Conditional quantile estimation with auxiliary information for left-truncated and dependent data

In this paper, the empirical likelihood method is used to define a new estimator of conditional quantile in the presence of auxiliary information for the left-truncation model. The asymptotic normality of the estimator is established when the data exhibit some kind of dependence. It is assumed that the lifetime observations with multivariate covariates form a stationary α‐mixing sequence. The result shows that the asymptotic variance of the proposed estimator is not larger than that of standard kernel estimator. Finite sample behavior of the estimator is investigated via simulations too.     

A comparative study on a permutation statistic

We studied asymptotic distribution and finite sample properties of a randomly weighted permutation statistic. The asymptotic normality and the finite sample simulations derived from our studies provided theoretical and numerical justifications for distributional assumption of many useful test statistics used in identifying spatial autocorrelations of mapped data. We compared a new method in computing the mean and the approximated variance of the randomly weighted D statistic, a special permutation statistic, with the Walter’s conditional method. In the numerical illustration of the method, we calculated the standardized values of the D statistic by subtracting the mean from the D statistic and dividing the difference by the standard deviation for the standardized mortality ratios (SMRs) and the life expectancies among the 48 states of the continental USA. Spatial autocorrelations of the SMRs and the life expectancies were found to be statistically significant.     

Generalized choice models for categorical time series

The NDARMA models of Jacobs and Lewis (1983) allow the modeling of categorical processes with an ARMA-like serial dependence structure. These models can be represented through a backshift mechanism, and we analyze marginal and bivariate properties of the resulting backshift process. Motivated by this backshift mechanism, we define the new class of generalized choice (GC) models, which include the usual NDARMA models as a special case, and we derive results describing the marginal and bivariate distribution of the GC model. We discuss implications concerning DMA(∞) models and the serial dependence structure of NDARMA models. Examples show that the family of GC models allows creating sparsely parametrized models for categorical processes with different types of serial dependence structure.     

A note on Läuter-type robust design for polynomial regression models

In this paper, we consider the problem of model robust design for simultaneous parameter estimation among a class of polynomial regression models with degree up to k. A generalized D-optimality criterion, the Ψ_α‐optimality criterion, first introduced by Läuter (1974) is considered for this problem. By applying the theory of canonical moments and the technique of maximin principle, we derive a model robust optimal design in the sense of having highest minimum Ψ_α‐efficiency. Numerical comparison indicates that the proposed design has remarkable performance for parameter estimation in all of the considered rival models.     

Robust multivariate Bayesian forecasting under functional distortions in the -metric

This paper investigates robustness of multivariate forecasting in the Bayesian framework. The minimax approach is used to construct robust statistical procedures under deviations from hypothetical assumptions. The deviations are defined as functional distortions using the χ²-pseudo-metric. Two cases of deviations are considered: distortions of parameter distribution and distortions of joint distribution of observations and parameters. Explicit forms for the guaranteed upper risk functional are obtained and integral equations for robust prediction statistics are given for both cases.     

On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N⁻¹ for the multivariate sample skewness and approximate χ²${\chi }^2$ test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ²${\chi }^2$ test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the obtained expectation, variance and improved approximate χ²${\chi }^2$ test statistic. Furthermore, upper and lower percentiles of χ²${\chi }^2$ test statistic derived in this paper are compared with those of χ²${\chi }^2$ test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).     

Empirical Bayes vs. fully Bayes variable selection

For the problem of variable selection for the normal linear model, fixed penalty selection criteria such as AIC, _Cp${\mathrm{C}}_p$, BIC and RIC correspond to the posterior modes of a hierarchical Bayes model for various fixed hyperparameter settings. Adaptive selection criteria obtained by empirical Bayes estimation of the hyperparameters have been shown by George and Foster [2000. Calibration and Empirical Bayes variable selection. Biometrika 87(4), 731–747] to improve on these fixed selection criteria. In this paper, we study the potential of alternative fully Bayes methods, which instead margin out the hyperparameters with respect to prior distributions. Several structured prior formulations are considered for which fully Bayes selection and estimation methods are obtained. Analytical and simulation comparisons with empirical Bayes counterparts are studied.     

A note on the queue with a random set-up time under a restricted admissibility policy with a Bernoulli vacation schedule

This paper deals with the queue size distribution of an M^x/G/1 queue with a random set-up time and with a Bernoulli vacation schedule under a restricted admissibility policy. This generalizes the model studied by Madan and Choudhury [Sankhyá 66 (2004) 175–193].     

The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

Sensitivity analysis of longitudinal data with intermittent missing values

Several models for longitudinal data with nonrandom missingness are available. The selection model of Diggle and Kenward is one of these models. It has been mentioned by many authors that this model depends on untested modelling assumptions, such as the response distribution, from the observed data. So, a sensitivity analysis of the study’s conclusions for such assumptions is needed. The stochastic EM algorithm is proposed and developed to handle continuous longitudinal data with nonrandom intermittent missing values when the responses have non-normal distribution. This is a step in investigating the sensitivity of the parameter estimates to the change of the response distribution. The proposed technique is applied to real data from the International Breast Cancer Study Group.     

Lower bounds of the wrap-around -discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L₂-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L₂-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L₂-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

The two-sample -test with a known ratio of variances

We consider the two-sample t-test where error variances are unknown but with known relationships between them. This situation arises, for example, when two measuring instruments average different number of replicates to report the response. In particular we compare our procedure with the usual Satterthwaite approximation in the two sample t-test with variances unequal. Our procedure uses the knowledge of a known ratio of variances while the Satterthwaite approximation assumes only that the two variances are unequal. Simulations show that our procedure has both better size and better power than the Satterthwaite approximation. Finally, we consider an extension of our results to the General Linear Model.     

Nonparametric control charts based on runs and Wilcoxon-type rank-sum statistics

In this article, we introduce three new distribution-free Shewhart-type control charts that exploit run and Wilcoxon-type rank-sum statistics to detect possible shifts of a monitored process. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of these charts is that, due to their nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the charts for some typical FAR values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations.     

Multivariate analysis of variance with additional data from an unknown subset of the populations

In this paper we consider the problem of testing the means of k multivariate normal populations with additional data from an unknown subset of the k populations. The purpose of this research is to offer test procedures utilizing all the available data for the multivariate analysis of variance problem because the additional data may contain valuable information about the parameters of the k populations. The standard procedure uses only the data from identified populations. We provide a test using all available data based upon Hotelling' s generalized T²statistic. The power of this test is computed using Betz's approximation of Hotelling' s generalized T²statistic by an F-distribution. A comparison of the power of the test and the standard test procedure is also given.     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.