On the problem of the unique characterization of discrete distributions by single regression of weak records

We consider the problem of the unique identification of discrete probability distributions by the single regression function of non-adjacent discrete weak record values. We present a new approach to this problem and we show that the uniqueness of the characterization is equivalent to the uniqueness of solution to a corresponding difference equation or an appropriate system of difference equations. This result is applied to obtain known as well as new characterizations of discrete distributions.     

New tests for stochastic order with application to case control studies

It is shown that if a binary regression function is increasing then retrospective sampling induces a stochastic ordering of the covariate distributions among the responders, which we call cases, and the non-responders, which we call controls. We also show that if the covariate distributions are stochastically ordered then the regression function must be increasing. This means that testing whether the regression function is monotone is equivalent to testing whether the covariate distributions are stochastically ordered. Capitalizing on these new probabilistic observations we proceed to develop two new non-parametric tests for stochastic order. The new tests are based on either the maximally selected, or integrated, chi-bar statistic of order one. The tests are easy to compute and interpret and their large sampling distributions are easily found. Numerical comparisons show that they compare favorably with existing methods in both small and large samples. We emphasize that the new tests are applicable to any testing problem involving two stochastically ordered distributions.     

A sequential procedure for testing the existence of a random walk model in finite samples

Given the random walk model, we show, for the traditional unrestricted regression used in testing stationarity, that no matter what the initial value of the random walk is or its drift or its error standard deviation, the sampling distributions of certain statistics remain unchanged. Using Monte Carlo simulations, we estimate, for different finite samples, the sampling distributions of these statistics. After smoothing the percentiles of the empirical sampling distributions, we come up with a new set of critical values for testing the existence of a random walk, if each statistic is being used on an individual base. Combining the new sets of critical values, we finally suggest a general methodology for testing for a random walk model.     

Regression Diagnostic under Model Misspecification

We propose two novel diagnostic measures for the detection of influential observations for regression parameters in linear regression. Traditional diagnostic statistics focus on the effect of deletion of data points either on parameter estimates, or on predicted values. A data point is regarded as influential by the new methods if its inclusion determines a significantly different likelihood function for the parameter of interest. The concerned likelihood function is asymptotically valid for practically all underlying distributions whose second moments exist.     

Characterizations via regression of generalized order statistics

In this paper, we present some characterizations of distributions based on the regression of generalized order statistics. In the case of adjacent generalized order statistics, the conditional expectation of one generalized order statistic given the other one completely characterizes distributions depending on the type of regression function. In the case of non-adjacent generalized order statistics, the characterization of distributions using conditional expectations becomes more complicated. The results presented in the paper unify and extend some of the existing results involving order statistics and record values.     

Regression via order statistics and their concomitants

We consider a five-dimensional normal distribution and derive the exact joint distribution one variable, linear combinations of order statistics from two other variables, and linear combinations of the corresponding concomitants of these order statistics. We show that this joint distribution is a mixture of trivariate unified skew-normal distributions. This mixture representation enables us to predict one variable based on linear combinations of order statistics from two other variables and linear combinations of the corresponding concomitants. We finally illustrate the usefulness of these results by using a real data.     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

The bivariate linear model: a geometris approach

The geometric approach to the general linear model is in accessible to the majorityof statistics students be cause the computations require matrix algebra.This article presents the geometric approach for the special case of the bivariate linear model,for which the only tool require dis the in ner product.The geometric approach is introduced by showing the dual2-dimensional and5-dimensional representations of several bivariate samples x of size5.The assumptions of the bivariate model are stated geometrically,and the distributions of the regression coefficient sare derived.Theanalysis of variance(ANOVA)right triangle is pictured and the sides of the triang leare associated with their corresponding entries in the ANOVA table.     

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n ^?3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n ^?1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.     

OPERATOR METHODS IN ORDER STATISTICS

This paper presents operator methods, based on difference and differential operators, for establishing identities satisfied by distributions of order statistics. In addition to deducing many known identities, these operator methods may also be used to derive several new identities for distributions of order statistics.     

