Bootstrap Confidence Limits Research Articles

On a Class of m out of n Bootstrap Confidence Intervals

Summary It is widely known that bootstrap failure can often be remedied by using a technique known as the ‘m out of n’ bootstrap, by which a smaller number, m say, of observations are resampled from the original sample of size n. In successful cases of the bootstrap, the m out of n bootstrap is often deemed unnecessary. We show that the problem of constructing nonparametric confidence intervals is an exceptional case. By considering a new class of m out of n bootstrap confidence limits, we develop a computationally efficient approach based on the double bootstrap to construct the optimal m out of n bootstrap intervals. We show that the optimal intervals have a coverage accuracy which is comparable with that of the classical double-bootstrap intervals, and we conduct a simulation study to examine their performance. The results are in general very encouraging. Alternative approaches which yield even higher order accuracy are also discussed.

A graphical approach to obtaining confidence limits ofCpk

The process capability index Cpk has been widely used as a process performance measure. In practice this index is estimated using sample data. Hence it is of great interest to obtain confidence limits for the actual index given a sample estimate. In this paper we depict graphically the relationship between process potential index (Cp), process shift index (k) and percentage non-conforming (p). Based on the monotone properties of the relationship, we derive two-sided confidence limits for k and Cpk under two different scenarios. These two limits are combined using the Bonferroni inequality to generate a third type of confidence limit. The performance of these limits of Cpk in terms of their coverage probability and average width is evaluated by simulation. The most suitable type of confidence limit for each specific range of k is then determined. The usage of these confidence limits is illustrated via examples. Finally a performance comparison is done between the proposed confidence limits and three non-parametric bootstrap confidence limits. The results show that the proposed method consistently gives the smallest width and yet provides the intended coverage probability. © 1997 John Wiley & Sons, Ltd.

Stopping Rules in Principal Components Analysis: A Comparison of Heuristical and Statistical Approaches

Approaches to determining the number of components to interpret from principal components analysis were compared. Heuristic procedures included: retaining components with eigenvalues (λ) > 1 (i.e., Kaiser—Guttman criterion); components with bootstrapped λ > 1 (bootstrapped Kaiser—Guttman); the scree plot; the broken—stick model; and components with λ totalling to a fixed amount of the total variance. Statistical approaches included: Bartlett's test of sphericity; Bartlett's test of homogeneity of the correlation matrix, Lawley's test of the second λ bootstrapped confidence limits on successive λ (i.e., significant differences between λ); and bootstrapped confidence limits on eigenvector coefficients (i.e., coefficients that differ significantly from zero). All methods were compared using simulated data matrices of uniform correlation structure, patterned matrices of varying correlation structure and data sets of lake morphometry, water chemistry, and benthic invertebrate abundance. The most consistent results were obtained from the broken—stick model and a combined measure using bootstrapped λ and associated eigenvector coefficients. The traditional and bootstrapped Kaiser—Guttman approaches over—estimated the number of nontrivial dimensions as did the fixed—amount—of—variance model. The scree plot consistently estimated one dimension more than the number of simulated dimensions. Barlett's test of sphericity showed inconsistent results. Both Bartlett's test of homogeneity of the correlation matrix and Lawley's test are limited to testing for only one and two dimensions, respectively.

Biochemcial systematics of the North American Pteronarcys (Pteronarcyidae: Plecoptera)

Allozyme data from eight species of Pteronarcyidae were used to determine phylogenetic relationhips within the family and genus Pteronarcys. Data were analysed as allele frequency data using the BIOSYS-1 computer package and as discrete characters using the PENNY (parsimony) and MIX (mixed parsimony) options of the PHYLIP computer program. Certain analyses yielded species relationships which are in agreement with phylogenies based upon morphological characters from various life-history stages. Bootstrap confidence limits on phylogenies based on allozyme as well as morphological data are discussed. The age of the family likely contributes to the lack of concordance with previously proposed phylogenies.

SAMPLE SIZE REQUIREMENTS FOR THE DETERMINATION OF CHANGES IN SOIL NUTRIENT POOLS

In long-term field studies, the number of soil samples required to detect a specified change is often determined by using the standard error of a soil property from a pilot study. When a soil property is highly variable, estimates of its standard error will also be highly variable. Therefore, it is useful to have confidence limits for the standard error in sample size calculations. We explored the empirical distribution of the standard error for several soil properties (total carbon pools, exchangeable cation pools, and forest floor element pools) using data collected from a northern hardwood forest site in central New Hampshire. We used a bootstrapping routine to simulate data sets for sample sizes ranging from 6 to 60 soil pits and computed the mean, 95%, and 98% confidence limits for the percentage change detectable by a specified number of sample observations. Using the 95% confidence limit for standard error resulted in as much as a twofold increase in the sample size requirement compared with the average standard error. For normally distributed data, the bootstrap confidence limits of standard error agree with those computed analytically. In the presence of modest nonnormality in the data, however, the analytic confidence limits do not agree with the bootstrap limits. Since the bootstrap procedure is free of parametric assumptions, it is a useful tool for general application in soil science.

The automatic percentile method: Accurate confidence limits in parametric models

AbstractThe problem of constructing confidence limits for a scalar parameter is considered. Under weak conditions, Efron's accelerated bias‐corrected bootstrap confidence limits are correct to second order in parametric familles. In this article, a new method, called the automatic percentile method, for setting approximate confidence limits is proposed as an attempt to alleviate two problems inherent in Efron's method. The accelerated bias‐corrected method is not fully automatic, since it requires the calculation of an analytical adjustment; furthermore, it is typically not exact, though for many situations, particularly scalar‐parameter familles, exact answers are available. In broader generality, the proposed method is exact when exact answers exist, and it is second‐order accurate otherwise. The automatic percentile method is automatic, and for scalar parameter models it can be iterated to achieve higher accuracy, with the number of computations being linear in the number of iterations. However, when nuisance parameters are present, only second‐order accuracy seems obtainable.

On bootstrap confidence limits for possibly skew distributed statistics

Statistics for which confidence limits or tests are calculated by bootstrap techniques frequently have asymmetric distributions. Approaches based only on boot-strapped variance are inadequatein such cases. In a Mte. Carlo study with a markedly skew X2-distributed statistic an approach by Edgeworth expansions using bootstrapped estimates of variance and skewness of the statistic's distribution performed well with respect to size and power and is proposed for variaus applications.