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A multivariate model involving differences of independent gamma distributed com-ponents was introduced by Arnold (2020) The present paper provides a more detailed discussion of this model.

This paper provides a Bayesian framework for testing the number of modes in a two-component Gaussian mixture. The test is done by first setting up conjugate priors and computing the corresponding posteriors, which are then integrated over the restricted subspace of unimodal parameter space for the mixture distribution, thus obtaining the prior and posterior probabilities of unimodality. Monte Carlo and Gibbs sampling methods are employed to numerically compute these probabilities due to the difficulty in finding analytical solutions. A conclusion on unimodality for the given data is arrived at based on the Bayes factor. Effectiveness of the proposed Bayes test is demonstrated via simulations, and applied to a practical data set on adult human heights in order to answer the question whether the combined height data for men and women is bimodal.

KEYWORDS: Gaussian mixture and unimodal/bimodal distribution and Bayesian test and Monte Carlo method and Gibbs sampling and Bayes factor and human heights.

A numerical approach to optimal adaptive integer pulse control of stochastic non-linear systems is presented. For most stochastic nonlinear systems, optimal adaptive control rules cannot be derived with analytical methods. A robust optimization al-gorithm is created. The complete nonlinear adaptively controlled stochastic system is simulated during 100 years, for 100 alternative sequences of stochastic distur-bances, for every feasible integer combination of adaptive control rules. The optimal adaptive control rules that maximize the expected value of the objective function are selected as the optimal adaptive control rules. The method is very general and can easily be applied to most adaptive nonlinear stochastic control problems, from technology, management or other fields. The method is tested and applied to the wolf-moose predator prey system. The parameters of this stochastic nonlinear dy-namical system have recently been estimated from empirical data from Isle Royale in Lake Superior, USA. The objective function is the expected total present value of all hunting net revenues and the environmental value of preserving the wolf population. The value of the wolf population is a strictly increasing and strictly concave function of the population level. Periodically, the region is visited and the population levels are determined. If the population levels, one for each species, exceed the optimal control limits, then the populations are reduced to the control limits, via hunting. Then, the system is left to develop until the next period. Optimal population control limits and objective function values are determined for alternative levels of the wolf population value function. The average optimal moose hunting level is a decreasing function of the wolf population value parameter and an increasing function of the level of risk in the predator prey system. The average optimal wolf population level is an increasing function of the wolf population value parameter and a decreasing function of the level of risk in the predator prey system.

KEYWORDS: Adaptive Optimization; Stochastic Nonlinear System; Predator Prey; Moose; Wolf.

In this paper, a study on evaluating measure of modified rotatability for second degree polynomial using a pair of balanced incomplete block designs ( 5 v 15 : v - number of factors) is suggested which enables us to assess the degree of modified rotatability for a given response surface design.

KEYWORDS: Response surface methodology; modified rotatability; measure.

In this article it is shown that the new unit Lindley distribution of Mazuchelli et al. [2] is in fact complementary of the original unit Lindley distribution of Mazuchelli et al. [1] and as such almost all properties of the former directly follows from those of the later quite easily. Some illustrative derivation of properties of new unit Lindley distribution from those of the unit Lindley distribution is also presented.

KEYWORDS: Complementary random variable; Unit-Lindley distribution ; Exponential family

The concept of Time To Test Transform (TTT) is well known for its applications in different fields of study such as reliability analysis, econometrics, stochastic mod-eling and ordering distributions. In this article, we estimate the TTT for the Lo-max function based on censored sample. The Bayes estimates are evaluated under squared error, entropy, precautionary loss functions. The empirical evaluation of the estimates is done using a simulation study.

KEYWORDS: Time To Test Transform; Censored sampling; Entropy loss function; Lomax distribution; Precautionary loss function; Prior distribution; Squared error loss function.