In the literature on statistical inference in software reliability, the assumptions of parametric models and random sampling of bugs have been pervasive. We argue that both assumptions are problematic, the first because of robustness concerns and the second due to logical and practical difficulties. These considerations motivate the approach taken in this paper. We propose a nonparametric software reliability model based on the order-statistic paradigm. The objective of the work is to estimate, from data on discovery times observed within a type I censoring framework, both the underlying distribution F from which discovery times are generated and N, the unknown number of bugs in the software. The estimates are used to predict the next time to failure. The approach makes use of Bayesian nonparametric inference methods, in particular, the beta-Stacy process. The proposed methodology is illustrated on both real and simulated data

A linear (circular) consecutive k-out-of-n:F system is a system of n linearly (circularly) ordered components which fails if and only if at least k consecutive components fail. We use recursive relationships on the reliability of such systems with independent identically distributed components to show that for any fixed k, the lifetime of a (linear or circular) consecutive k-out-of-n:F system is stochastically decreasing in n. This result also holds for linear systems when the components are independent and not necessarily identically distributed, but not in general for circular systems.

LetIbe the Riemann integral over [0,A] of a realization of a random processX(t)with stationary independent increments (SII), finite second moments, andX(0) = 0. The optimal design, consisting of n distinct observation times, for estimatingIrelative to a mean-squared error optimality criterion is shown to be the design such thatt_{i} = 2iA/(2n + 1), i = 1, \cdots,n, independent of the parameters of the process. The mean-squared error of the optimal estimate is displayed. The optimal design for estimating the integral over a proper subinterval of [0,A] is also derived.

The multiple decision problem of identifying the mode of a discrete (k-valued) random variable is examined under a time-varying finite memory constraint. Under the assumption that the distribution of the observed random variable has a unique mode, the partial information model, in which it is assumed that the two largest probabilities can be separated by a known number, is shown to be2kstate perfectly achievable. The general model, in which no assumptions are made on the distribution save the fact that the mode is unique, is shown to be4k (k - 1)state perfectly achievable.

Asymptotically\varepsilon-optimal automata were developed by Hellman and Cover [4] for testing simple hypotheses concerning the parameter of an independent identically distributed sequence of Bernoulli random variables. These automata permit transitions only between adjacent states and employ artificial randomization only at extreme states. In this paper we study the problem of approximating the optimal Hellman-Cover automaton in fixed-sample-size problems. It is shown that the optimal level of the parameter, which regulates the probability of transitions out of an extreme state, tends to zero at the rate\ln n/nin symmetric testing problems wherenis the sample size. We develop an approximation for the optimal parameter value valid fornsufficiently large.

The purpose of this correspondence is to display an inductive proof of an optimality result for testing symmetric simple hypotheses concerning a binomial parameter with a three-state memory andnobservations. Specifically, it is shown that if one restricts attention to three-state automata allowing transitions only between adjacent states, then no artificial randomization is required in the middle state of the optimal machine. The success in this problem of what will be called simultaneous induction suggests the possibility of obtaining characterizations of optimal automata in more general problems by similar techniques.

This article treats the asymptotic theory of estimating a binomial parameterpwith time-invariant finite memory. The approach taken to this problem is as follows. A decision rule is a pair(t,a)in whichtfixes the transition function of a finite automaton, andais a vector of estimates ofp. Attention is restricted to automata whose transition functions allow transitions only between adjacent states. Rules(t,a)for whichtsatisfies this restriction are termed tridiagonal. For the class of prior distributions on [0,1] which have continuous density functions, we study the performance of a corresponding class of tridiagonal rules\{ (t^{\ast},a^{\ast}) \}relative to quadratic loss functions. These rules display sensitivity to the shape of the prior, and have the advantage that the Bayes estimatea^{\ast}(givent^{\ast}) is easily computed. Within the class of all tridiagonal rules, a particular rule(t^{\ast},a^{\ast})is shown, for memory size up to 30, to be locally admissible and minimax as well as locally Bayes with respect to the uniform prior.