Empirical study of some non-parametric tests for dispersion of correlated data

Results of a computer simulation study of power and robustness of three competitor tests for comparing scales, for use with correlated data: Rothstein, Richardson and Bell (RRB), Arvesen, and Pitman, are presented. It is found that unless one could ímprove the approximate null distributions for Arvesen's and Pitman's test, RRB's procedure is best, having simulated probabilities of Type I error closest to the test's nominal α and being reasonably robust and powerful, for all distributions considered.     

Application of box-cox transformations to discrimination for the two-class problem

The performance of Box-Cox power transformations in classification using Hinkley's (1975) method is studied. Misclassification probabilities before and after transformation are compared. It is found that the use of Box-Cox transformations can sometimes substantially reduce the error probabilities. Estimates of error probabilities are obtained and certain properties are derived. Examples for a number of distributions are given.     

Modelling ordinal assessments: fit is not sufficient

Assessments in ordered categories are ubiquitous in the social sciences. These assessments are assigned ordinal counts and analyzed with probabilistic models. If the counts fit the model, it is assumed that no unaccounted for factors govern the distribution and that it is a random error distribution. However, because tests of fit utilize parameter estimates from the data, the data may fit the model even when the modeled distributions cannot be random error distributions. This paper applies the additional criterion of strict unimodality, common to all random error distributions, to decide if a modeled distribution is not a random error distribution. However, not only are common random error distributions strictly unimodal, they are also strictly log-concave, a stronger form of unimodality which ensures smooth transitions between probabilities of adjacent counts. The paper shows that the operation for determining the strict unimodality also ensures that the distribution is locally strictly log-concave around the measure of the entity of assessment.     

A Bayesian Hierarchical Model for Multiple Comparisons in Mixed Models

We propose a Bayesian hierarchical model for multiple comparisons in mixed models where the repeated measures on subjects are described with the subject random effects. The model facilitates inferences in parameterizing the successive differences of the population means, and for them, we choose independent prior distributions that are mixtures of a normal distribution and a discrete distribution with its entire mass at zero. For the other parameters, we choose conjugate or vague priors. The performance of the proposed hierarchical model is investigated in the simulated and two real data sets, and the results illustrate that the proposed hierarchical model can effectively conduct a global test and pairwise comparisons using the posterior probability that any two means are equal. A simulation study is performed to analyze the type I error rate, the familywise error rate, and the test power. The Gibbs sampler procedure is used to estimate the parameters and to calculate the posterior probabilities.     

Parametric bootstrap inferences for unbalanced panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients of panel data regression models with incomplete panels. Some simulation results are presented to compare the performance of the PB approaches with the approximate inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly better than the approximate methods with respect to the coverage probabilities and the Type I error rates. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the multi-way error component regression models with unbalanced panels. Finally, the proposed approaches are illustrated by using a real data example.     

Skew-probit measurement error models

In this paper we extend the structural probit measurement error model by considering the unobserved covariate to follow a skew-normal distribution. The new model is termed the structural skew-normal probit model. As in the normal case, the likelihood function is obtained analytically, and can be maximized by using existing statistical software. A Bayesian approach using Markov chain Monte Carlo techniques for generating from the posterior distributions is also developed. A simulation study demonstrates the usefulness of the approach in avoiding attenuation which arises with the naive procedure. Moreover, a comparison of predicted and true success probabilities indicates that it seems to be more efficient to use the skew probit model when the distribution of the covariate (predictor) is skew. An application to a real data set is also provided.     

A NONPARAMETRIC TEST OF SYMMETRY VERSUS ASYMMETRY FOR RANKED-SET SAMPLES

This paper introduces a nonparametric test of symmetry for ranked-set samples to test the asymmetry of the underlying distribution. The test statistic is constructed from the Cramér-von Mises distance function which measures the distance between two probability models. The null distribution of the test statistic is established by constructing symmetric bootstrap samples from a given ranked-set sample. It is shown that the type I error probabilities are stable across all practical symmetric distributions and the test has high power for asymmetric distributions.     

Multi-step variance minimization in sequential tests

We introduce a multi-step variance minimization algorithm for numerical estimation of Type I and Type II error probabilities in sequential tests. The algorithm can be applied to general test statistics and easily built into general design algorithms for sequential tests. Our simulation results indicate that the proposed algorithm is particularly useful for estimating tail probabilities, and may lead to significant computational efficiency gains over the crude Monte Carlo method.     

How non-normality affects the quadratic discriminant function

The quadratic discriminant function is commonly used for the two group classification problem when the covariance matrices in the two populations are substantially unequal. This procedure is optimal when both populations are multivariate normal with known means and covariance matrices. This study examined the robustness of the QDF to non-normality. Sampling experiments were conducted to estimate expected actual error rates for the QDF when sampling from a variety of non-normal distributions. Results indicated that the QDF was robust to non-normality except when the distributions were highly skewed, in which case relatively large deviations from optimal were observed. In all cases studied the average probabilities of misclassification were relatively stable while the individual population error rates exhibited considerable variability.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

The Effects of Misconceptions on the Properties of Friedman's Test

Friedman's test is a widely used rank-based alternative to the analysis of variance (ANOVA) F-test for identifying treatment differences in a randomized complete block design. Many texts provide incomplete or misleading information about when Friedman's test may be appropriately applied. We discuss the assumptions needed for the test and common misconceptions. We show via simulation that when the variance or skew of the treatment distributions differ, application of Friedman's test to detect differences in treatment location can result in Type I error probabilities larger than the nominal α, and even when α is unaffected, the power of the test can be less than expected.     

