Computing factors for exact two-sided tolerance limits for a normal distribution

A self-contained FORTRAN subroutine is provided which computes factors for Wald-Wolfowitz type tolerance limits allowing arbitrary combinations of sample size n and degrees of freedom ν. The exact calculations from our program reveal inadequacies of two existing approximations, especially when ν ? n. Numerous applications where ν ≠ n ? 1 are cited; two of these are discussed and illustrated.     

Use of a two-sided tolerance interval in the design and evaluation of biosimilarity in clinical studies

In assessing biosimilarity between two products, the question to ask is always “How similar is similar?” Traditionally, the equivalence of the means between products is the primary consideration in a clinical trial. This study suggests an alternative assessment for testing a certain percentage of the population of differences lying within a prespecified interval. In doing so, the accuracy and precision are assessed simultaneously by judging whether a two-sided tolerance interval falls within a prespecified acceptance range. We further derive an asymptotic distribution of the tolerance limits to determine the sample size for achieving a targeted level of power. Our numerical study shows that the proposed two-sided tolerance interval test controls the type I error rate and provides sufficient power. A real example is presented to illustrate our proposed approach.     

Minimum-area confidence sets for a normal distribution

When making inference on a normal distribution, one often seeks either a joint confidence region for the two parameters or a confidence band for the cumulative distribution function. A number of methods for constructing such confidence sets are available, but none of these methods guarantees a minimum-area confidence set. In this paper, we derive both a minimum-area joint confidence region for the two parameters and a minimum-area confidence band for the cumulative distribution function. The minimum-area joint confidence region is asymptotically equivalent to other confidence regions in the literature, but the minimum-area confidence band improves on existing confidence bands even asymptotically.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Probabilistic tolerance design for a subsystem under burr distribution using taguchi's loss functions

Taguchi (1986) has derived tolerances for subcomponents, subsystems, parts and materials in which the relationship between a higher-level (Y) and a lower-level (X) quality characteristic is assumed to be deterministic and linear, namely, Y=_α+βX, without an error term. Tsai (1990) developed a probabilistic tolerance design for a subsystem in which a bivariate normal distribution between the above two quality characteristics as well as Taguchi's quadratic loss function were considered together to develop a closed form solution of the tolerance design for a subsystem. The Burr family is very rich for fitting sample data, and has positive domain. A bivariate Burr distribution can describe a nonlinear relationship between two quality characteristics, hence, it is adopted instead of a bivariate normal distribution and the simple solutions of three probabilistic tolerance desings for a subsystem are obtained for three cases of “nominal-is-best”, “smaller-is-berrer”, and “larger-is-beter” quality characteristics, by using Taguchi’ los functions, respectively.     

Modified two‐sided normal tolerance intervals for batch acceptance of dose uniformity

We consider modified tolerance intervals for assessing content (dose) uniformity from one‐ and two‐stage designs. This work was motivated by a proposal from the International Pharmaceutical Aerosol Consortium on Regulation and Science (IPAC‐RS) to modify tolerance intervals to reduce their known conservatism away from the centre of the acceptance interval. IPAC‐RS's original proposal suffers from anti‐conservatism, but we find this is due to the choice of constants, not the form of the acceptance rule. We propose an alternative basis for choosing the constants in the acceptance rules and compare attained levels to those of other choices, including revised constants proposed by IPAC‐RS. These alternate choices of the constants lead to modified tolerance intervals that are either conservative at the sample sizes for which they were chosen or may have slight anti‐conservatism, within the limits of the simulations. Copyright © 2004 John Wiley & Sons, Ltd.     

Enhancement for two commonly-used approximations for the inverse cumulative function of the normal distribution

Two commonly used approximations for the inverse distribution function of the normal distribution are Schmeiser's and Shore's. Both approximations are based on a power transformation of either the cumulative density function (CDF) or a simple function of it. In this note we demonstrate, that if these approximations are presented in the form of the classical one-parameter Box-Cox transformation, and the exponent of the transformation is expressed as a simple function of the CDF, then the accuracy of both approximations may be considerably enhanced, without losing much in algebraic simplicity. Since both approximations are special cases of more general four-parameter systems of distributions, the results presented here indicate that the accuracy of the latter, when used to represent non-normal density functions, may also be considerably enhanced.     

