A diathered k-s staticstic

The Kolmogorov-Sruimov test depends for its distribution-free property on the observation that, if F() is the CDF of continuous random variable A then random variable F(X) is uniform on the interval [0,1]. That observation is false if F() is not continuous, a restriction on the applicability of the KS test. This paper introduces a generalization of the KS statistic that works for all distributions, including discrete ones.     

ON THE PERFORMANCE OF KOLMOGOROV-SMIRNOV TESTS ON STABLE RANDOM VARIABLES1

This paper provides two procedures to perform the Kolmogorov–Smirnov (K-S) tests on stable random variables. One utilizes the integral representation of the cumulative distribution function of this random variable to perform the K-S test; the other utilizes the Gil-Pelaez-Roseń transformation of data (Csörg(1983)) to test uniformity on (0,1) for the transformed sample. Both procedures are examined and evaluated by a simulation study.     

A method to generate autocorrelated uniform random numbers

We present a simple, fast method to generate autocorrelated uniform random numbers. The “sum of uniforms” method adds a pair of U(0,1) random numbers, transforms the sum to a third U(0,1) random number, and uses this third random number as one member of the next pair. The method produces any desired level of positive or negative correlation between successive random numbers.     

A unified approach to binomial-type distributions of order k

In this paper, we consider the distribution of the number of "1"-runs of length k in a sequence of {0,1}-valued random variables of length n by using a new (unified) counting scheme called l-overlapping counting. Here, k and n are positive integers with k ≦ and l is an integer less than k. We obtain the prohabi!ity generating function of the distribution of the number of eoverlapping "in-runs of iength k in the sequence, even when the underiying sequence is a dependent sequence such as a highcr order Markov chaic.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

Logistic processes involving markovian minimization

The class of logistic processes involving geometric minimization introduced by Arnold (1989) is extended by replacing the Bernoulli sequence used in the definition of the process by a Markovian (0,1) sequence. A more flexible range of dependence structures is thus introduced. Parameter estimation and related processes are discussed.     

A Hybrid Method for Random Pattern Sequence Classification

We present a hybrid method for random pattern sequence classification that takes into account the random structural properties of the sequence. The method works in two steps. A segmentation step, dividing the original sequence into segments, such that all observations in a same segment belong to a unique class, and a classification step, where each segment is classified by a neural network classifier.     

A simple generalization of the Box–Muller method for obtaining a pair of correlated standard normal variables

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given     

Proportional hazards model for discrete data: Some new developments

ABSTRACT

Many times, a product lifetime can be described through a non negative integer valued random variable. In this article, we propose a proportional hazards model for discrete data analogous to the version for continuous data. Some ageing properties of the model are discussed. Stochastic comparison of pair of random variables that follow the model are also made. A new test based on U-statistics is developed for testing that the proportionality parameter in the proposed model is 1. The asymptotic properties of the proposed test are studied. We present some numerical results to asses the performance of the test procedure.     

Omnibus testing and gene filtration in microarray data analysis

When thousands of tests are performed simultaneously to detect differentially expressed genes in microarray analysis, the number of Type I errors can be immense if a multiplicity adjustment is not made. However, due to the large scale, traditional adjustment methods require very stringen significance levels for individual tests, which yield low power for detecting alterations. In this work, we describe how two omnibus tests can be used in conjunction with a gene filtration process to circumvent difficulties due to the large scale of testing. These two omnibus tests, the D-test and the modified likelihood ratio test (MLRT), can be used to investigate whether a collection of P-values has arisen from the Uniform(0,1) distribution or whether the Uniform(0,1) distribution contaminated by another Beta distribution is more appropriate. In the former case, attention can be directed to a smaller part of the genome; in the latter event, parameter estimates for the contamination model provide a frame of reference for multiple comparisons. Unlike the likelihood ratio test (LRT), both the D-test and MLRT enjoy simple limiting distributions under the null hypothesis of no contamination, so critical values can be obtained from standard tables. Simulation studies demonstrate that the D-test and MLRT are superior to the AIC, BIC, and Kolmogorov-Smirnov test. A case study illustrates omnibus testing and filtration.     

