Joint (k,l)-hurdle random effects models for mixed longitudinal power series and normal outcomes

A random effects model for analyzing mixed longitudinal normal and count outcomes with and without the possibility of non ignorable missing outcomes is presented. The count response is inflated in two points (k and l) and the (k, l)-Hurdle power series is used as its distribution. The new distribution contains, as special submodels, several important distributions which are discussed, such as (k, l)-Hurdle Poisson and (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions among others. Random effects are used to take into account the correlation between longitudinal outcomes and inflation parameters. A full likelihood-based approach is used to yield maximum likelihood estimates of the model parameters. A simulation study is performed in which for count outcome (k, l)-Hurdle Poisson, (k, l)-Hurdle negative binomial and (k, l)-Hurdle binomial distributions are considered. To illustrate the application of such modelling the longitudinal data of body mass index and the number of joint damage are analyzed.     

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.     

Identifying Variables Contributing to Outliers in Phase I

When a process is monitored with a T ² control chart in a Phase II setting, the MYT decomposition is a valuable diagnostic tool for interpreting signals in terms of the process variables. The decomposition splits a signaling T ² statistic into independent components that can be associated with either individual variables or groups of variables. Since these components are T ² statistics with known distributions, they can be used to determine which of the process variable(s) contribute to the signal. However, this procedure cannot be applied directly to Phase I since the distributions of the individual components are unknown. In this article, we develop the MYT decomposition procedure for a Phase I operation, when monitoring a random sample of individual observations and identifying outliers. We use a relationship between the T ² statistic in Phase I with the corresponding T ² statistic resulting when an observation is omitted from this sample to derive the distributions of these components and demonstrate the Phase I application of the MYT decomposition.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

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.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

Monitoring Variation in a Multivariate Process When the Dimension is Large Relative to the Sample Size

A control procedure is presented for monitoring changes in variation for a multivariate normal process in a Phase II operation where the subgroup size, m, is less than p, the number of variates. The methodology is based on a form of Wilk' statistic, which can be expressed as a function of the ratio of the determinants of two separate estimates of the covariance matrix. One estimate is based on the historical data set from Phase I and the other is based on an augmented data set including new data obtained in Phase II. The proposed statistic is shown to be distributed as the product of independent beta distributions that can be approximated using either a chi-square or F-distribution. An ARL study of the statistic is presented for a range of conditions for the population covariance matrix. Cases are considered where a p-variate process is being monitored using a sample of m observations per subgroup and m < p. Data from an industrial multivariate process is used to illustrate the proposed technique.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

MULTIVARIATE GENERALIZED POLYA DISTRIBUTION OF ORDER k

ABSTRACT

An order k (or cluster) generalized Polya distribution and a multivariate generalized Polya-Eggenberger one where derived in (Sen, K.; Jain, R. Cluster Generalized Negative Binomial Distribution. In Probability Models and Statistics, A. J. Medhi Festschrift on the Occasion of his 70th Birthday; Borthakur, A.C. et al., Eds.; New Age International Publishers: New Delhi, 1996; 227–241 and Sen, K.; Jain, R. A Multivariate Generalized Polya-Eggenberger Probability Model-First Passage Approach. Communications in Statistics—Theory and Methods 1997, 26, 871–884). Presently, both distributions are generalized to a multivariate generalized Polya distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases new multivariate Polya and inverse Polya distributions of order k and the multivariate generalized negative binomial distribution of the same order derived recently in (Tripsiannis, G.A.; Philippou, A.N.; Papathanasiou, A.A. Multivariate Generalized Distributions of Order k. Medical Statistics Technical Report #41: Democritus University of Thrace, Greece, 2001). Limiting cases are considered and applications are indicated.     

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT

We present an alternative sampling scheme for the Hotelling's T² control chart with variable parameters (VP T²) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Estimation and testing of P(Y > X) in two-parameter exponential distributions

In this article, we obtain the UMVUE of the reliability function ξ=P(Y>X) and the UMVUE of ξ ^k =[P(Y>X)] ^k in the two-parameter exponential distributions with known scale parameters. We also derive the distribution of the UMVUE of ξ and further considering the tests of hypotheses regarding the reliability function ξ.     

Estimation of system reliability

In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common stress which is independent of the strengths of these k components. If (X ₁,X ₂,…,X _k ) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability is given byR=P[Y<X _(k−s+1)] whereX _(k−s+1) is (k−s+1)-th order statistic of (X ₁,…,X _k ). We estimate R when (X ₁,…,X _k ) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution of the proposed estimator.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.