Phase-I monitoring of general linear profiles in multistage processes

In this article, the general linear profile-monitoring problem in multistage processes is addressed. An approach based on the U statistic is first proposed to remove the effect of the cascade property in multistage processes. Then, the T² chart and a likelihood ratio test (LRT)-based scheme on the adjusted parameters are constructed for Phase-I monitoring of the parameters of general linear profiles in each stage. Using simulation experiments, the performance of the proposed methods is evaluated and compared in terms of the signal probability for both weak and strong autocorrelations, for processes with two and three stages, as well as for two sample sizes. According to the results, the effect of the cascade property is effectively removed and hence each stage can be monitored independently. In addition, the result shows that the LRT approach provides significantly better results than the T² method and outperforms it under different shift and autocorrelation scenarios. Moreover, the proposed methods perform better when larger sample sizes are used in the process. Two illustrative examples, including a real case and a simulated example, are used to show the applicability of the proposed methods.     

Goodness-of-fit test for uniform stochastic ordering among several distributions

The likelihood-ratio test (LRT) is considered as a goodness-of-fit test for the null hypothesis that several distribution functions are uniformly stochastically ordered. Under the null hypothesis, H₁ : F₁ ? F₂ ?···? F_N, the asymptotic distribution of the LRT statistic is a convolution of several chi-bar-square distributions each of which depends upon the location parameter. The least-favourable parameter configuration for the LRT is not unique. It can be two different types and depends on the number of distributions, the number of intervals and the significance level α. This testing method is illustrated with a data set of survival times of five groups of male fruit flies.     

A Two Sample Test for Mean Vectors with Unequal Covariance Matrices

In this paper, we consider testing the equality of two mean vectors with unequal covariance matrices. In the case of equal covariance matrices, we can use Hotelling’s T² statistic, which follows the F distribution under the null hypothesis. Meanwhile, in the case of unequal covariance matrices, the T² type test statistic does not follow the F distribution, and it is also difficult to derive the exact distribution. In this paper, we propose an approximate solution to the problem by adjusting the degrees of freedom of the F distribution. Asymptotic expansions up to the term of order N^{? 2} for the first and second moments of the U statistic are given, where N is the total sample size minus two. A new approximate degrees of freedom and its bias correction are obtained. Finally, numerical comparison is presented by a Monte Carlo simulation.     

On stochastic processes for quantitative trait locus mapping under selective genotyping

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval [0, T] representing a chromosome. The originality of this study is that we are under selective genotyping: only the individuals with extreme phenotypes are genotyped. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on [0, T] and under local alternatives with a QTL at t^☆ on [0, T]. We show that the LRT process is asymptotically the square of a ‘non-linear interpolated and normalized Gaussian process’. We have an easy formula in order to compute the supremum of the square of this interpolated process. We prove that we have to genotype symmetrically and that the threshold is exactly the same as in the situation where all the individuals are genotyped.     

The effect of parameter estimation on phase II monitoring of poisson regression profiles

ABSTRACT

The effect of parameters estimation on profile monitoring methods has only been studied by a few researchers and only the assumption of a normal response variable has been tackled. However, in some practical situation, the normality assumption is violated and the response variable follows a discrete distribution such as Poisson. In this paper, we evaluate the effect of parameters estimation on the Phase II monitoring of Poisson regression profiles by considering two control charts, namely the Hotelling’s T² and the multivariate exponentially weighted moving average (MEWMA) charts. Simulation studies in terms of the average run length (ARL) and the standard deviation of the run length (SDRL) are carried out to assess the effect of estimated parameters on the performance of Phase II monitoring approaches. The results reveal that both in-control and out-of-control performances of these charts are adversely affected when the regression parameters are estimated.     

The geometrical interpretation of statistical tests in multivariate linear regression

A geometrical interpretation of the classical tests of the relation between two sets of variables is presented. One of the variable sets may be considered as fixed and then we have a multivariate regression model. When the Wilks’ lambda distribution is viewed geometrically it is obvious that the two special cases, theF distribution and the HotellingT ² distribution are equivalent. From the geometrical perspective it is also obvious that the test statistic and thep-value are unchanged if the responses and the predictors are interchanged.     

The Design and Performance of the Multivariate Synthetic-T 2 Control Chart

One of the objectives of research in statistical process control is to obtain control charts that show few false alarms but, at the same time, are able to detect quickly the shifts in the distribution of the quality variables employed to monitor a productive process. In this article, the synthetic-T ² control chart is developed, which consists of the simultaneous use of a CRL chart and a Hotelling's T ² control chart. The ARL is calculated employing Markov chains for steady and zero-state scenarios. A procedure of optimization has been developed to obtain the optimum parameters of the synthetic-T ², for zero and steady cases, given the values of in-control ARL and magnitude of shift which needs to be detected rapidly. A comparison between (standard T ², MEWMA, T ² with variable sample size, and T ² with double sampling) charts reveals that the synthetic-T ² chart always performs better than the standard T ² chart. The comparison with the remaining charts demonstrate in which cases the performance of this new chart makes it interesting to employ in real applications.     

