Hypothesis testing on the correlation coefficient

A simple and accurate test on the value of the correlation coefficient in normal bivariate populations is here proposed. Its accuracy compares favourably with any previous approximations.     

A robust test for testing the correlation coefficient

A test based on Tiku's MML (modified maximum likelihood) estimators is developed for testing that the population correlation coefficient is zero. The test is compared with various other tests and shown to have good Type I error robustness and power for numerous symmetric and skew bivariate populations.     

Confidence interval estimation and hypothesis testing for a common correlation coefficient

Several procedures have been proposed for testing the hypothesis that all off-diagonal elements of the correlation matrix of a multivariate normal distribution are equal. If the hypothesis of equal correlation can be accepted, it is then of interest to estimate and perhaps test hypotheses for the common correlation. In this paper, two versions of five different test statistics are compared via simulation in terms of adequacy of the normal approximation, coverage probabilities of confidence intervals, control of Type I error, and power. The results indicate that two test statistics based on the average of the Fisher z-transforms of the sample correlations should be used in most cases. A statistic based on the sample eigenvalues also gives reasonable results for confidence intervals and lower-tailed tests.     

Empirical likelihood-based serial correlation testing in partially varying coefficient single-index models

ABSTRACT

Partially varying coefficient single-index models (PVCSIM) are a class of semiparametric regression models. One important assumption is that the model error is independently and identically distributed, which may contradict with the reality in many applications. For example, in the economical and financial applications, the observations may be serially correlated over time. Based on the empirical likelihood technique, we propose a procedure for testing the serial correlation of random error in PVCSIM. Under some regular conditions, we show that the proposed empirical likelihood ratio statistic asymptotically follows a standard χ² distribution. We also present some numerical studies to illustrate the performance of our proposed testing procedure.     

A robust procedure for testing an assumed value of the population correlation coefficient

For testing that the population correlations coefficientp Q, Tiku and Balakrishnan1986) developed a robust test. This test is extended here to situcitions where one wants to test that p p , p being a specified non-zero value of p. o o     

The asymptotic distribution of the correlation coefficient in testing fit to the exponential distribution

The asymptotic distribution of certain tests of fit to the exponential distribution is obtained. The tests are based on regression of the order statistics on their expectations under a standard exponential distribution. Asymptotic normality at the rate (log n)^1/2 is obtained for a family of statistics including the correlation coefficient.     

Hypothesis testing in log-Birnbaum-Saunders regressions

The log-Birnbaum-Saunders regression model introduced by Rieck and Nedelman (1991 Rieck, J. R., Nedelman, J. R. (1991). A log-linear model for the Birnbaum-Saunders distribution. Technometrics 33:51–60. [Google Scholar]) is useful for modeling lifetimes of materials and equipments subject to different conditions. Our goal in this article is twofold. First, we numerically evaluate the finite sample performances of the likelihood ratio, score and Wald tests in the log-Birnbaum-Saunders regression model. Second, we introduce a RESET-like misspecification test for that model. The null hypothesis is that the model is correctly specified which is tested against the alternative hypothesis of model misspecification. The power of the test is evaluated using Monte Carlo simulations. Bootstrap-based inference is also considered. An empirical application is presented and discussed.     

Combined estimators for the correlation coefficient

Three combined estimators for the bivariate normal correlation parameter are considered. The data consist of k independent sample correlation coefficients and it is assumed that the underlying correlation parameters are all equal to ρ. Based upon the joint density function of the sample correlations a combined estimator of ρ is obtained as an approximation to the maximum likelihood solution. Two linearly combined estimators are also considered. One of them is based on Fisher's z-transformation of the sample correlations and the other on an unbiased estimator of ρ. The comparison of these three estimators indicates that the combined (approximate) MLE has a slightly smaller estimated mean squared error relative to the other two combined methods of estimation, but it does so at the expense of a relatively larger bias.     

