Rank covariance matrix estimation of a partially known covariance matrix

Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the affine equivariant rank covariance matrix (RCM) that has been studied in Visuri et al. [2003. Affine equivariant multivariate rank methods. J. Statist. Plann. Inference 114, 161–185]. In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is affine equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and hence reduce the problem of estimating the original RCM to estimating marginal rank covariance matrices. This is a great computational advantage when the dimension of the original data vector is large.     

Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

Partial differential equations for hypergeometric functions 3F2 of matrix argument

Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions _pF_q of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial differential equations for the functions of ₂F₁ of one argument matrix and two argument matrices, respectively. Such differential equations have been used to obtain asymptotic expansions of the functions (Muirhead [7], [8], [9], Sugiura [10]). The purpose of this paper is to derive partial differential equations for the functions ₃F₂ (a₁ a₂, a₃; b₁, b₂, R) and ₃F₂ (a₁, a₂, a₃; b₁, b₂; R, S). Differential equations for ₂F₂ are also obtained.     

Influential observations in principal component analysis:a case study

A number of results have been derived recently concerning the influence of individual observations in a principal component analysis. Some of these results, particularly those based on the correlation matrix, are applied to data consisting of seven anatomical measurements on students. The data have a correlation structure which is fairly typical of many found in allometry. This case study shows that theoretical influence functions often provide good estimates of the actual changes observed when individual observations are deleted from a principal component analysis. Different observations may be influential for different aspects of the principal component analysis (coefficients, variances and scores of principal components); these differences, and the distinction between outlying and influential observations are discussed in the context of the case study. A number of other complications, such as switching and rotation of principal components when an observation is deleted, are also illustrated.     

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

Covariance matrices, or in general matrices of sums of squares and cross-products, are used as input in many multivariate analyses techniques. The eigenvalues of these matrices play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of a covariance matrix. The relationship between the eigenvalues of the covariance matrix computed from all data and the eigenvalues of the perturbed covariance matrix (a covariance matrix computed after a small subset of the observations has been deleted) cannot in general be written in closed-form. Two methods for approximating the eigenvalues of a perturbed covariance matrix have been suggested by Hadi (1988) and Wang and Nyquist (1991) for the case of a perturbation by a single observation. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem. We also compare the two improved approximations in terms of their accuracies.     

Linear model estimation errors with estimated design parameters

The problem of error estimation of parameters b in a linear model,Y = Xb+ e, is considered when the elements of the design matrix X are functions of an unknown ‘design’ parameter vector c. An estimated value c is substituted in X to obtain a derived design matrix [Xtilde]. Even though the usual linear model conditions are not satisfied with [Xtilde], there are situations in physical applications where the least squares solution to the parameters is used without concern for the magnitude of the resulting error. Such a solution can suffer from serious errors.

This paper examines bias and covariance errors of such estimators. Using a first-order Taylor series expansion, we derive approximations to the bias and covariance matrix of the estimated parameters. The bias approximation is a sum of two terms:One is due to the dependence between ? and Y; the other is due to the estimation errors of ? and is proportional to b, the parameter being estimated. The covariance matrix approximation, on the other hand, is composed of three omponents:One component is due to the dependence between ? and Y; the second is the covariance matrix ∑_b corresponding to the minimum variance unbiased b, as if the design parameters were known without error; and the third is an additional component due to the errors in the design parameters. It is shown that the third error component is directly proportional to bb'. Thus, estimation of large parameters with wrong design matrix [Xtilde] will have larger errors of estimation. The results are illustrated with a simple linear example.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Modified linear discriminant analysis using block covariance matrix in high-dimensional data

Among many classification methods, linear discriminant analysis (LDA) is a favored tool due to its simplicity, robustness, and predictive accuracy but when the number of genes is larger than the number of observations, it cannot be applied directly because the within-class covariance matrix is singular. Also, diagonal LDA (DLDA) is a simpler model compared to LDA and has better performance in some cases. However, in reality, DLDA requires a strong assumption based on mutual independence. In this article, we propose the modified LDA (MLDA). MLDA is based on independence, but uses the information that has an effect on classification performance with the dependence structure. We suggest two approaches. One is the case of using gene rank. The other involves no use of gene rank. We found that MLDA has better performance than LDA, DLDA, or K-nearest neighborhood and is comparable with support vector machines in real data analysis and the simulation study.     

Spacing of information in polynomial regression:a simple solution

Consider a k polynomial regression on a single real variable. If n uncorrelated observations are to be taken in a design with support on more than k+1 points, there is an approximate experiment, ν, with support on k+1 points and n observations such that both designs have the same information matrix for the model. A proof of this result is provided. A method to obtain the approximate design ν is given and illustrated by an example. The source of disagreement between Kiefer (1959) and De La Garza (1954) in the solution of this problem is clarified.     

Independent attributes control chart for Markov-dependent processes

Statistical process monitoring (SPM) is a very efficient tool to maintain and to improve the quality of a product. In many industrial processes, end product has two or more attribute-type quality characteristics. Some of them are independent, but the observations are Markovian dependent. It is essential to develop a control chart for such situations. In this article, we develop an Independent Attributes Control Chart for Markov Dependent Processes based on error probabilities criterion under the assumption of one-step Markov dependency. Implementation of the chart is similar to that of Shewhart-type chart. Performance of the chart has been studied using probability of detecting shift criterion. A procedure to identify the attribute(s) responsible for out-of-control status of the process is given.     

