An alternative approach to reliability analysis of cold standby systems

ABSTRACT

An alternative approach is applied for reliability analysis of standby systems on the basis of matrix renewal function. In this regard, a single-server, two identical unit cold standby systems with an imperfect switch is considered as a three-state semi-Markov process. Several important reliability measures such as availability, mean time to failure, expected number of failures, etc., are obtained for general lifetime distributions. Also, the main results have been treated to the case of exponential lifetimes and explicit formulas obtained for this case in addition of some numerical illustrations. This approach can easily be extended to more general standby systems with different configurations.     

Some remarks on a ridge-type-estimator and good prior means

Pliskin (1987) compared modified ridge regression estimators based on prior information with respect to their mean square error matrices. A further characterization of good prior mean is given here, and the case of different ridge parameters is also considered.     

Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

In the investigation of the restricted linear model ? _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.     

Fluid Flow Models and Queues—A Connection by Stochastic Coupling

We establish in a direct manner that the steady state distribution of Markovian fluid flow models can be obtained from a quasi birth and death queue. This is accomplished through the construction of the processes on a common probability space and the demonstration of a distributional coupling relation between them. The results here provide an interpretation for the quasi-birth-and-death processes in the matrix-geometric approach of Ramaswami and subsequent results based on them obtained by Soares and Latouche.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

An algorithm to test divisibility of continuous distributions,applied to the uniform distribution

We present an algorithm to test if a continuous random variable is n-divisible. Through the algorithm, testing for divisibility is put in the format of numerical analysis amenable to computer processing. This is illustrated with the uniform distribution which is known to be non-divisible. New evidence and new proofs arising out of the insight developed with this style are given. This algorithm, along with the analytical methods should provide much needed versatility to tackle problems in this area. The fact that Bondesson [1987] works on the divisibility of the Half Cauchy Distribution gives evidence that there are many distributions which can use the combination of the analytical and the algorithmic tools to settle the divisibility issue.     

Sampling from a Discrete Distribution While Preserving Monotonicity

This article describes a cutpoint sampling method for efficiently sampling from an n-point discrete distribution that preserves the monotone relationship between a uniform deviate and the random variate it generates. This property is useful for developing a sampling plan to reduce variance in a Monte Carlo or simulation study. The expected number of comparisons with this method is derived and shown to be bounded above by (m + n ?1)/n, where m denotes the number of cut-points. The alias sampling method, which is regarded as the most efficient table sampling technique, generally lacks the monotone property and requires 2n storage locations, whereas the proposed cutpoint sampling method requires m + n storage locations. The article describes two modifications for cases in which n is large and possibly infinite. It is shown that circumstances arise in which the cutpoint method requires fewer comparisons on average than the alias method does for exactly the same space requirement. The article also describes an algorithm to implement the proposed method.     

Dispersion-diminishing transformations

Let D(σ) consist of matrices congruent to and dominated by a given matrix σ , and let T(σ) be the corresponding congruent transformations. These classes are characterized and their properties studied when σ is positive definite. Dispersion orderings are considered, including dispersion-diminishing linear transformations, concentration properties of which are shown. Arbitrary linear transformations are decomposed into contractions, isometries and dilations on subspaces relative to Mahalanobis norms. Applications are noted in statistical process control and linear inference     

Recombinative Generalization of Tacts Through Matrix Training with Individuals with Autism Spectrum Disorder

Foss (Journal of Experimental Psychology, 76, 450–459, 1968a; Journal of Experimental Psychology, 77, 341–344, 1968b) compared overlap and non-overlap instruction to promote recombinative response generalization using a matrix training procedure. In the present study, we used a similar set of procedures to teach tacting of kitchen items and prepositions (i.e., relational autoclitics) to three females ages 13–20, diagnosed with autism spectrum disorder. We taught some kitchen items/prepositions as tacts (e.g., “the strainer is to the right of the box”) according to a non-overlap instructional sequence. Subsequently, we taught more combinations in an overlap instructional sequence. Each training procedure was followed by probes of untrained relations. Two participants demonstrated recombinative generalization of untrained combinations following the first non-overlap phase, while the third participant demonstrated some response generalization of untrained relations after a few additional training sequences. All three participants demonstrated generalized tacting of object components while two participants showed generalized tacting of preposition components.     

Performance of Interval Estimators for the Inverse Hypergeometric Distribution

The inverse hypergeometric distribution is of interest in applications of inverse sampling without replacement from a finite population where a binary observation is made on each sampling unit. Thus, sampling is performed by randomly choosing units sequentially one at a time until a specified number of one of the two types is selected for the sample. Assuming the total number of units in the population is known but the number of each type is not, we consider the problem of estimating this parameter. We use the Delta method to develop approximations for the variance of three parameter estimators. We then propose three large sample confidence intervals for the parameter. Based on these results, we selected a sampling of parameter values for the inverse hypergeometric distribution to empirically investigate performance of these estimators. We evaluate their performance in terms of expected probability of parameter coverage and confidence interval length calculated as means of possible outcomes weighted by the appropriate outcome probabilities for each parameter value considered. The unbiased estimator of the parameter is the preferred estimator relative to the maximum likelihood estimator and an estimator based on a negative binomial approximation, as evidenced by empirical estimates of closeness to the true parameter value. Confidence intervals based on the unbiased estimator tend to be shorter than the two competitors because of its relatively small variance but at a slight cost in terms of coverage probability.