Testing equality of a large number of densities

Abstract

The problem of testing equality of a large number of densities is considered. The classical k-sample problem compares a small, fixed number of distributions and allows the sample size from each distribution to increase without bound. In our asymptotic analysis the number of distributions tends to infinity but the size of individual samples remains fixed. The proposed test statistic is motivated by the simple idea of comparing kernel density estimators from the various samples to the average of all density estimators. However, a novel interpretation of this familiar type of statistic arises upon centering it. The asymptotic distribution of the statistic under the null hypothesis of equal densities is derived, and power against local alternatives is considered. It is shown that a consistent test is attainable in many situations where all but a vanishingly small proportion of densities are equal to each other. The test is studied via simulations, and an illustration involving microarray data is provided.

Inference for high-dimensional split-plot-designs: A unified approach for small to large numbers of factor levels

“…In the current paper, we follow this approach and extend the results of [31] to general heteroscedastic split-plot designs with a independent groups of repeated measurements. To even allow for a large number of groups as in [3,4] or [39], we do not only consider the case with a fixed number a ∈ N of samples but additionally allow for situations with a → ∞. The latter case is of particular interest if most groups are rather small (as in screening trials) such that a classical test would essentially possess no power for fixed a.…”

A nonparametric test of stationarity for independent data

“…The test of stationarity proposed in this paper is an adaptation of the test of Zhan and Hart (2014). Their test is for a setting where one has a large number of small data sets and wishes to test whether all these data sets come from the same distribution.…”

“…Their test is for a setting where one has a large number of small data sets and wishes to test whether all these data sets come from the same distribution. In the current setting, one may partition the N observations into, say, p smaller data sets of equal size, and then apply the test of Zhan and Hart (2014) to these p sets. The statistic of Zhan and Hart (2014) is analogous to one proposed by Lehmann (1951).…”

“…In the current setting, one may partition the N observations into, say, p smaller data sets of equal size, and then apply the test of Zhan and Hart (2014) to these p sets. The statistic of Zhan and Hart (2014) is analogous to one proposed by Lehmann (1951). Let F i be the empirical distribution function (edf) for the ith small data set, i = 1, .…”

“…The statistic of Zhan and Hart (2014) is an analog of (1) that compares kernel density estimates rather than edfs. A number of authors, including Eubank et al (1994), Martínez-Camblor and de UñaÁlvarez (2009) and Rayner et al (2009), have made the case that goodness-of-fit tests based on density estimates are generally more powerful than ones based on edfs.…”