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Mixture models are useful in describing a wide variety of random phenomena because of their flexibility in modeling. They have continued to receive increasing attention over the years from both a practical and theoretical point of view. In their applications, estimating the number of mixture components is often the main research objective or the first step toward it. Estimation of the number of mixture components heavily depends on the underlying distribution. As an extension of normal mixture models, we introduce a skew-normal mixture model and adapt the reversible jump Markov chain Monte Carlo algorithm to estimate the number of components with some applications to biological data. The reversible jump algorithm is also applied to the Cox proportional hazard model with frailty. We consider a regression model for the variance components in the proportional hazards frailty model. We propose a Bayesian model averaging procedure with a reversible jump Markov chain Monte Carlo step which selects the model automatically. The resulting regression coefficient estimates ignore the model uncertainty from the frailty distribution. Finally, the proposed model and the estimation procedure are illustrated with simulated example and real data.

By far the most popular method to account for dependencies in the transportation network analysis literature is the use of the multivariate normal (MVN) distribution. While in certain cases there is some theoretical underpinning for the MVN assumption, in others there is none. This can lead to misleading results: results do not only depend on whether dependence is modeled, but also how dependence is modeled. When assuming the MVN distribution, one is limiting oneself to a specific set of dependency structures, which can substantially limit validity of results. In this report an existing, more flexible, correlation-based approach (where just marginal distributions and their correlations are specified) is proposed, and it is demonstrated that, in simulation studies, such an approach is a generalization of the MVN assumption. The need for such generalization is particularly critical in the transportation network modeling literature, where oftentimes there exists no or insufficient data to estimate probability distributions, so that sensitivity analyses assuming different dependence structures could be extremely valuable. However, the proposed method has its own drawbacks. For example, it is again not able to exhaust all possible dependence forms and it relies on some not-so-known properties of the correlation coefficient. / text

The problem under consideration is simultaneous monitoring of the means of two or more correlated variables of a process, by collecting a small fixed random sample at fixed time intervals. The target values are considered known, whereas the variance covariance matrix of the data must be estimated. A typical parametric chart to monitor this process would involve the assumption that the data follow a multivariate normal distribution. If this assumption is not reasonable or if it is difficult to verify, for example in a short production run, a multivariate control chart based on classical nonparametric statistics could be used. Control charts based on the sign and signed rank statistics are explored. Past sample information for each variable is retained through an exponentially weighted moving average statistic (EWMA) in order to increase the sensitivity of the charts to detect small shifts from the target. The properties of the charts are evaluated using simulation. Such charts are not distribution-free in the nonparametric sense, but they are more robust than the parametric equivalent chart because, among other reasons, they require only covariance estimates. Nonparametric charts are less efficient than the parametric equivalent chart if the measurements follow a normal distribution, but they improve significantly if the measurements follow a distribution with heavier tails. / Ph. D.

Suppose two or more variables are jointly normally distributed. If there is a common relationship between these variables it would be very important to quantify this relationship by a parameter called the correlation coefficient which measures its strength, and the use of it can develop an equation for predicting, and ultimately draw testable conclusion about the parent population. This research focused on the correlation coefficient ρ for the bivariate and trivariate normal distribution when equal variances and equal covariances are considered. Particularly, we derived the maximum Likelihood Estimators (MLE) of the distribution parameters assuming all of them are unknown, and we studied the properties and asymptotic distribution of . Showing this asymptotic normality, we were able to construct confidence intervals of the correlation coefficient ρ and test hypothesis about ρ. With a series of simulations, the performance of our new estimators were studied and were compared with those estimators that already exist in the literature. The results indicated that the MLE has a better or similar performance than the others.

In this paper, we prove that the joint distribution of random vectors Z 1 and Z 2 and the distribution of Z 2 are skew normal provided that Z 1 is skew normally distributed and Z 2 conditioning on Z 1 is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.

Thesis (M. Sc. (Mathematical Statistics)) -- University of the North, 1978 / Refer to the document

Let π1, ..., πk be k(&gt;_2) populations such that πi, i = 1, 2, ..., k, is characterized by the normal distribution with unknown mean and ui variance aio2 , where ai is known and o2 may be unknown. Suppose that on the basis of independent samples of size ni from π (i=1,2,...,k), we are interested in selecting a random-size subset of the given populations which hopefully contains the population with the largest mean.Based on likelihood ratios, several new procedures for this problem are derived in this report. Some of these procedures are compared with the classical procedure of Gupta (1956,1965) and are shown to be better in certain respects. / <p>Ny rev. utg.</p><p>This is a slightly revised version of Statistical Research Report No. 1978-6.</p> / digitalisering@umu

Shewhart X bar control charts with estimated control limits are widely used in practice. When the sample size is not fixed,we propose seven statistics to estimate the standard deviation sigma . These estimators are applied to estimate the control limits of Shewhart X bar control chart. The estimated results through simulated computation are given and discussed. Finally, we investigate the performance of the Shewhart X bar control charts based on the seven estimators of sigma via its simulated average run length (ARL).