Homogeneity test based on ranked set samples

Abstract

The homogeneity hypothesis is investigated in a location family of distributions. A moment-based test is introduced based on data collected from a ranked set sampling scheme. The asymptotic distribution of the proposed test statistic is determined and the performance of the test is studied via simulation. Furthermore, for small sample sizes, the bootstrap procedure is used to distinguish the homogeneity of data. An illustrative example is also presented to explain the proposed procedures in this paper.     

Parameter estimation from interval-valued data using the expectation-maximization algorithm

This paper investigates on the problem of parameter estimation in statistical model when observations are intervals assumed to be related to underlying crisp realizations of a random sample. The proposed approach relies on the extension of likelihood function in interval setting. A maximum likelihood estimate of the parameter of interest may then be defined as a crisp value maximizing the generalized likelihood function. Using the expectation-maximization (EM) to solve such maximizing problem therefore derives the so-called interval-valued EM algorithm (IEM), which makes it possible to solve a wide range of statistical problems involving interval-valued data. To show the performance of IEM, the following two classical problems are illustrated: univariate normal mean and variance estimation from interval-valued samples, and multiple linear/nonlinear regression with crisp inputs and interval output.     

Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

A modified Mack–Wolfe test for the umbrella alternative problem

ABSTRACT

The Mack–Wolfe test is the most frequently used non parametric procedure for the umbrella alternative problem. In this paper, modifications of the Mack–Wolfe test are proposed for both known peak and unknown peak umbrellas. The exact mean and variance of the proposed tests in the null hypothesis are also derived. We compare these tests with some of the existing tests in terms of the type I error rate and power. In addition, a real data example is presented.     

On the ratio of two independent skewnesses

In many fields, the researchers are interested in making inferences about the ratio of skewnesses in two independent populations. In the present paper, the asymptotic distribution for the ratio of the sample skewnesses in two independent populations is established. Then the asymptotic distribution is used to derive the asymptotic confidence interval and to test the hypothesis for the ratio of population's skewnesses. Finally, the applicability of the proposed method is investigated through Monte Carlo simulations.     

Test for Trend With a Multinomial Outcome

ABSTRACT

There is no established procedure for testing for trend with nominal outcomes that would provide both a global hypothesis test and outcome-specific inference. We derive a simple formula for such a test using a weighted sum of Cochran–Armitage test statistics evaluating the trend in each outcome separately. The test is shown to be equivalent to the score test for multinomial logistic regression, however, the new formulation enables the derivation of a sample size formula and multiplicity-adjusted inference for individual outcomes. The proposed methods are implemented in the R package multiCA.     

Interval robust design under contaminated and non normal data

Abstract

Robust parameter design (RPD) is an effective tool, which involves experimental design and strategic modeling to determine the optimal operating conditions of a system. The usual assumptions of RPD are that normally distributed experimental data and no contamination due to outliers. And generally the parameter uncertainties in response models are neglected. However, using normal theory modeling methods for a skewed data and ignoring parameter uncertainties can create a chain of degradation in optimization and production phases such that misleading fit, poor estimated optimal operating conditions, and poor quality products. This article presents a new approach based on confidence interval (CI) response modeling for the process mean. The proposed interval robust design makes the system median unbiased for the mean and uses midpoint of the interval as a measure of location performance response. As an alternative robust estimator for the process variance response modeling, using biweight midvariance is proposed which is both resistant and robust of efficiency where normality is not met. The results further show that the proposed interval robust design gives a robust solution to the skewed structure of the data and to contaminated data. The procedure and its advantages are illustrated using two experimental design studies.     

Approximating Symmetric Distributions via Sampling and Coefficient of Variation

Abstract

The Coefficient of Variation is one of the most commonly used statistical tool across various scientific fields. This paper proposes a use of the Coefficient of Variation, obtained by Sampling, to define the polynomial probability density function (pdf) of a continuous and symmetric random variable on the interval [a, b]. The basic idea behind the first proposed algorithm is the transformation of the interval from [a, b] to [0, b-a]. The chi-square goodness-of-fit test is used to compare the proposed (observed) sample distribution with the expected probability distribution. The experimental results show that the collected data are approximated by the proposed pdf. The second algorithm proposes a new method to get a fast estimate for the degree of the polynomial pdf when the random variable is normally distributed. Using the known percentages of values that lie within one, two and three standard deviations of the mean, respectively, the so-called three-sigma rule of thumb, we conclude that the degree of the polynomial pdf takes values between 1.8127 and 1.8642. In the case of a Laplace (μ, b) distribution, we conclude that the degree of the polynomial pdf takes values greater than 1. All calculations and graphs needed are done using statistical software R.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

On Hotelling’s T2 test in a special paired sample case

In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.     

Point estimation of lower quantiles based on two-sampling scheme

ABSTRACT

Consider a two-sampling scheme in which an initial sample is first taken from the underlying population and then by assuming a suitable restriction on this sample, some more data points are observed as a new restricted sample. This sampling scheme is used to do inference about the lower quantiles of the underlying distribution. The results are compared with those of simple random sampling in view of mean squared error and Pitman’s measure of closeness criteria for exponential and uniform distributions. It will be shown that the proposed sampling scheme would improve the performance of the point estimators of the lower quantiles of the population.     