Integer-valued moving average (INMA) process

A simple model for a stationary sequence of dependent integer-valued random variables {X_n} is given. The sequence to be called integer-valued moving average (INMA) process, is taken as the “survivals” of i.i.d. non-negative integervalued random variables. It is argued that the model’s structure reflects to some extent the mechanism generating real life data for many counting process and consequently it is useful for modelling such processes. Various properties for the special case in which {X_n} is Poisson INMA (1) process, such as the joint distribution, regression, time reversibility, along with the conditional and partial correlations, are discussed in details. Extension of the INMA of first order to higher order moving average is considered.     

A test for randomness based on a complexity measure

A test for randomness based on a statistic related to the complexity of finite sequences is presented. Simulation of binary sequences under different stochastic models provides estimates of the power of the test. The results show that the test is sensitive to a variety of alternatives to randomness and suggest that the proposed test statistic is a reasonable measure of the stochastic complexity of a finite sequence of discrete random variables.     

A modified harmonic mean test procedure for variance components

A complete class of tests of variance components is characterized within the class of tests statistics of the form of a ratio of a linear combination of chi-squared random variables to an independent chi-squared random variable. This result is used in the context of general unbalanced mixed models to show that the harmonic mean method results in an inadmissible test of the random treatment effects. The harmonic mean procedure is then modified in such a way that the modified test uniformly dominates the original test. Two competitive tests are the LMP (locally most powerful) and Wald's tests, which have optimal power properties against small and large alternatives, respectively. A Monte Carlo simulation study reveals that the modified test outperforms both the LMP and Wald's tests in badly unbalanced designs and that it is a viable alternative in less unbalanced designs.     

Shapiro–Wilk test for skew normal distributions based on data transformations

A probability property that connects the skew normal (SN) distribution with the normal distribution is used for proposing a goodness-of-fit test for the composite null hypothesis that a random sample follows an SN distribution with unknown parameters. The random sample is transformed to approximately normal random variables, and then the Shapiro–Wilk test is used for testing normality. The implementation of this test does not require neither parametric bootstrap nor the use of tables for different values of the slant parameter. An additional test for the same problem, based on a property that relates the gamma and SN distributions, is also introduced. The results of a power study conducted by the Monte Carlo simulation show some good properties of the proposed tests in comparison to existing tests for the same problem.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Sampling efficiency for an alternating poisson process

The efficiency of schemes for sampling an alternating Poisson process (0,1 observations) is evaluated by the inverse ratio of the variance of the proportion estimate, p, to the binomial variance. The variance ratio presented by D.R. Cox (in Renewal Theory) for fixed interval sampling is generalized to accommodate random sampling and random sampling after a time delay equal to a fixed proportion, γ , of the mean time between observations, δ. The result is a sampling design tool that provides quantifications for the effect of various spacings between observations and of fixed vs. random sampling. Direct application is made to thes field of work sampling.     

Testing hypotheses when there could be an initial effect

In many treatment-versus-control experiments, the observed random variables can be written as the product of a Bernoulli and a continuous random variable. The treatment can affect the distribution of the observations in two ways.

1. the probability that the observation is 0 could be altered.

2. the distribution of the nonzero observations could be changed.

We may also want to measure the combined effect of the treatment.

3. the expected value of control and treated units may differ.

A method is presented for testing for the presence of the combined effect when the general form of the distribution function of the continuous observations is known. For the case when this distribution function is from the family of gamma distributions, a previously proposed test criterion for the combined effect has poor power properties. In this paper, we discuss a test criterion that has improved power properties.     

Estimated Generalized Least Squares for Random Coefficient Regression Models

ABSTRACT. This paper considers a general class of random coefficient regression (RCR) models to represent pooled cross-sectional and time series data. A new method is given to estimate the covariance matrix of the error component in these RCR models. Also, the asymptotic and small sample properties of the estimated generalized least squares estimator of the regression coefficient vector are established. Procedures for testing a linear restriction on the mean vector of the random coefficients are derived. Finally, a test for non-randomness in the RCR model is devised, and the asymptotic distribution of the test statistic is obtained.     

A Simple Test For The Independence of Sets of Variables

A test for the mutual independence of subvectors of the p-dimensional random vector X , distributed as N( 0, S? ), is described. The test is based on the maximum likelihood estimates (MLEs) of the off-(block) diagonal elements of S?. It is shown that the resulting test statistic is much easier to compute than the likelihood ratio (LR) test statistic while retaining the same asymptotic power properties in view of the general properties of tests based on the MLEs (ML test) and the likelihood ratio (LR test).