Generalized portmanteau statistics and tests of randomness

This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented in the paper were derived by usig a symbolic manipulation program.     

Efficiency of nonparametric tests for scale shift at an unknown time point

Recently Hsieh considered three nonparametric tests for the scalechange problem and showed them to be asymptotically as efficient as their parametric competitors. Here the class of so called sum-type statistics is studied which contains the statistics of Hsieh. It is proved that max-type statistics can be constructed that are at least as efficient in the sense of Bahadur as the sum-type statistics.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

A bartlett correction factor for a likelihood ratio statistic in multivariate bioassay

A more accurate Bartlett correction factor is proposed for a likelihood ratio statistic in multivariate bioassay.     

Distribution-free tests in randomized blocks

Classes of distribution-free tests are proposed for testing homogeneity against order restricted as well as unrestricted alternatives in randomized block designs with multiple observations per cell. Allowing for different interblock scoring schemes, these tests are constructed based on the method of within block rankings. Asymptotic distributions (cell sizes tending to infinity) of these tests are derived under the assumption of homogeneity. The Pitman asymptotic relative efficiencies relative to the least squares statistics are studied. It is shown that when blocks are governed by different distributions, adaptive choice of scores within each block results in asymptotically more efficient tests as compared with methods that ignore such information. Monte Carlo simulations of selected designs indicate that the method of within block rankings is more power robust with respect to differing block distributions.     

Bounds on the regression coefficients when a covariate is categorized

This paper considers the problem of a linear regression model in which a continuous covariate is categorized. In such a case one can obtain bounds on the OLS-estimate which would be calculated if the complete information were available. These bounds give an indication of the information loss due to grouping and show what can be inferred from the data without additional assumptions.     

A nonparametric estimator of the number of classes based on a stratified random sample

Under stratified random sampling, we develop a k^th-order bootstrap bias-corrected estimator of the number of classes θ which exist in a study region. This research extends Smith and van Belle’s (1984) first-order bootstrap bias-corrected estimator under simple random sampling. Our estimator has applicability for many settings including: estimating the number of animals when there are stratified capture periods, estimating the number of species based on stratified random sampling of subunits (say, quadrats) from the region, and estimating the number of errors/defects in a product based on observations from two or more types of inspectors. When the differences between the strata are large, utilizing stratified random sampling and our estimator often results in superior performance versus the use of simple random sampling and its bootstrap or jackknife [Burnham and Overton (1978)] estimator. The superior performance is often associated with more observed classes, and we provide insights into optimal designation of the strata and optimal allocation of sample sectors to strata.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).