Percentile bounds and tolerance limits for the birnbaum-saunders distribution

In this paper, a confidence interval for the lOOpth percentile of the Birnbaum-Saunders distribution is constructed. Conservative two-sided tolerance limits are then obtained from the confidence limits. These results are useful for reliability evaluation when using the Birnbaum-Saunders model. A simple scheme for generating Birnbaum-Saunders random variates is derived. This is used for a simulation study on investigating the effectiveness of the proposed confidence interval in terms of its coverage probability.     

Tables of percentage points of the k-variate normal distribution for large values of k

This paper gives tabulations of the upper α percentage points of the maximum absolute value of the k-variate normal distribution with common correlation ρ for values of k as high as 500. The tables are useful for performing multiple comparison procedures in experiments with large numbers of treatments.     

Simultaneous one-sided tolerance intervals for sets of contrasts from normal populations

This paper considers the problem of constructing simultaneous prediction and tolerance intervals for sets of contrasts of normal variables in situations where simultaneous intervals are available. Tables are given with critical values used in simultaneous tolerance bounds for two classes of contrasts: pairwise many-one and profile type.     

Tables of confidence limits on the tail area of the normal distribution

Tables are given of confidence limits on tail areas, γ, of the normal distribution, where γ = P{Y ≥ L}, and where L is a given number, and Y is normally distributed with unknown mean, μ, and unknown variance, σ².     

Two normal approximations to the binomial distribution

Two new normal approximations are proposed for the cumulative binomial distribution when the mean-is reasonably large. Their adequacy is compared with that of certain well-known approximations- The first is recommended for its simplicity and accuracy relative to the standard and Gram-Charlier approximations. The second Is shown to be more accurate than all known approximations for a certain range of the probability of success.     

Second-order properties of improved two-stage procedure for a multivariate normal distribution

We consider the classical two problems: (I) fixed-size confidence region estimation problem, and (II) bounded risk point estimation problem, for the mean vector μ in the N_p(μ, ∑) population when ∑ is unknown and positive definite. Healy's (1956) solution given by a two-stage procedure for the prob¬lem (I) is improved in terms of the sample size. The second-order properties of the solution to each problem are verified under the assumptions that the maximum latent root which is simple of ∑ is bounded below for (I) and tr(∑) is bounded below for (II), by respective known and positive number.     

New approximation of the normal distribution function

Two new and simple expressions, one for 0 ≤ x ≤ 2 and another for x > 2, for the normal distribution function, are developed which can be easily computed on desk calculators. They are also comparable in accuracy to the one developed by Patry and Keller (1964).     

An intermediate-precision approximation of the inverse cumulative normal distribution

Accurate methods used to evaluate the inverse of the standard normal cumulative distribution function at probability ρ commonly used today are too cumbersome and/or slow to obtain a large number of evaluations reasonably quickly, e.g., as required in certain Monte Carlo applications. Previously reported simple approximations all have a maximum absolute error ε_m > 10^-4 for a ρ-range of practical concern, such as Min[ρ,l?ρ]≥10_?6. An 11-term polynomial-based approximationis presented for which ε_m > 10^-6 in this range.     

The two-sided power distribution for the treatment of the uncertainty in PERT

The aim of this paper is to include the Two-Sided Power (TSP) distribution in the PERT methodology making use of the advantages that this four-parameter distribution offers. In order to be completely determined, a distribution of this type needs, the same as the beta distribution, a new datum apart from the three usual values a (pessimistic), m (most likely) and b (optimistic). To solve this question, when using the beta distribution in the PERT context, we are looking for the maximum similarity with the normal and so it is required that the distribution has the same variance as the normal or its same kurtosis, giving rise to the constant variance and mesokurtic families, respectively. Nevertheless, while this approach can be only applied to the beta distribution for some values in the range of the standardized mode, in the case of the TSP distribution this methodology leads always to a solution. A detailed analysis comparing the beta and TSP distribution based on their PERT means and variances is presented indicating better results for the second. We are very grateful for the comments and suggestions of two anonymous referees.     

Shortest tolerance intervals controlling both tails of the exponential distribution based on record values

This paper deals with computing shortest width tolerance intervals controlling both tails of the exponential distribution on the basis of record values. Equal-tailed and shortest tolerance factors are derived. The expected widths of these tolerance intervals are evaluated via a Monte Carlo simulation study. Finally, two illustrative examples are also included.