Chi-Square quantile-based multivariate variance monitoring for individual observations

Under the normality assumption, some statistics for monitoring a multivariate process variance for individual observations can be used to detect a variance shift, but the distribution of their in-control run length has a high variance as well as the median that is extremely smaller than the mean, which leads to many false alarms in the in-control process. In this paper, we propose a chi-square quantile-based monitoring statistic which is free of the problems. The numerical experiments show that the proposed monitoring statistics outperform the existing monitoring statistics in terms of the detection of a shift for the variance.     

Simple pilot procedures for the avoidance of disconnected experimental designs

Summary.  Two simple pilot procedures are proposed for avoiding the problem of dealing with a disconnected experimental design. Both procedures should be carried out on the selected design before any experimentation is considered. The first procedure is a check that the suggested design is connected with respect to treatments. This makes use of the information matrix for the model and provides feed-back on a disconnected design. The second procedure specifies which observations are influential in causing a connected design to become disconnected, with respect to any set of parameter effects, if these observations are lost during the experimental period. This specification is found by examining the projection matrix for the model. These pilot procedures are illustrated by several examples.     

Evidential inference for diffusion-type processes

This article analyses diffusion-type processes from a new point-of-view. Consider two statistical hypotheses on a diffusion process. We do not use a classical test to reject or accept one hypothesis using the Neyman–Pearson procedure and do not involve Bayesian approach. As an alternative, we propose using a likelihood paradigm to characterizing the statistical evidence in support of these hypotheses. The method is based on evidential inference introduced and described by Royall [Royall R. Statistical evidence: a likelihood paradigm. London: Chapman and Hall; 1997]. In this paper, we extend the theory of Royall to the case when data are observations from a diffusion-type process instead of iid observations. The empirical distribution of likelihood ratio is used to formulate the probability of strong, misleading and weak evidences. Since the strength of evidence can be affected by the sampling characteristics, we present a simulation study that demonstrates these effects. Also we try to control misleading evidence and reduce them by adjusting these characteristics. As an illustration, we apply the method to the Microsoft stock prices.     

Exact intervals and tests for mean of symmetrical population when one "sample" value possibly an outlier

The (continuous) data are n observations that are believed to be a random sample from a symmetrical population. Confidence intervals and significance tests for the population mean are desired. There is, however, the possibility that either the smallest observation or the largest observation is an outlier. That is, the population providing this observation differs from the symmetrical population providing the other n - 1 observations. If this occurs, intervals and tests are desired for the mean of the population providing the other n - 1 observations. Some investigation difficulties can be overcome if intervals and tests can be developed that are simultaneously usable for all of these three situations (a confidence coefficient, or significance level, has the same value for all three situations). Two kinds of intervals and tests with this property are developed. These results always involve both the next to smallest observations and should have at least moderately high efficiencies. Also, some extensions are considered, such as allowing each observation to be from a different population.     

The exponential–Weibull lifetime distribution

In this paper, we propose a new three-parameter model called the exponential–Weibull distribution, which includes as special models some widely known lifetime distributions. Some mathematical properties of the proposed distribution are investigated. We derive four explicit expressions for the generalized ordinary moments and a general formula for the incomplete moments based on infinite sums of Meijer's G functions. We also obtain explicit expressions for the generating function and mean deviations. We estimate the model parameters by maximum likelihood and determine the observed information matrix. Some simulations are run to assess the performance of the maximum likelihood estimators. The flexibility of the new distribution is illustrated by means of an application to real data.     

Analysis of Variance by Randomization when Variances are Unequal

If there are significant factor and interaction effects with analysis of variance using ran-domization inference, they can be detected by tests that compare the F -statistics for the real data with the distributions of these statistics obtained by randomly allocating either the original observations or the residuals to the various factor combinations. Such tests involve the assumption that the effect of factors or interactions is to shift the observations for a factor combination by a fixed amount, without changing the amount of variation at that combination. In reality the expected amount of variation at each factor combination, as measured by the variance, may not be constant, which may upset the properties of the tests for the effects of factors and interactions. This paper discusses several possible methods for adjusting the randomization procedure to allow for this type of problem, including generalizations of methods that have been proposed for comparing the means of several samples when there is unequal variance but no factor structure. A simulation study shows that the best of the methods examined is one for which the randomized sets of data are designed to approximate the distributions of F -statistics when unequal variance is present.     

On Bayes' theorem for fuzzy data

There are some ideas concerning a generalization of Bayes' theorem to the situation of fuzzy data. Some of them are given in the references [1], [5], and [7]. But the proposed methods are not generalizations in the sense of the probability content of Bayes' theorem for precise data. In the present paper a generalization of Bayes' theorem to the case of fuzzy data is described which contains Bayes' theorem for precise data as a special case and allows to use the information in fuzzy data in a coherent way. Moreover a generalization of the concept of HPD-regions is explained which makes it possible to model and analyze the situation of fuzzy data. Also a generalization of the concept of predictive distributions is given in order to calculate predictive densities based on fuzzy sample information.     

An optimal control chart for finite matrix sequences at some unknown change point

We present a new measure for evaluating the performance of control charts to detect abrupt changes of finite matrix sequences. The objective is to minimize the probability that the control chart fails to raise the alarm at unknown change point time for a given in-control average run length. We construct and prove the optimal control chart with dynamic control limits in different pre- and post-change distributions. We validate the optimality of the proposed chart by conducting exhaustive experiments on both simulation study and real-world data.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.