New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Lower bounds for confidence coefficients for confidence intervals for finite population quantiles

Assuming stratified simple random sampling, a confidence interval for a finite population quantile may be desired. Using a confidence interval with endpoints given by order statistics from the combined stratified sample, several procedures to obtain lower bounds (and approximations for the lower bounds) for the confidence coefficients are presented. The procedures differ with respect to the amount of prior information assumed about the var-iate values in the finite population, and the extent to which sample data is used to estimate the lower bounds.     

Approximated non parametric confidence regions for the ratio of two percentiles

In the wood industry, it is common practice to compare in terms of the ratio of the same-strength properties for lumber of two different dimensions, grades, or species. Because United States lumber standards are given in terms of population fifth percentile, and strength problems arise from the weaker fifth percentile rather than the stronger mean, so the ratio should be expressed in terms of the fifth percentiles rather than the means of two strength distributions. Percentiles are estimated by order statistics. This paper assumes small samples to derive new non parametric methods such as percentile sign test and percentile Wilcoxon signed rank test, construct confidence intervals with covergage rate 1 – αx for single percentiles, and compute confidence regions for ratio of percentiles based on confidence intervals for single percentiles. Small 1 – αx is enough to obtain good coverage rates of confidence regions most of the time.     

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.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

Empirical likelihood confidence regions for comparison distributions and roc curves

Abstract: The authors derive empirical likelihood confidence regions for the comparison distribution of two populations whose distributions are to be tested for equality using random samples. Another application they consider is to ROC curves, which are used to compare measurements of a diagnostic test from two populations. The authors investigate the smoothed empirical likelihood method for estimation in this context, and empirical likelihood based confidence intervals are obtained by means of the Wilks theorem. A bootstrap approach allows for the construction of confidence bands. The method is illustrated with data analysis and a simulation study.     

Simulation of weibull and gamma autoregressive stationary process

This paper presents two simple non-Gaussian first-order autoregressive markovian processes which are easy to simulate via a computer. The autoregressive Gamma process {X_n:} is constructed according to the stochastic difference equation X_n:=V_n:X_n?1+?_n:, where {?_n:} is an i.i.d. Exponential sequence and {V_n:} is i.i.d. with Power-function distribution defined on the interval [0,1). The autoregressive Weibull process {X_n:} is constructed from the probabilistic model X_n:= k.min (X_n?1:, Y_n:) where {Y_n:} is an i.i.d. Weibull sequence and k > 1.     

Simple bounds on availability in a model with unknown life and repair distributions

We take a fresh look at the classic model of a device supported by a single statistically identical spare and provision for repairs, with system failure resulting whenever the currently operating unit fails before the repair of the previously failed unit is completed to allow it to become a spare. The limiting availability A(F,G) of this system depends on the life distribution F and repair time distribution G through α=∫GdF and the expected downtime. In this paper we derive several computable and sharp bounds on A(F,G) when F,G have suitable life distribution characteristics in the sense of reliability theory but are otherwise unknown except for at most two moments. Among other results, we find a sharp bound which involves the MTBF, MTTR and the second moment of the life-distribution of the device through its coefficient of variation. This leads to a maximin result for DFR repairs and DMRL lives.     

Automatic selection of the proper family for simultaneous confidence intervals

Statisticians often employ simultaneous confidence intervals to reduce the likelihood of their drawing false conclusions when they must make a number of comparisons. To do this properly, it is necessary to consider the family of comparisons over which simultaneous confidence must be assured. Sometimes it is not clear what family of comparisons is appropriate. We describe how computer software can monitor the types of contrasts a user examines, and select the smallest family of contrasts that is likely to be of interest. We also describe how to calculate simultaneous confidence intervals for these families using a hybrid of the Bonferroni and Scheffé methods. Our method is especially suitable for problems with discrete and continuous predictors.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

The starship for point estimates and confidence intervals on a mean and for percentiles

The starship procedure for transformations to normality described by Owen (1988) is implemented using the Johnson (1949) System of transformations, the Slifker-Shapiro (1980) technique for choosing a transformation, and the Shapiro-Wilk (1965) test for normality. This procedure was applied to obtain maximum likelihood point estimates of a mean, to obtain confidence intervals on a mean, and to estimate percentiles of a distribution based on a sample. Simulations of three distributions show that the starship has many desirable properties, and can be compared very favorably with the bootstrap procedures of Efron (1987).     

