A unified approach to permutation testing for equivalence

Statistical Methods & Applications - The notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control....     

Pairwise testing of group mean vectors in MANOVA with small samples

Several, multivariate, pairwise, multiple comparison procedures are proposed as follow-ups for a significant multivariate analysis of variance. The Peritz procedure is generalized from univariate to several multivariate applications. Procedures are evaluated using overall power, any-pair power and all-pairs power applied to mean vectors with common sample sizes of 4, 5, and 9. Monte Carlo simulation demonstrated greater power than previously proposed univariate procedures in many conditions especially for all-pairs power. The multivariate Peritz procedure based on the Lawley–Hotelling trace was found to be most powerful in many conditions.     

Multiple comparisons of mean vectors with large dimension under general conditions

ABSTRACT

Multiple comparisons for two or more mean vectors are considered when the dimension of the vectors may exceed the sample size, the design may be unbalanced, populations need not be normal, and the true covariance matrices may be unequal. Pairwise comparisons, including comparisons with a control, and their linear combinations are considered. Under fairly general conditions, the asymptotic multivariate distribution of the vector of test statistics is derived whose quantiles can be used in multiple testing. Simulations are used to show the accuracy of the tests. Real data applications are also demonstrated.     

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T² _R, based on Tiku's (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T² _R is one the whole considerably more powerful than the prominenet Hotelling T² statistics. We also develop a robust statistic T² _D for testing that two multivariate distributions (skew or symmetric) are identical; T² _D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σ_k)f((x-μ_k)/σ_k) and the means and variances of these marginal distributions exist.     

A unified approach of testing for discrete and continuous Pareto laws

New tests are proposed for the Pareto distribution as well as its discrete version, the so called Zipf’s law. In both cases the discrepancy between the empirical moment of arbitrary negative order and its theoretical counterpart is utilized in a weighted integral test statistic. If the weight function is of exponential rate of decay interesting limit statistics are obtained. The tests are shown to be consistent under fixed alternatives and a Monte Carlo study is drawn to investigate the performance of the proposed procedures in small samples. Furthermore a bootstrap procedure is proposed to cope with the case of unknown shape parameter. We conclude with applications to real data.     

A unified empirical likelihood approach for testing MCAR and subsequent estimation

For an estimation with missing data, a crucial step is to determine if the data are missing completely at random (MCAR), in which case a complete‐case analysis would suffice. Most existing tests for MCAR do not provide a method for a subsequent estimation once the MCAR is rejected. In the setting of estimating means, we propose a unified approach for testing MCAR and the subsequent estimation. Upon rejecting MCAR, the same set of weights used for testing can then be used for estimation. The resulting estimators are consistent if the missingness of each response variable depends only on a set of fully observed auxiliary variables and the true outcome regression model is among the user‐specified functions for deriving the weights. The proposed method is based on the calibration idea from survey sampling literature and the empirical likelihood theory.     

A robust procedure for testing the equality of mean vectors of two bivariate populations with unequal covariance matrices

A robust test is developed for testing equality of the mean vectors of two bivariate (multivariate) populations when the variance-covariance matrices are not necessarily equal. The test is an extension of the univariate robust test given by Tiku and Singh (1981).     

A unified approach to multilevel sample selection models

Abstract

We propose a unified approach for multilevel sample selection models using a generalized result on skew distributions arising from selection. If the underlying distributional assumption is normal, then the resulting density for the outcome is the continuous component of the sample selection density and has links with the closed skew-normal distribution (CSN). The CSN distribution provides a framework which simplifies the derivation of the conditional expectation of the observed data. This generalizes the Heckman’s two-step method to a multilevel sample selection model. Finite-sample performance of the maximum likelihood estimator of this model is studied through a Monte Carlo simulation.     

A testing approach to estimation

Results from the theory of uniformly most powerful invariant tests are used to develop a new parameter estimation procedure. The procedure is used to derive parameter estimators for several important distributions. Results of simulation studies comparing the performances of the new estimators and maximum likelihood estimators are presented.     

A unified approach to binomial-type distributions of order k

In this paper, we consider the distribution of the number of "1"-runs of length k in a sequence of {0,1}-valued random variables of length n by using a new (unified) counting scheme called l-overlapping counting. Here, k and n are positive integers with k ≦ and l is an integer less than k. We obtain the prohabi!ity generating function of the distribution of the number of eoverlapping "in-runs of iength k in the sequence, even when the underiying sequence is a dependent sequence such as a highcr order Markov chaic.     

