minimum $φ-$divergence estimator (MD$φ$DE) is generally non robust and behaves similarly to the maximum likelihood estimator. We give some arguments supporting the non robustness property, and give insights on how to modify the classical approach. An alternative class of $φ-$divergences robust estimators based on the dual representation is presented. We study consistency and robustness properties from an influence function point of view of the new estimators. In a second part, we invoke the Basu-Lindsay approach for approximating $φ-$divergences and provide a comparison between these approaches. The so called dual $φ-$divergence is also discussed and compared to our new estimator. A full simulation study of all these approaches is given in order to compare efficiency and robustness of all mentioned estimators against the so-called minimum density power divergence, showing encouraging results in favor of our new class of minimum dual $φ-$divergences.

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