Comparison of confidence intervals for the Behrens-Fisher problem

The Behrens–Fisher problem concerns the inferences for the difference between means of two independent normal populations without the assumption of equality of variances. In this article, we compare three approximate confidence intervals and a generalized confidence interval for the Behrens–Fisher problem. We also show how to obtain simultaneous confidence intervals for the three population case (analysis of variance, ANOVA) by the Bonferroni correction factor. We conduct an extensive simulation study to evaluate these methods in respect to their type I error rate, power, expected confidence interval width, and coverage probability. Finally, the considered methods are applied to two real dataset.     

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.     

Cohpasisons of improved bonferroni and sidak/slepian bounds with applications to normal markov processes

The recent literature contains theorems improving on both the standard Bonferroni inequality (Hoover (1990)) and the Sidak/Slepian inequalities (Glaz and Johnson (1984)), The application of these improved theorems to upper bounds for non coverage of simultaneous confidence intervals on multivariate normal variables is explored. The improved Bonferroni upper bounds always hold, while improved Sidak/Slepian bounds only apply to special cases. It is shown that improved Sidak/Slepian bounds will always hold for Normal Markov Processes, a commonly occuring and easily identifiable class of multivariate normal variables. The improved Sidak/Slepian upper bound, if it applies, is proven to be superior to the computationally equivalent improved Bonferroni bound. This improvement, however, is not great when both methods are used to determine upper bounds for Type I error in the range of .01 to .10.     

Modified cramer-von mises and anderson-darling tests for weibull distributions with unknown location and scale parameters

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.     

Confidence curves and improved exact confidence intervals for discrete distributions

The author describes a method for improving standard “exact” confidence intervals in discrete distributions with respect to size while retaining correct level. The binomial, negative binomial, hypergeometric, and Poisson distributions are considered explicitly. Contrary to other existing methods, the author's solution possesses a natural nesting condition: if α < α', the 1 ‐ α' confidence interval is included in the 1 ‐ α interval. Nonparametric confidence intervals for a quantile are also considered.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Bootstrap confidence intervals for the coefficient of quartile variation

ABSTRACT

In non-normal populations, it is more convenient to use the coefficient of quartile variation rather than the coefficient of variation. This study compares the percentile and t-bootstrap confidence intervals with Bonett's confidence interval for the quartile variation. We show that empirical coverage of the bootstrap confidence intervals is closer to the nominal coverage (0.95) for small sample sizes (n = 5, 6, 7, 8, 9, 10 and 15) for most distributions studied. Bootstrap confidence intervals also have smaller average width. Thus, we propose using bootstrap confidence intervals for the coefficient of quartile variation when the sample size is small.     

Simultaneous confidence bands and regions for log-location-scale distributions with censored data

In many areas of application, especially life testing and reliability, it is often of interest to estimate an unknown cumulative distribution (cdf). A simultaneous confidence band (SCB) of the cdf can be used to assess the statistical uncertainty of the estimated cdf over the entire range of the distribution. Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] presented an approach to construct an SCB for the cdf of a continuous random variable. For the log-location-scale family of distributions, they gave explicit forms for the upper and lower boundaries of the SCB based on expected information. In this article, we extend the work of Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] in several directions. We study the SCBs based on local information, expected information, and estimated expected information for both the “cdf method” and the “quantile method.” We also study the effects of exceptional cases where a simple SCB does not exist. We describe calibration of the bands to provide exact coverage for complete data and type II censoring and better approximate coverage for other kinds of censoring. We also discuss how to extend these procedures to regression analysis.