Bayesian two-stage design for drug screening trials with switching hypothesis tests based on continuous endpoints

Abstract

In this paper, we propose a Bayesian two-stage design with changing hypothesis test by bridging a single-arm study and a double-arm randomized trial in one phase II clinical trial based on continuous endpoints rather than binary endpoints. We have also calibrated with respect to frequentist and Bayesian error rates. The proposed design minimizes the Bayesian expected sample size if the new candidate has low or high efficacy activity subject to the constraint upon error rates in both frequentist and Bayesian perspectives. Tables of designs for various combinations of design parameters are also provided.     

ROBUSTNESS PROPERTIES OF LOGNORMAL CONFIDENCE INTERVALS FOR LOGNORMAL AND GAMMA DISTRIBUTED DATA

ABSTRACT

The performances of six confidence intervals for estimating the arithmetic mean of a lognormal distribution are compared using simulated data. The first interval considered is based on an exact method and is recommended in U.S. EPA guidance documents for calculating upper confidence limits for contamination data. Two intervals are based on asymptotic properties due to the Central Limit Theorem, and the other three are based on transformations and maximum likelihood estimation. The effects of departures from lognormality on the performance of these intervals are also investigated. The gamma distribution is considered to represent departures from the lognormal distribution. The average width and coverage of each confidence interval is reported for varying mean, variance, and sample size. In the lognormal case, the exact interval gives good coverage, but for small sample sizes and large variances the confidence intervals are too wide. In these cases, an approximation that incorporates sampling variability of the sample variance tends to perform better. When the underlying distribution is a gamma distribution, the intervals based upon the Central Limit Theorem tend to perform better than those based upon lognormal assumptions.     

基于正态分布点值化的区间主成分评价法及应用

针对区间数据点值化过程中所存在的“代表性不足”的缺陷,提出了基于正态分布的点值化方法并将之应用于区间主成分评价法。通过与基于中心点值化的区间主成分法的比较,得到三个主要结论：第一,基于正态分布的点值化方法能将各样品的点值化结果导向指标均值,而非区间值的中心点;第二,基于正态分布的点值化结果增加了数据信息量;第三,基于正态分布点值化的区间主成分评价法提高了数据降维效果,具有更好的因子命名能力。应用结果表明,在考虑正态分布情况下,对区间数据的点值化处理方法具有较好的效果,基于正态分布点值化的方法可推广至基于区间数的评价和决策问题。     

Testing for structural changes in linear regressions with time-varying variance

Abstract

This article proposes a nonparametric test for structural changes in linear regression models that allows for serial correlation, autoregressive conditional heteroskedasticity and time-varying variance in error terms. The test requires no trimming of the boundary region near the end points of the sample period, and requires no prior information on the alternative, what it requires is the transformed OLS residuals under the null hypothesis. We show that the test has a limiting standard normal distribution under the null hypothesis, and is powerful against single break, multiple breaks and smooth structural changes. The Monte Carlo experiment is conducted to highlight the merits of the proposed test relative to other popular tests for structural changes.     

Semiparametric inference on partially linear single-index model

ABSTRACT

We consider semiparametric inference on the partially linearsingle-index model (PLSIM). The generalized likelihood ratio (GLR) test is proposed to examine whether or not a family of new semiparametric models fits adequately our given data in the PLSIM. A new GLR statistic is established to deal with the testing of the index parameter α₀ in the PLSIM. The newly proposed statistic is shown to asymptotically follow a χ²-distribution with the scale constant and the degrees of freedom being independent of the nuisance parameters or function. Some finite sample simulations and a real example are used to illustrate our proposed methodology.     

Testing the Martingale Difference Hypothesis

Abstract

In this paper we consider testing that an economic time series follows a martingale difference process. The martingale difference hypothesis has typically been tested using information contained in the second moments of a process, that is, using test statistics based on the sample autocovariances or periodograms. Tests based on these statistics are inconsistent since they cannot detect nonlinear alternatives. In this paper we consider tests that detect linear and nonlinear alternatives. Given that the asymptotic distributions of the considered tests statistics depend on the data generating process, we propose to implement the tests using a modified wild bootstrap procedure. The paper theoretically justifies the proposed tests and examines their finite sample behavior by means of Monte Carlo experiments.     

TESTS FOR EQUALITY OF TWO DISTRIBUTIONS WITH CORRELATED INTERVAL CENSORED LIFE-TIMES

ABSTRACT

Large sample properties of Life-Table estimator are discussed for interval censored bivariate survival data. We restrict our attention to the situation where response times within pairs are not distinguishable, and the univariate survival distribution is the same for any individual within any pair. The large sample properties are applied to test for equality of two distributions with correlated response times where treatments are applied to different independent sets of cohorts. Data, which can be separated into two independent sets, from an angioplasty study where more than one procedure is performed on some patients are used to illustrate this methodology.