This paper introduces a new class of prior distributions for reliability growth tests with binomial data under the monotone model. The proposed prior has a conditional form, which accords well with various actual situations in reliability growth tests. The expressions of the corresponding means and variances for all stages with and without conditioning are obtained, and the relationship between the shape of the prior distributions and their parameters are discussed. These results are helpful in order to incorporate expert opinions. The posterior density and the Bayesian lower bound of the relibility at the end of the test, and a computation method for them are given. The new family of prior distributions includes the uniform prior used by Smith (1977) and the ordered Dirichlet priors presented by Mazzuchi and Soyer (1992, 1993) as special cases. Comparisons are made by two examples, which show the limitations of the later two.
In actual applications of regression analysis, users face two difficult problems. One is to find the most appropriate functional form, while the other is to search for the best subset derivable from a given set of all possible explanatory variables. Variable selection for the Box-Cox transformation may be useful to concurrently solve both problems. The purpose of this paper is to (1) concretely formulate the j-th OLS-best subset problem for the Box-Cox transformation, (2) introduce a knowledge-based computational method to solve it and (3) propose a solution to the (first) OLS-best subset problem (j=1) or one selected by a user among solutions to the first j OLS-best subset problems (j>1) solved in a run of a computer as a solution to a variable selection problem for the Box-Cox transformation. The integer j, specified by the user, depends on his scientific knowledge, criteria for statistical and data-analytic tests and model-building experience.
A new influence measure is proposed to assess the influence of individual observations on prediction mean square errors (PMSE) in variable selection problems. It is based on the estimated PMSE which consists of Cook’s distance and Mallows’ CP statistic. Another interpretation of Cook’s distance is also given through the expression of the new influence measure. Illustrative examples show the effectiveness of the new influence measure.
In this paper, we consider the effects of nonnormality on the upper percentiles of T2max statistic in elliptical distributions. Some approximations based on the Bonferroni inequalities and asymptotic expansion procedure are given under the elliptical distribution setup. In order to achieve the purpose, asymptotic expansions for the distributions of univariate and bivariate Hotelling’s T2 type statistics are derived by a perturbation method when each population has the elliptical distribution. Finally, the accuracy of the approximations is investigated by Monte Carlo simulations for selected vales of parameters.
This paper considers the competing risks problem with randomly right-censored data. Let F(j)(t) be the cause-specific cumulative incidence function of a cause j, which is the probability of death due to a cause j by time t in the presence of other acting causes. The Aalen-Johansen estimator F(j)n is a nonparametric maximum likelihood estimator of F(j). Under the assumption that all F(j)’s and a censoring distribution are continuous, asymptotic properties of the Aalen-Johansen integral s(j)n=∫φdF(j)n are investigated. Let F be the overall lifetime distribution. We show that for any F-integrable function φ, the Aalen-Johansen integral s(j)n converges almost surely as n→∞. It is also shown that under some mild integrability assumptions for φ, the joint distribution of √¯ns(j)n’s for all causes is asymptotically multivariate normal.
Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let ¯Pk denote the probability that the first device will survive the k shocks and ¯Qk denote such a probability for second device. Let ¯F(t) and ¯G(t) denote the survival functions of the first and second device respectively. In this paper we show that some new partial ordering, namely dual (D), dual stochastic (DST), dual weak likelihood ratio (DWLR), increasing failure rate (IFR), dual mean residual lives (DMRL) and dual convex (DCX) orderings between the shock survival probabilities ¯Pk and ¯Qk are preserved by the corresponding survival function ¯F(t) and ¯G(t). We also obtain sufficient condition under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.
In the framework of disclosure control of a microdata set, a unique record is at risk of being identified. Even if a record is not unique in the microdata set, it may be considered risky if the frequency k of the cell, in which the record falls, is small. The notion of minimum unsafe combination introduced by Willenborg and de Waal (1996) is important in this respect. The purpose of this paper is to clarify the logical relationships between this notion and other closely related notions, and give an algorithm for obtaining relevant combinations of variables. We will define the minimum k-unsafe and maximum k-safe sets of variables for each record. We also give an illustration to show the usefulness of the proposed technique for the purpose of disclosure risk assessment and control.