Reference Priors For Non-Normal Two-Sample Problems

The reference prior algorithm (Berger and Bernardo, 1992) is applied to locationscale models with any regular sampling density. A number of two-sample problems is analyzed in this general context, extending the dierence, ratio and product of Normal means problems outside Normality, while explicitly considering possibly dierent sizes for each sample. Since the reference prior turns out to be improper in all cases, we examine existence of the resulting posterior distribution and its moments under sampling from scale mixtures of Normals. In the context of an empirical example, it is shown that a reference posterior analysis is numerically feasible and can display some sensitivity to the actual sampling distributions. This illustrates the practical importance of questioning the Normality assumption.Behrens-Fisher problem;Fieller-Creasy problem;Gibbs sampling;Jeffreys' prior;location-scale model;posterior existence;product of means;scale mixtures of normals;skewness

Cram\'{e}r-type moderate deviations for Studentized two-sample UU-statistics with applications

Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cram\'{e}r-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample $t$-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.Comment: Published at http://dx.doi.org/10.1214/15-AOS1375 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric Confidence Intervals for the One- and Two-Sample Problems

Confidence intervals for the mean of one sample and the difference in means of two independent samples based on the ordinary-t statistic suffer deficiencies when samples come from skewed distributions. In this article, we evaluate several existing techniques and propose new methods to improve coverage accuracy. The methods examined include the ordinary-t, the bootstrap-t, the biased-corrected acceleration (BCa) bootstrap, and three new intervals based on transformation of the t-statistic. Our study shows that our new transformation intervals and the bootstrap-t intervals give best coverage accuracy for a variety of skewed distributions; and that our new transformation intervals have shorter interval lengths

Common Functional Principal Components

Functional principal component analysis (FPCA) based on the Karhunen-Lo`eve decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the logreturns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. Firstly we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.Functional Principal Components, Nonparametric Regression, Bootstrap, Two Sample Problem

VERYFICATION OF LOCATION PROBLEM IN ECONOMIC RESEARCH

In economic research very often the location problem in the single sample or estimation  of the difference in two samples location is commonly tested by experimental economists. Usually the used tests are Wilcoxon test for single sample location or Wilcoxon – Mann – Whitney for two samples location problem. Unfortunately those tests have some disadvantages such as robustness against assumptions or week efficiency. In the paper, some less known procedures, which allow avoid those problems, will be presented. Considered methods will be illustrated on the example of the data  analysis from real-estate market

Distribution-Free Tests for Two-Sample Location Problems Based on Subsamples

Nonparametric tests for location problems have received much attention in the literature. Many nonparametric tests have been proposed for one, two and several samples location problems. In this paper a class of test statistics is proposed for two sample location problem when the underlying distributions of the samples are symmetric. The class of test statistics proposed is linear combination of U-statistics whose kernel is based on subsamples extrema. The members of the new class are shown to be asymptotically normal. The performance of the proposed class of tests is evaluated using Pitman Asymptotic Relative Efficiency. It is observed that the members of the proposed class of tests are better than the existing tests in the literature