An approach to the construction of parametric confidence bands on cumulative distribution functions

Kanofsky, Paul; Srinivasan, R.

doi:10.1093/biomet/59.3.623

Cited by 28 publications

(16 citation statements)

References 6 publications

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“…r 14, a0) passing through (x0, Y0) such that r ~0)cC~. We mention in passing that C~ differs markedly in this respect from the two-sided band BL developed in [3] which was not full and was considerably whittled down.…”

Section: Introductionmentioning

confidence: 78%

“…The methods employed in our construction have already been outlined in [3] and [8] where a two-sided confidence band for r o) is developed. Reference may also be made to [7] where similar results are given for the exponential c.d.f., and [6] which is concerned with the Weibull c.d.f.…”

Section: Introductionmentioning

confidence: 99%

“…Using certain results of [3] it is possible to obtain a simple expression for L~ + which is much more manageable than (2.2). For this purpose consider the function The distribution of L~ + can now be obtained from (2.9) and (2.10) via Monte Carlo methods using repeated samples of size n of standard Table 1 for several commonly used values of a; the entries are believed to be accurate to two decimal places.…”

Section: Introductionmentioning

confidence: 99%

“…

SummaryBased on a random sample from the normal cumulative distribution function r a) with unknown parameters /~ and o, one-sided confidence contours for r ;/~, z), -oo ~ x ~ co, and simultaneous confidence intervals for r o)-r ;/~, ~), -co ~ x ~ y ~ co, are constructed using the method outlined in [3]. Small sample and asymptotic distributions of the relevant statistics are provided so that the construction could be completely carried out in any practical situation.

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confidence: 99%

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Further results on simultaneous confidence intervals for the normal distribution

Srinivasan

Wharton

1976

Ann Inst Stat Math

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

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confidence: 99%

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Further results on simultaneous confidence intervals for the normal distribution

Srinivasan

Wharton

1976

Ann Inst Stat Math

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“…Even if the general approach is used in a parametric setting, it makes the band unnecessarily broad in the tails. Kanofsky and Srinivasan (1972) overcome the problem under normal, exponential and uniform models by using the maximum absolute difference between the true function and an estimator of it (similar to the Kolmogorov-Smirnov statistics) and by adjusting the resulting band to obtain the required confidence level. Using the Wald statistics with (expected) Fisher information, Cheng and Iles (1983) provide an alternative general procedure that can be applied to construct simultaneous bands for any continuous function g(·; θ) of the parameters θ.…”

Section: Literature Reviewmentioning

confidence: 99%

Parametric Simultaneous Confidence Bands for Cumulative Distributions From Censored Data

Jeng

Meeker

2001

Technometrics

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This article describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function. Our results are built on extensions of previous work by Cheng and Iles for two-sided and one-sided bands, respectively. Cheng and Iles used Wald statistics with (expected) Fisher information. We consider three alternatives-Wald statistics with observed Fisher information, Wald statistics with local information, and likelihood ratio statistics. We compare standard large-sample approximate methods with simulation or bootstrap-calibrated versions of the same methods. For (log-)location-scale distributions with complete or failure (Type II) censoring, the bootstrap methods have the correct coverage probability. A simulation for the Weibull distribution and time-censored (Type I) data shows that bootstrap methods provide coverage probabilities that are closer to nominal than those based on the usual large-sample approximations. We illustrate the methods with examples from product-life analysis and nondestructive evaluation probability of detection. KeywordsBootstrap, Life data, Likelihood ratio, Probability of detection, Reliability, Simultaneous confidence band, Wald Disciplines Statistics and Probability CommentsThis preprint was published in Technometrics 43 (2001) AbstractThis paper describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function (cdf). Our results are built on extensions of previous work by Cheng and Iles (1983, 1988). Cheng and Iles use Wald statistics with (expected) Fisher information and provide different approaches to find one-sided and two-sided simultaneous confidence bands. We consider three statistics, Wald statistics with Fisher information, Wald statistics with local information, and likelihood ratio statistics. Unlike pointwise confidence intervals, it is not possible to combine two 95% one-sided simultaneous confidence bands to get a 90% two-sided simultaneous confidence band. We present a general approach for construction of two-sided simultaneous confidence bands on a cdf for a continuous parametric model from complete and censored data. Both two-sided and one-sided simultaneous confidence bands for the location-scale parameter model are discussed in detail including situations with complete and censored data.We start by using standard large-sample approximations and then extend and compare these to corresponding simulation or bootstrap calibrated versions of the same methods. The results show that bootstrap methods provide more accurate coverage probabilities than those based on the usual large sample approximations. A simulation for the Weibull distribution and Type I censored data is used to compare finite sample properties. For the location-scale model with complete or Type II censoring, the bootstrap methods are exact. Simulation results show that, with Type I censoring, a bootstrap method based on the Wald statistic with local information provid...

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Representing Uncertainty by Possibility Distributions Encoding Confidence Bands, Tolerance and Prediction Intervals

Hamed

Serrurier

Durand

2012

Lecture Notes in Computer Science

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Representing uncertainty by possibility distributions encoding confidence bands, tolerance and prediction intervals.Abstract. For a given sample set, there are already different methods for building possibility distributions encoding the family of probability distributions that may have generated the sample set. Almost all the existing methods are based on parametric and distribution free confidence bands. In this work, we introduce some new possibility distributions which encode different kinds of uncertainties not treated before. Our possibility distributions encode statistical tolerance and prediction intervals (regions). We also propose a possibility distribution encoding the confidence band of the normal distribution which improves the existing one for all sample sizes. In this work we keep the idea of building possibility distributions based on intervals which are among the smallest intervals for small sample sizes. We also discuss the properties of the mentioned possibility distributions.

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An approach to the construction of parametric confidence bands on cumulative distribution functions

Cited by 28 publications

References 6 publications

Further results on simultaneous confidence intervals for the normal distribution

Further results on simultaneous confidence intervals for the normal distribution

Parametric Simultaneous Confidence Bands for Cumulative Distributions From Censored Data

Representing Uncertainty by Possibility Distributions Encoding Confidence Bands, Tolerance and Prediction Intervals

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