A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

Randomized statistical inference: A unified statistical inference frame of frequentist,fiducial, and Bayesian inference

We propose randomized inference(RI), a new statistical inference approach. RI may be realized through a randomized estimate(RE) of a parameter vector, which is a random vector that takes values in the parameter space with a probability density function(PDF) that depends on the sample or sufficient statistics,such as the posterior distributions in Bayesian inference. Based on the PDF of an RE of an unknown parameter,we propose a framework for both the vertical density representation(VDR) test and the construction of a confidence region. This approach is explained with the aid of examples. For the equality hypothesis of multiple normal means without the condition of variance homogeneity, we present an exact VDR test, which is shown as an extension of one-way analysis of variance(ANOVA). In the case of two populations, the PDF of the Welch statistics is given by using the RE. Furthermore, through simulations, we show that the empirical distribution function, the approximated t, and the RE distribution function of Welch statistics are almost equal. The VDR test of the homogeneity of variance is shown to be more efficient than both the Bartlett test and the revised Bartlett test. Finally, we discuss the prospects of RI.     

两个回归函数相等的Bootstrap检验

两个回归参数相等性检验一直是统计界感兴趣的问题之一．在这篇文章中，四个检验统计量被用于度量两曲线的差异，在原假设下统计量的分布采用向量数据的重复抽样来逼近，并给出了—些模拟结果．     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

Tests d'égalité contre l'alternative de croisement de deux fonctions de survie

We propose two tests to test the hypothesis of the equality of two survival functions against the alternative of crossing of these functions for right-censured data. The asymptotic laws of test statistics are given. The powers of the tests are calculated. To cite this article: V. Bagdonavi?ius et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The geometry of spheres in free abelian groups

We study word metrics on ${\mathbb{Z}^d}$ by developing tools that are fine enough to measure dependence on the generating set. We obtain counting and distribution results for the words of length n. With this, we show that counting measure on spheres always converges to cone measure on a polyhedron (strongly, in an appropriate sense). Using the limit measure, we can reduce probabilistic questions about word metrics to problems in convex geometry of Euclidean space. We give several applications to the statistics of ??size-like?? functions.     

Testing hypotheses about the equality of several risk measure values with applications in insurance

This paper explores statistical tests about the equality of risk measure values obtained using a distortion-based risk measure. We consider both the case in which the risk measure value is specified in the null hypothesis and the case in which it is not. In the former case, one- and two-sided alternatives are considered, and in the latter case, ordered and unordered alternatives are considered. Asymptotically most powerful tests are obtained, and asymptotic distributions of the test statistics are found using results about the asymptotic distributions of the risk measure values. Finally, we consider a numerical example and conclude the paper with notes on when the results of the paper could, or could not, be safely used.     

两总体下双参数指数分布门限参数的比较

本文研究了当刻度参数未知时,双参数指数分布门限参数的检验问题.利用了似然比检验的方法,获得了检验统计量及检验水平的上下界.该方法应用在两个截尾数相等情形下,所得到的检验水平是真实的.     

Confidence limits for the mean of exponential distribution in any time-sequential samples

We present the general results determining confidence limits for the mean of exponential distribution in any time-sequential samples,which are obtained in any sequen- tial life tests with replacement or without replacement.Especially,we give the best lower confidence limits in the case of no failure data.     

Testing the equality of error distributions from k independent GARCH models

In this paper we study the problem of testing the null hypothesis that errors from $k$ independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the $k$ residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data.     

Computational aspects of statistical intervals based on two Type-II censored samples

In this paper, we propose an efficient branch and bound procedure to compute exact nonparametric statistical intervals based on two Type-II right censored data sets. The procedure is based on some recurrence relations for the distribution and density functions of progressively Type-II censored order statistics which can be applied to compute the coverage probabilities. We illustrate the method for both confidence and prediction intervals of a given level.     

Asymptotics on Semiparametric Analysis of Multivariate Failure Time Data Under the Additive Hazards Model

Many survival studies record the times to two or more distinct failures on each subject. The failures may be events of different natures or may be repetitions of the same kind of event. In this article, we consider the regression analysis of such multivariate failure time data under the additive hazards model. Simple weighted estimating functions for the regression parameters are proposed, and asymptotic distribution theory of the resulting estimators are derived. In addition, a class of generalized Wald and generalized score statistics for hypothesis testing and model selection are presented, and the asymptotic properties of these statistics are examined.     

Exact two-sample nonparametric test for quantile difference between two populations based on ranked set samples

Creation of a ranked set sample, by its nature, involves judgment ranking error within set units. This ranking error usually distorts statistical inference of the population characteristics. Tests may have inflated sizes, confidence intervals may have incorrect coverage probabilities, and the estimators may become biased. In this paper, we develop an exact two-sample nonparametric test for quantile shift between two populations based on ranked set samples. This test is based on two independent exact confidence intervals for the quantile of interest corresponding to the two populations and rejects the null hypothesis of equal quantiles if these intervals are disjoint. It is shown that a pair of 83 and 93% confidence intervals provide a 5 and 1% test for the equality of quantiles. The proposed test is calibrated for the effect of judgment ranking error so that the test has the correct size even under a wide range of judgment ranking errors. A small scale simulation study suggests that the test performs quite well for cycle sizes as small as 2.     

K-sample problem using strong approximations of empirical copula processes

In this article we explore asymptotic properties of some statistics based on K-sample extensions of multivariate empirical copula processes. These statistics can be used to test the equality of copulas pertaining to K independent samples.     

Goodness-of-fit test for switching diffusion

We consider two Cramér–von Mises goodness-of-fit tests for hypotheses that the observed diffusion process has sign-type trend coefficient based on empirical distribution function and empirical density function. It is shown that the limit distributions of the proposed tests statistics are defined by the integral type functionals of continuous Gaussian processes. We study the behavior of these statistics under the alternative hypothesis and we prove that the tests are consistent. We provide the Karhunen-Loève expansion on \mathbbR{\mathbb{R}} of the corresponding limiting processes and we show that the eigenfunctions in these expansions have expressions in term of Bessel functions.     

Global fluctuations in general Dyson’s Brownian motion

We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Universality limits for random matrices and de Branges spaces of entire functions

We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [−1,1].     

Large Deviation Probabilities for Square-Gaussian Stochastic Processes

We investigate properties of square-Gaussian stochastic processes. These processes are formed by quadratic forms of Gaussian processes or by limits in the mean square of quadratic forms of Gaussian processes. Special classes of these processes are determined and investigated. For processes from these classes estimates of large deviation probability are obtained. These estimates we use to estimate the probability that Gaussian vector-valued process leave some region on some interval of time. We construct asymptotic confidence regions for estimates of covariance functions of vector-valued Gaussian processes. Criterion of hypothesis testing on covariance functions of these processes is constructed.