Goodness-of-fit test statistics that dominate the Kolmogorov statistics

Berk, Robert H.; Jones, D. H.

doi:10.1007/bf00533250

Cited by 139 publications

(133 citation statements)

References 10 publications

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“…These distribution-free test studied by Berk and Jones (1979). In this case, the test statistic is given by…”

Section: Introductionmentioning

confidence: 99%

Optimal distribution-free confidence bands for a distribution function

Frey

2008

Journal of Statistical Planning and Inference

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Distribution-free confidence bands for a distribution function are typically obtained by inverting a distribution-free hypothesis test. We propose an alternate strategy in which the upper and lower bounds of the confidence band are chosen to minimize a narrowness criterion. We derive necessary and sufficient conditions for optimality with respect to such a criterion, and we use these conditions to construct an algorithm for finding optimal bands. We also derive uniqueness results, with the Brunn-Minkowski Inequality from the theory of convex bodies playing a key role in this work. We illustrate the optimal confidence bands using some galaxy velocity data, and we also show that the optimal bands compare favorably to other bands both in terms of power and in terms of area enclosed.

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“…These distribution-free test studied by Berk and Jones (1979). In this case, the test statistic is given by…”

Section: Introductionmentioning

confidence: 99%

Optimal distribution-free confidence bands for a distribution function

Frey

2008

Journal of Statistical Planning and Inference

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“…Since this estimator is well-known and widely used to estimate the survivor function when right censoring occurs, the resulting exact confidence bands should have widespread applicability. The method we describe relies on inverting a modified version of the likelihood-based test statistic first proposed by Berk and Jones [2]. Although the optimality properties of the Berk-Jones statistic are no longer assured, because our method is likelihood-based, the resulting confidence bands inherit the familiar, attractive properties of the likelihood approach -they are range preserving, transformation invariant and their "shape," that is, their symmetry or asymmetry, is determined by the data.…”

Section: Discussionmentioning

confidence: 99%

“…His approach involves inverting a nonparametric likelihood test of uniformity first described by Berk and Jones [2], using the recursive algorithm of Noé [3] to calculate joint probabilities from the exact null distribution of the test statistic for all sample sizes up to 1,000. An attractive feature of Owen's method is that the BerkJones test statistic on which his approach relies is more efficient (in Bahadur's sense) than any weighted Kolmogorov-Smirnov test statistic at any alternative to the uniform distribution.…”

Section: Introductionmentioning

confidence: 99%

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Exact Nonparametric Confidence Bands for the Survivor Function

Matthews

2013

The International Journal of Biostatistics

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A method to produce exact simultaneous confidence bands for the empirical cumulative distribution function that was first described by Owen, and subsequently corrected by Jager and Wellner, is the starting point for deriving exact nonparametric confidence bands for the survivor function of any positive random variable. We invert a nonparametric likelihood test of uniformity, constructed from the KaplanMeier estimator of the survivor function, to obtain simultaneous lower and upper bands for the function of interest with specified global confidence level. The method involves calculating a null distribution and associated critical value for each observed sample configuration. However, Noe recursions and the Van Wijngaarden-Decker-Brent root-finding algorithm provide the necessary tools for efficient computation of these exact bounds. Various aspects of the effect of right censoring on these exact bands are investigated, using as illustrations two observational studies of survival experience among non-Hodgkin's lymphoma patients and a much larger group of subjects with advanced lung cancer enrolled in trials within the North Central Cancer Treatment Group. Monte Carlo simulations confirm the merits of the proposed method of deriving simultaneous interval estimates of the survivor function across the entire range of the observed sample.

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“…Berk and Jones (1979) proposed an intuitively appealing method of testing the simple goodness of fit null hypothesis F = F 0 for some specified continuous F 0 in the one dimensional iid situation. It is also based on the empirical CDF, and quite a bit of useful work has been done on finite sample distributions of the Berk-Jones test statistic.…”

Section: Some Important Inequalitiesmentioning

confidence: 99%

Classical Nonparametrics

Dasgupta

Asymptotic Theory of Statistics and Probability

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The development of nonparametrics originated from a concern about the approximate validity of parametric procedures based on a specific narrow model when the model is questionable. Procedures which are reasonably insensitive to the exact assumptions that one makes are called robust. Such assumptions may be about a variety of things. They may be about an underlying common density assuming that the data are iid; they may be about the dependence structure of the data itself; in regression problems, they may be about the form of the regression function, etc. For example, if we assume that our data are iid from a certain N (θ, 1) density, then we have a specific parametric model for our data. Statistical models are always, at best, an approximation. We do not believe that the normal model is the correct model. There is a trade-off.As a simple example, consider the t-test for the mean µ of a normal distribution.If normality holds, then under the null hypothesis, H 0 : µ = µ 0 ,for all n, µ 0 , and σ. However, if the population is not normal, neither the size nor the power of the t-test remains the same as under the normal case. If these hange substantially, we have a robustness problem. However, as we will later see, by making a minimal number of assumptions (specifically, no parametric assumptions) we can develop procedures with some sort of a safety net. Such methods would qualify for being called nonparametric methods.

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Goodness-of-fit test statistics that dominate the Kolmogorov statistics

Cited by 139 publications

References 10 publications

Optimal distribution-free confidence bands for a distribution function

Optimal distribution-free confidence bands for a distribution function

Exact Nonparametric Confidence Bands for the Survivor Function

Classical Nonparametrics

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