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In credit risk, debtors with different creditworthiness are divided into rating classes. One problem is to define the borders of the rating classes. A natural way to estimate these breakpoints from default observations comes out of the field of change point analysis. In order to account for dependency between the debtors, the literature proposes a combination of a breakpoint model with a one-factor model. One finds strongly consistent estimators for the threshold of the rating classes and the corresponding default probabilities, also called risk levels. But an investigation of the inherent model properties is as yet missing. For this reason we derive the default correlation and study its relationship to the model parameters, i.e., the breakpoint, the risk levels, and a new correlation term, named score correlation, appearing in a simulation study. Eventually, we check the magnitude of the score correlation used in the simulation study.

We usually assume that counts of nonconforming items have a binomial distribution with parameters ( n , p ), where n and p represent the sample size and the fraction nonconforming, respectively. The non-negative, discrete and usually skewed character and the target mean (np0)${(np_0)}$ of this distribution may prevent the quality control engineer to deal with a chart to monitor p with: a pre-specified in-control average run length (ARL), say α-1${\alpha ^{-1}}$; a positive lower control limit; the ability to control not only increases but also decreases in p in an expedient fashion. Furthermore, as far as we have investigated, the n p - and p -charts proposed in the Statistical Process Control literature are ARL-biased, in the sense that they take longer, in average, to detect some shifts in the fraction nonconforming than to trigger a false alarm. Having all this in mind, this paper explores the notions of uniformly most powerful unbiased tests with randomization probabilities to eliminate the bias of the ARL function of the n p -chart and to bring its in-control ARL exactly to α-1${\alpha ^{-1}}$.

In this paper, an attempt is made to derive the most efficient economic reliability sampling plans for accepting a lot containing identical units having exponentially distributed lifetime with parameter θ. We consider two types of sampling plans, namely, (a) sequential sampling plan (t1,t2)${(t_1,t_2)}$ and (b) repetitive group sampling plan (n,t1,t2)${(n,t_1,t_2)}$. Under plan (a), the lot is rejected when the time between successive failures (Yr)${(Y_r)}$ is less than t 1 , and accepted when Yr≥t2${Y_r\ge t_2}$. The testing will continue for t1≤Yr<t2${t_1\le Y_r&lt; t_2}$. Also we formulate an optimization problem that minimizes the total expected testing cost. Under plan (b), four different criteria are used to derive the sampling plan. The optimization problem formulated under each criterion is solved using a genetic algorithm to obtain the plan parameters (n,t1,t2)${(n,t_1,t_2)}$. Several numerical examples are discussed to illustrate our plans. In addition, a real example is also considered to demonstrate our plan. Finally, we compare the cost of our plan with that of an existing plan in the literature. Our plan has significant potential to reduce the testing cost by about 50%.

In this article, acceptance sampling plans are developed for the exponentiated Fréchet distribution based on percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given customer's risk and producer's risk simultaneously. The operating characteristic values of the sampling plans are presented. One example with real data set is also given as an illustration.

We suggest a ranked set sample method to improve Bayesian prediction intervals. The paper deals with the Bayesian prediction intervals in the context of an ordered ranked set sample from a certain class of exponential-type distributions. A proper general prior density function is used and the predictive cumulative function is obtained in the two-sample case. The special case of linear exponential distributed observations is considered and completed with numerical results.

In this paper, a special case of Schröder's functional equation is solved. Then, the general solution of the equation is utilized to determine the class of distributions by their α-quantile past lifetime function.