On the Flexibility of Metropolis–Hastings Acceptance Probabilities in Auxiliary Variable Proposal Generation

Abstract. Use of auxiliary variables for generating proposal variables within a Metropolis–Hastings setting has been suggested in many different settings. This has in particular been of interest for simulation from complex distributions such as multimodal distributions or in transdimensional approaches. For many of these approaches, the acceptance probabilities that are used turn up somewhat magic and different proofs for their validity have been given in each case. In this article, we will present a general framework for construction of acceptance probabilities in auxiliary variable proposal generation. In addition to showing the similarities between many of the proposed algorithms in the literature, the framework also demonstrates that there is a great flexibility in how to construct acceptance probabilities. With this flexibility, alternative acceptance probabilities are suggested. Some numerical experiments are also reported.     

A new mixture model for capture heterogeneity

Summary.  We propose a mixture of binomial and beta–binomial distributions for estimating the size of closed populations. The new mixture model is applied to several real capture–recapture data sets and is shown to provide a convenient, objective framework for model selection. The new model is compared with three alternative models in a simulation study, and the results shed light on the general performance of models in this area. The new model provides a robust flexible analysis, which automatically deals with small capture probabilities.     

Partitioning a distribution

Suppose it is desired to partition a distribution into k groups (classes) using squared error or absolute error as the mea¬sure of information retained. An algorithm to obtain the optimal boundaries (or class probabilities) is giTen. For the case of squared error optimal class probabilities were obtained for k = 2 to 15 for beta (for various values of the parameters), chi-square (12 d.f.) exponential, normals and uniform distributions. Results obtained are compared and analysed in light of existing papers, Special attention is given to the case k =5, corresponding to the assignment of the letter grades A, B, C, D9 P in courses, and to the case k = 9 corresponding to stanines.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

Bayesian Variable Selection Under Collinearity

In this article, we highlight some interesting facts about Bayesian variable selection methods for linear regression models in settings where the design matrix exhibits strong collinearity. We first demonstrate via real data analysis and simulation studies that summaries of the posterior distribution based on marginal and joint distributions may give conflicting results for assessing the importance of strongly correlated covariates. The natural question is which one should be used in practice. The simulation studies suggest that posterior inclusion probabilities and Bayes factors that evaluate the importance of correlated covariates jointly are more appropriate, and some priors may be more adversely affected in such a setting. To obtain a better understanding behind the phenomenon, we study some toy examples with Zellner’s g-prior. The results show that strong collinearity may lead to a multimodal posterior distribution over models, in which joint summaries are more appropriate than marginal summaries. Thus, we recommend a routine examination of the correlation matrix and calculation of the joint inclusion probabilities for correlated covariates, in addition to marginal inclusion probabilities, for assessing the importance of covariates in Bayesian variable selection.     

Comparison of adjusted and unadjusted marginal probabilities for estimating binomial parameters in observational studies

In this study, we demonstrate how generalized propensity score estimators (Imbens’ weighted estimator, the propensity score weighted estimator and the generalized doubly robust estimator) can be used to calculate the adjusted marginal probabilities for estimating the three common binomial parameters: the risk difference (RD), the relative risk (RR), and the odds ratio (OR). We further conduct a simulation study to compare the estimated RD, RR, and OR using the adjusted and the unadjusted marginal probabilities in terms of the bias and mean-squared error (MSE). Although there is no clear winner in terms of the MSE for estimating RD, RR, and OR, simulation results surprisingly show thatthe unadjusted marginal probabilities produce the smallest bias compared with the adjusted marginal probabilities in most of the estimates. Hence, in conclusion, we recommend using the unadjusted marginal probabilities to estimate RD, RR, and OR, in practice.     

Utilizing auxiliary variables in sequential ratio and regression estimation schemes

Ratio and regression estimators for a mean are considered in conjunction with certain sequential sampling schemes. An auxiliary variable is assumed present and both fixed-cost and fixed- width confidence interval stopping rules are investigated. The asymptotic distributions of the estimators are derived as well as optimal probabilities pertinent to the schemes. Comparisons are made with results of certain double sampling procedures. Estimation of the ratio of two means is also considered and the results of a Monte Carlo simulation are included.     

The Effect of Mis-Specification Between the Lognormal and Weibull Distributions on the Interval Estimation of a Quantile for Complete Data

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.     

Multinomial Probability Estimation by Wavelet Thresholding

In this article, we introduce a wavelet threshold estimator to estimate multinomial probabilities. The advantages of the estimator are its adaptability to the roughness and sparseness of the data. The asymptotic behavior of the estimator is investigated through an often-used criteria: the mean sum of squared error (MSSE). We show that the MSSE of the estimator achieves the optimal rate of convergence. Its performance on finite samples is examined through simulation studies which show favorable results for the new estimator over the commonly used kernel estimator.