A unified approach to regression analysis under double-sampling designs

We propose a unified approach to the estimation of regression parameters under double-sampling designs, in which a primary sample consisting of data on the rough or proxy measures for the response and/or explanatory variables as well as a validation subsample consisting of data on the exact measurements are available. We assume that the validation sample is a simple random subsample from the primary sample. Our proposal utilizes a specific parametric model to extract the partial information contained in the primary sample. The resulting estimator is consistent even if such a model is misspecified, and it achieves higher asymptotic efficiency than the estimator based only on the validation data. Specific cases are discussed to illustrate the application of the estimator proposed.     

A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.     

A unified approach to characterization problems using conditional expectations

In recent years characterization problems have become of increasing interest. It is well known that mean residual life e(x) = E(X - x|X ⩾x) and right-censored mean function m_R(x) = E(X | X ⩽ x), uniquely determine the distribution function F(x) = P(X ⩽ x). In this paper, we study characterizations problems for general distributions, using the doubly censored mean function m(x, y) = E(X | x⩽X⩽y). We show that m(x, y) characterizes F(x), obtaining the explicit expression of F(x) from m(x, y). Moreover, we give properties that any function must verifies to be a doubly censored mean function and we obtain stability theorems for these characterizations.     

A mean deviation-based approach to monitor process variability

The study proposes a Shewhart-type control chart, namely an MD chart, based on average absolute deviations taken from the median, for monitoring changes (especially moderate and large changes – a major concern of Shewhart control charts) in process dispersion assuming normality of the quality characteristic to be monitored. The design structure of the proposed MD chart is developed and its comparison is made with those of two well-known dispersion control charts, namely the R and S charts. Using power curves as a performance measure, it has been observed that the design structure of the proposed MD chart is more powerful than that of the R chart and is very close competitor to that of the S chart, in terms of discriminatory power for detecting shifts in the process dispersion. The non-normality effect is also examined on design structures of the three charts, and it has been observed that the design structure of the proposed MD chart is least affected by departure from normality.     

A unified approach to stochastic comparisons of multivariate mixture models

In this paper, we consider a unified approach to stochastic comparisons of random vectors corresponding to two general multivariate mixture models. These stochastic comparisons are made with respect to multivariate hazard rate, reversed hazard rate and likelihood ratio orders. As an application, results are presented for stochastic comparisons of generalized multivariate frailty models.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

Bootstrap approach to test the homogeneity of order restricted mean vectors when the covariance matrices are unknown

Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are unknown, arbitrary positive definite and unequal are considered. This problem of testing has been studied to some extent, for example, by Kulatunga and Sasabuchi (1984 Kulatunga, D. D. S., Sasabuchi, S. (1984). A test of homogeneity of mean vectors against multivariate isotonic alternatives. Mem Fac Sci, Kyushu Univ Ser A Mathemat 38:151–161. [Google Scholar]) when the covariance matrices are known and also Sasabuchi et al. (2003 Sasabuchi, S., Tanaka, K., Tsukamodo, T. (2003). Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown. Annals of Statistics. 31(5):1517–1536.[Web of Science ®] , [Google Scholar]) and Sasabuchi (2007 Sasabuchi, S. (2007). More powerful tests for homogeneity of multivariate normal mean vectors under an order restriction. Sankhya 69(4):700–716. [Google Scholar]) when the covariance matrices are unknown but common. In this paper, a test statistic is proposed and because of the main advantage of the bootstrap test is that it avoids the derivation of the complex null distribution analytically, a bootstrap test statistic is derived and since the proposed test statistic is location invariance the bootstrap p-value defined logical and some steps are presented to estimate it. Our numerical studies via Monte Carlo simulation show that the proposed bootstrap test can correctly control the type I error rates. The power of the test for some of the p-dimensional normal distributions is computed by Monte Carlo simulation. Also, the null distribution of test statistic is estimated using kernel density. Finally, the bootstrap test is illustrated using a real data.     

Cure rate models: A unified approach

The authors propose a novel class of cure rate models for right‐censored failure time data. The class is formulated through a transformation on the unknown population survival function. It includes the mixture cure model and the promotion time cure model as two special cases. The authors propose a general form of the covariate structure which automatically satisfies an inherent parameter constraint and includes the corresponding binomial and exponential covariate structures in the two main formulations of cure models. The proposed class provides a natural link between the mixture and the promotion time cure models, and it offers a wide variety of new modelling structures as well. Within the Bayesian paradigm, a Markov chain Monte Carlo computational scheme is implemented for sampling from the full conditional distributions of the parameters. Model selection is based on the conditional predictive ordinate criterion. The use of the new class of models is illustrated with a set of real data involving a melanoma clinical trial.