Hypothesis testing in the presence of nuisance parameters

How to deal with nuisance parameters is an important problem in econometrics because of the non-experimental nature of economic data. This paper suggests a new approach to dealing with such parameters in the context of hypothesis testing. It involves calculating p-values conditional on values for key nuisance parameters and then taking a weighted average of these values with the weights reflecting the likelihood or posterior probabilities of these values being true. Two specific applications are discussed. These are testing linear regression coefficients in the presence of first-order autoregressive (AR(1)) disturbances and testing for AR(1) disturbances in the dynamic linear regression model. For the former testing problem, a Monte Carlo experiment demonstrates that the new procedure typically provides more accurate inferences than the accepted conventional tests.     

Hypothesis testing in models using the box-cox transformation

The small sample powers of two statistics, the likelihood ratio test, and a test based on the asymptotic normality of maximum likelihood estimators (z-test) were compared in a simulation experiment. Two models were specified, one containing the Box-Cox transformation on the dependent variable only, and one containing the Box Cox transformation on both the dependent and independent variables. The transformation parameter,λ was estimated 200 times, for each of six different values of z in each of three sample sizes foi both models. At each replication. 17 hypotheses were tested using both a likelihood ratio test and a z-test. Results indicate that w hiic both likelihood ratio tests and z-tests are unbiased, in small samples the z-test is generally preferable to the likelihood ratio test.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Hypothesis testing via a penalized-likelihood approach

It is illustrated in this paper that hypothesis testing procedures can be derived based on the penalized likelihood approach. Based on this point of view, many traditional hypothesis tests, including the two-sample mean test, score test, and Hotelling’s $T^2$ test are revisited under the penalized likelihood framework. Similar framework is also applicable to the empirical likelihood.     

Hypothesis testing in mixture regression models

Summary.  We establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models. Moreover, under specific and reasonable conditions, we show that the optimal convergence rate of n ^−1/4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators. We also derive the asymptotic distributions of two log-likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p -value. Simulation studies are conducted to investigate the empirical performance of the two test statistics. Finally, two real data sets are analysed to illustrate the application of our theoretical results.     

Bootstrap power of the generalized correlation coefficient

We present a bootstrap Monte Carlo algorithm for computing the power function of the generalized correlation coefficient. The proposed method makes no assumptions about the form of the underlying probability distribution and may be used with observed data to approximate the power function and pilot data for sample size determination. In particular, the bootstrap power functions of the Pearson product moment correlation and the Spearman rank correlation are examined. Monte Carlo experiments indicate that the proposed algorithm is reliable and compares well with the asymptotic values. An example which demonstrates how this method can be used for sample size determination and power calculations is provided.     

On intra-class correlation coefficient estimation

Consider the problem of estimating the intra-class correlation coefficient of a symmetric normal distribution. In a recent article (Pal and Lim (1999)) it has been shown that the three popular estimators, namely—the maximum likelihood estimator (MLE), the method of moments estimator (MME) and the unique minimum variance unbiased estimator (UMVUE), are second order admissible under the squared error loss function. In this paper we study the performance of the above mentioned estimators in terms of Pitman Nearness Criterion (PNC) as well as Stochastic Domination Criterion (SDC). We then apply the aforementioned estimators to two real life data sets with moderate to large sample sizes, and bootstrap bias as well as mean squared errors are computed to compare the estimators. In terms of overall performance the MME seems most appealing among the three estimators considered here and this is the main contribution of our paper. Formerly University of Southewestern Louisisna     

An improved concordance correlation coefficient

It is often required to compare two measurements in medicine and other experimental sciences. This problem covers a broad range of data, and examples can be found in different industries. In this paper, a new index on measuring agreement is proposed, which is similar to Lin's concordance correlation coefficient but derived from a criterion which is more conceptually appealing and which offers improvements. An example is used to demonstrate the benefit of using the new index. Copyright © 2003 John Wiley & Sons, Ltd.     

Notes on the MLE of correlation coefficient in meta analysis

We study the properties of two approximations to the MLE of the correlation coefficient based on estimates from several studies in meta analysis. Our work is based on an approximation to the density of a function of the sample product-moment estimate due to Dclury, Hsu, and Kraemer. Regarding this approximation, we point out and correct some mistakes in the literature.