Asymptotically Optimal Design Points for Rejection Algorithms

ABSTRACT

Very fast automatic rejection algorithms were developed recently which allow us to generate random variates from large classes of unimodal distributions. They require the choice of several design points which decompose the domain of the distribution into small sub-intervals. The optimal choice of these points is an important but unsolved problem. Therefore, we present an approach that allows us to characterize optimal design points in the asymptotic case (when their number tends to infinity) under mild regularity conditions. We describe a short algorithm to calculate these asymptotically optimal points in practice. Numerical experiments indicate that they are very close to optimal even when only six or seven design points are calculated.     

A Simple Method for Constructing Multidimensional Distributions of Correlated Categorical Data

We suggest a method for constructing a multidimensional distribution of correlated categorical data with fixed marginal distributions and specified degrees of association based on the log-linear models. A convex combination approach by Lee (1997 Lee , A. J. ( 1997 ). Some simple methods for generating correlated categorical variates . Computational Statistics & Data Analysis 26 : 133 – 148 .[Crossref], [Web of Science ®] , [Google Scholar]) is applied to get a joint distribution with fixed Pearson chi-square coefficient. By using the suggested method, we can generate three-dimensional distributions which have a fixed association among three variables. Therefore, the suggested method could be extended to higher dimensions.     

On Multi-Sided Permutation Tests

In fields such as clinical trials, the environment, epidemiology, genetics, pharmacology, social experiments, etc., situations in which the effect of a treatment can be positive on some individuals and negative on the rest may sometimes occur. In a two-sample design, this situation can be expressed by a response model where a random effect Δ in the alternative is such that , and . Such a situation is essentially different from the traditional two-sided test, in which the alternative is assumed to be active only on one of two directions. We consider alternatives in which two sub-alternatives (Δ < 0) and (Δ > 0) can be jointly active. In order to deal with such an atypical situation, we suggest to firstly apply two goodness-of-fit tests, one for the positive deviations from the empirical distribution function of control sample and the other for the negative, and then proceed with their nonparametric combination within a permutation framework. A simulation study inspects on properties of such a solution.     

Parameter Estimation for the Tail Distribution of a Random Sequence

The tail Y_t = X_t – u of a random sequence {X_t ∈ , t ∈ } with identically distributed X_t is approximated by the generalized Pareto distribution according to the extreme value theory, wherein Y_t occurs in clusters because of the dependence in the random sequence. Nevertheless, the parameters of the generalized Pareto distribution are estimated by the same methods as in the case of independent and identically distributed Y_t, provided that there is independence between the clusters of Y_t. The estimation variances and confidence intervals can be estimated by the jackknife method. The approaches are theoretically discussed and verified by extensive numerical researches.     

Fractional Regression Hot Deck Imputation Weight Adjustment

Fractional regression hot deck imputation (FRHDI) imputes multiple values for each instance of a missing dependent variable. The imputed values are equal to the predicted value plus multiple random residuals. Fractional weights enable variance estimation and preserve correlations. In some circumstances with some starting weight values, existing procedures for computing FRHDI weights can produce negative values. We discuss procedures for constructing non-negative adjusted fractional weights for FRHDI and study performance of the algorithm using simulation. The algorithm can be used effectively with FRDHI procedures for handling missing data in the context of a complex sample survey.     

Nonparametric Modeling Auxiliary Covariates in Random Coefficient Models

Random coefficient model (RCM) is a powerful statistical tool in analyzing correlated data collected from studies with different clusters or from longitudinal studies. In practice, there is a need for statistical methods that allow biomedical researchers to adjust for the measured and unmeasured covariates that might affect the regression model. This article studies two nonparametric methods dealing with auxiliary covariate data in linear random coefficient models. We demonstrate how to estimate the coefficients of the models and how to predict the random effects when the covariates are missing or mismeasured. We employ empirical estimator and kernel smoother to handle a discrete and continuous auxiliary, respectively. Simulation results show that the proposed methods perform better than an alternative method that only uses data in the validation data set and ignores the random effects in the random coefficient model.     

Analysis of Linear Models with One Factor Having Both Fixed and Random Levels

ABSTRACT

A general theory for a case where a factor has both fixed and random effect levels is developed under one-way treatment structure model. Estimation procedures for the fixed effects and variance components are consider for the model. The testing of fixed effects is considered when the variance–covariance matrix is known and unknown. Confidence intervals for estimable functions and prediction intervals for predictable functions are constructed. The computational procedures are illustrated using data from an on-farm trial.     

The Influence of Random Effects on Univariate and Bivariate Discrete Proportional Hazards Models

The random effects survival model has been widely used in the recent literature as a generalization of the continuous proportional hazards model. When a random effect is present, it is known that the hazard rates are generally underestimated in the context of continuous proportional hazards models. This article establishes theorems for the influence of random effects on both univariate and bivariate discrete proportional hazards models.     

Testing a Unit Root Based on Aggregate Time Series

Many time series encountered in practice are nonstationary, and instead are often generated from a process with a unit root. Because of the process of data collection or the practice of researchers, time series used in analysis and modeling are frequently obtained through temporal aggregation. As a result, the series used in testing for a unit root are often time series aggregates. In this paper, we study the effects of the use of aggregate time series on the Dickey–Fuller test for a unit root. We start by deriving a proper model for the aggregate series. Based on this model, we find the limiting distributions of the test statistics and illustrate how the tests are affected by the use of aggregate time series. The results show that those distributions shift to the right and that this effect increases with the order of aggregation, causing a strong impact both on the empirical significance level and on the power of the test. To correct this problem, we present tables of critical points appropriate for the tests based on aggregate time series and demonstrate their adequacy. Examples illustrate the conclusions of our analysis.     

Wavelets for Nonparametric Stochastic Regression with Mixing Stochastic Process

We propose a wavelet based stochastic regression function estimator for the estimation of the regression function for a sequence of mixing stochastic process with a common one-dimensional probability density function. Some asymptotic properties of the proposed estimator are investigated. It is found that the estimators have similar properties to their counterparts studied earlier in literature.