On goodness of fit for time series regression models

We Propose a Bayesian approach to chech the goodness of fit for time series regression models. The test statistics is proposed by Smith (1985) based on a sequence of random variables which are independently distributed standard normal if the model is correct. We estimate this sequence of random variables using several methods. The tests of goodness of fit are performed when either the error terms violate the Gaussian assumption, or the order is incorrect, or the model is misspecified. The methodology is illustrated using both a simulation study and three real date sets.     

A Unified Method for Checking Compatibility and Uniqueness for Finite Discrete Conditional Distributions

Checking compatibility for two given conditional distributions and identifying the corresponding unique compatible marginal distributions are important problems in mathematical statistics, especially in Bayesian inferences. In this article, we develop a unified method to check the compatibility and uniqueness for two finite discrete conditional distributions. By formulating the compatibility problem into a system of linear equations subject to constraints, it can be reduced to a quadratic optimization problem with box constraints. We also extend the proposed method from two-dimensional cases to higher-dimensional cases. Finally, we show that our method can be easily applied to checking compatibility and uniqueness for a regression function and a conditional distribution. Several numerical examples are used to illustrate the proposed method. Some comparisons with existing methods are also presented.     

NONPARAMETRIC TWO SAMPLE TESTS BASED ON AREA STATISTICS

ABSTRACT

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions against inequality or testing stochastic dominance of distributions in one or either direction against nondominance. Though three of the statistics considered have already been suggested in literature, two of them are new and deserve our interest. The finite sample distributions of the statistics (under F=G) can be calculated via recursion formulae. Two tables with critical values of the new statistics are included. The asymptotic distribution of the properly normalized versions of the area statistics are functionals of the Brownian bridge. The distribution functions and quantiles thereof are obtained by Monte Carlo simulation. Finally, the power functions of the two new tests based on area statistics are compared to the power functions of the tests based on the corresponding supremum statistics, i.e., statistics of the Kolmogorov–Smirnov type.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Testing order restricted hypotheses with proportional hazards

Inferences for survival curves based on right censored continuous or grouped data are studied. Testing homogeneity with an ordered restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods are obtained by applying order restricted procedures to the estimates of the regression coefficients. Ordered analogues to the log rank test which are based on the score statistics are considered also. Chi-bar-squared distributions, which have been studied extensively, are shown to provide reasonable approximations to the null distributions of these tests statistics. Using Monte Carlo techniques, the powers of these two types of tests are compared with those that are available in the literature.     

Properties and Inference for a New Class of Skew-t Distributions

In this article we introduce a new generalization of skew-t distributions, which contains the standard skew-t distribution, as a special case. This new class of distributions is an adequate model for modeling some dataset rather than the standard skew-t distributions. This kind of distributions can be represented as a scale-shape mixture of the extended skew-normal distributions. The main properties of this family of distributions are studied and a recurrence relation for the cumulative distribution functions (cdf) of them is presented. We derive the distribution of the order statistics from the trivariate exchangeable t-distribution in terms of our distribution and then an exact expression for the cdf of order statistics is derived. Likelihood inference for this distribution is also examined. The method is illustrated with a numerical example via a simulation study.     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

Stochastic comparisons of order statistics,with applications in relibility

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G^?1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described     

Do stock returns have an Archimedean copula?

The flexible class of Archimedean copulas plays an important role in multivariate statistics. While there is a large number of goodness-of-fit tests for copulas and parametric families of copulas, the question if a given data set belongs to an arbitrary Archimedean copula or not has not yet received much attention in the literature. This paper suggests a new, straightforward method to test whether a copula is an Archimedean copula without the need to specify its parametric family. We conduct Monte Carlo simulations to assess the power of the test. The approach is applied to (bivariate) joint distributions of stock asset returns. We find that, in general, stock returns may have Archimedean copulas.