In many statistical applications, it is often necessary to obtain an interval estimate for an unknown proportion or probability or, more generally, for a parameter whose natural space is the unit interval. The customary approximate two-sided confidence interval for such a parameter, based on some version of the central limit theorem, is known to be unsatisfactory when its true value is close to zero or one or when the sample size is small. A possible way to tackle this issue is the transformation of the data through a proper function that is able to make the approximation to the normal distribution less coarse. In this paper, we study the application of several of these transformations to the context of the estimation of the reliability parameter for stress-strength models, with a special focus on Poisson distribution. From this work, some practical hints emerge on which transformation may more efficiently improve standard confidence intervals in which scenarios. 1. Introduction In many fields of applied statistics, it is often necessary to obtain an interval estimate for an unknown proportion or probability or, more generally, for a parameter whose natural space is the unit interval [1]. If is the unknown parameter of a binomial distribution, the customary approximate two-sided confidence interval (CI) for is known to be unsatisfactory when its true value is close to zero or one or when the sample size is small. In fact, estimation can cause difficulties because the variance of the corresponding point estimator is dependent on itself and because its distribution can be skewed. A number of papers have been devoted to the development of more refined CIs for (see, e.g., [1–4]). Here, we will consider the estimation of the probability , where and are two independent rv’s. If represents the strength of a certain system and the stress on it, represents the probability that the strength overcomes the stress, and then the system works ( is then referred to as the “reliability” parameter). Such a statistical model is usually called the “stress-strength model” and in the last decades has attracted much interest from various fields [5, 6], ranging from engineering to biostatistics. In these works, inferential issues have been dealt with, mainly in the parametric context. The problem of constructing interval estimators for has been considered; when an exact analytical solution is not available, approximations based on the delta method and asymptotic normality of point estimators are carried out, some of them making use of some data transformation of the point
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