Testing the difference between two independent regression models

ABSTRACT

In some situations, for example, in biology or psychology studies, we wish to determine whether the linear relationship between response variable and predictor variables differs in two populations. The analysis of the covariance (ANCOVA) or, equivalently, the partial F-test approaches are the commonly used methods. In this study, the asymptotic distribution for the difference between two independent regression coefficients was established. The proposed method was used to derive the asymptotic confidence set for the difference between coefficients and hypothesis testing for the equality of the two regression models. Then a simulation study was conducted to compare the proposed method with the partial F method. The performance of the new method was comparable with that of the partial F method.     

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.     

Phase I monitoring of social network with baseline periods using poisson regression

Social network analysis is an important analytic tool to forecast social trends by modeling and monitoring the interactions between network members. This paper proposes an extension of a statistical process control method to monitor social networks by determining the baseline periods when the reference network set is collected. We consider probability density profile (PDP) to identify baseline periods using Poisson regression to model the communications between members. Also, Hotelling T² and likelihood ratio test (LRT) statistics are developed to monitor the network in Phase I. The results based on signal probability indicate a satisfactory performance for the proposed method.     

Partial residuals in cumulative regression models for ordinal data

We are concerned with cumulative regression models for an ordered categorical response variable Y. We propose two methods to build partial residuals from regression on a subset Z₁ of covariates Z., which take into regard the ordinal character of the response. The first method makes use of a multivariate GLM-representation of the model and produces residual measures for diagnostic purposes. The second uses a latent continuous variable model and yields new (adjusted) ordinal data Y^*. Both methods are illustrated by a data set from forestry.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.     

Inference with Bivariate Truncated Data

Inthis paper we build on previous work for estimation of the bivariatedistribution of the time variables T ₁ and T ₂when they are observable only on the condition that one of thetime variables, say T ₁, is greater than (left-truncation)or less than (right truncation) some observed time variable C ₁.In this paper, we introduce several results based on the InfluenceCurve (which we derive in this paper) of the NPMLE of the distributionF of (T ₁,T ₂) developed by van derLaan (van der Laan, 1996). Specifically we will: prove that theNPMLE is asymptotically equivalent to an estimator developedby Gürler (Gürler, 1997), derive the asymptotic distributionof the NPMLE based on its Influence Curve, present tests to determinethe amount of dependence between T ₁ and T ₂,present the results of simulation studies that compare the NPMLEand Gürler's estimator and evaluate the performance of boththe above mentioned tests and confidence intervals of Fbased on the asymptotic distribution of the NPMLE, and finallywe will apply the methods in a data analysis in which we alsopoint out practical issues that arise in the implementation ofthe estimator.     

Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator

It is demonstrated that integrals of the noncentral chi-square, noncentral F and noncentral T distributions can be evaluated on desk calculators. The same procedure can be used to compute probabilities for the distribution of the difference of two T-variables with equal degrees of freedom. The proposed method of computation can be used with any computer which yields probabilities for the chi-square and F distributions.     

Hotelling’s T2 control chart with double warning lines

Recent studies have shown that the T ² control chart with variable sampling intervals (VSI) and/or variable sample sizes (VSS) detects process shifts faster than the traditional T ² chart. This article extends these studies for processes that are monitored with VSI and VSS using double warning lines (T ² —DWL). It is assumed that the length of time the process remains in control has exponential distribution. The properties of T ² —DWL chart are obtained using Markov chains. The results show that the T ² —DWL chart is quicker than VSI and/or VSS charts in detecting almost all shifts in the process mean.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

THE CONVERGENCE OF AUTOCORRELATIONS AND AUTOREGRESSIONS1

For a general class of scalar stationary processes, essentially those for which the best linear predictor is the best predictor (in the mean square sense), it is shown that, under fairly minor additional conditions, the sample autocorrelations converge to the true values almost surely and hniformly in the lag, t, at a rate (T^-1log T)^1/2, where T is the sample size. For ARMA processes, if |t|(log T)^a, a < ∞, the rate is the best possible, namely (T^-1log log T)^1/2. In particular the somewhat implausible condition, on the innovations, that E{ε(t)²| F_t-l} is constant is avoided in these results. The theorems are used to discuss autoregressive approximation. When the stationary process is a vector process the condition on the innovation sequence, ε(t), that E{ε(t)ε(t)| F_t-l} be constant, cannot be entirely avoided in relation to autoregressive approximation. This is also discussed.     

Asymptotic calculations of functions of expected values and covariances of order statistics

Suppose m and V are respectively the vector of expected values and the covariance matrix of the order statistics of a sample of size n from a continuous distribution F. A method is presented to calculate asymptotic values of functions of m and V ^–1, for distributions F which are sufficiently regular. Values are given for the normal, logistic, and extreme-value distributions; also, for completeness, for the uniform and exponential distributions, although for these other methods must be used.