In this dissertation, we first develop new statistical methodologies for small area estimation problems with measurement errors. The prediction of small area means for the unit-level regression model with the functional measurement error in the area-specific covariate is considered. We obtain the James-Stein (JS) estimate of the true area-specific covariate. Consequently, we construct the pseudo Bayes (PB) and pseudo empirical Bayes (PEB) predictors of small area means and estimate the mean squared prediction error (MSPE) associated with each predictor. Secondly, we modify the point estimation of the true area-specific covariate obtained earlier such that the histogram of the predictors of the small area means gets closer to its true one. We propose the constrained Bayes (CB) estimate of the true area-specific covariate. We show the superiority of the CB over the maximum likelihood (ML) estimate in terms of the Bayes risk. We also show the PB predictor of the small area mean based on the CB estimate of the true area-specific covariate dominates its counterpart based on the ML estimate in terms of the Bayes risk. We compare the performance of different predictors of the small area means using measures such as sensitivity, specificity, positive predictive value, negative predictive value, and MSPE. We believe that using the PEB and pseudo hierarchical Bayes predictors of small area means based on the constrained empirical Bayes (CEB) and constrained hierarchical Bayes (CHB) offers higher precision in recognizing socio-economic groups which are in danger of the prehypertension. Clustering the small areas to understand the behavior of the random effects better and accordingly, to predict the small area means is the final problem we address. We consider the Fay-Herriot model for this problem. We design a statistical test to evaluate the assumption of the equality of the variance components in different clusters. In the case of rejection of the null hypothesis of the equality of the variance components, we implement a modified version of Tukey's method. We calculate the MSPE to evaluate the effect of the clustering on the precision of predictors of the small area means. We apply our methodologies to real data sets. / February 2017

In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.

Au cours des deux dernières décennies, des travaux actifs ont été menés pour étendre des résultats classiques liés au groupe symétrique à d'autres groupes plus généraux. Cette thèse a pour objectif d’étendre aux produits en couronne les résultats concernant le tableau de différence d’Euler. Elle est divisée en cinq chapitres. Le tableau de différence d’Euler lié à la suite {n!} conduit naturellement à la formule du nombre de dérangements. Nous étudions dans les deux premiers chapitres, le tableau de différence d’Euler associé à la suite {rnn!} et la généralisation du problème de dérangements. Pour les coefficients de ce dernier tableau, nous donnons des interprétations combinatoires en termes de k-successions sur les produits en couronne. Clarke et al. ont introduit un q-analogue du tableau de différence d’Euler sur le groupe symétrique. Dans le troisième chapitre, nous étendons leurs résultats sur les produits en couronne. En généralisant leur bijection, nous montrons que « (fix, exc, fmaj) » et « (fix, exc, fmaf) » sont équidistribués sur les produits en couronne où «fmaf» est une nouvelle statistique mahonienne. D’autre part, Foata et Han ont récemment construit deux transformations. Nous prouvons dans le quatrième chapitre que ses bijections fournissent une factorisation de la bijection de Clarke et al.. Dans le cinquième chapitre nous donnons une extension de la seconde transformation fondamentale de Foata sur les mots r-colorés. Nous prouvons l’équidistribution sur les produits en couronne de « (fmaj , des*) » et « (finv , col) » où « col » est la somme des couleurs et « des* » une nouvelle statistique. / In the last two decades, much effort has been made to extend various enumerative results on symmetric groups to other more general groups. The main objective of this thesis is to extend to wreath products the results that concern the Euler's difference table. It is divided into five chapters. Euler's difference table associated to the sequence {n!} leads naturally to the counting formula for the derangements. In the first two chapters, we study Euler's difference table associated to the sequence {rnn!} and the generalized derangement problem. For the coefficients appearing in the later table, we give the combinatorial interpretations in terms of k-successions on wreath products. Clarke et al. introduced a q-analogue of Euler's difference table on symmetric group. In the third chapter, we extend their results to wreath products. By generalizing their bijection, we prove the equidistribution of the triple statistics “(fix, exc, fmaj)” and “(fix, exc, fmaf)” on wreath products, where “fmaf” is a new mahonian statistic on wreath products. On the other hand, Foata and Han have recently constructed two new transformations. We prove in fourth chapter that their two bijections provide a factorization of Clarke et al.'s bijection. In the fifth chapter we give an extension of Foata’s second fundamental transformation on r-colored words. We show that the bistatistics “(fmaj , des*)” and “(finv , col)” are equidistributed on wreath products, where “col” is the sum of color and “des*” a new statistic.