Journal Article10.2307/2298005
Consistent Nonparametric Entropy-Based Testing
TL;DR: In this paper, a new class of tests for nonparametric hypotheses, with special reference to the problem of testing for independence in time series in the presence of a non-parametric marginal distribution under the null, is proposed.
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Abstract: form the test statistic. In order to obtain a normal null limiting distribution, a form of weighting is employed. The test is also shown to be consistent against a class of alternatives. The exposition focusses on testing for serial independence in time series, with a small application to testing the random walk hypothesis for exchange rate series, and tests of some other hypotheses of econometric interest are briefly described. This paper proposes a new class of tests for nonparametric hypotheses, with special reference to the problem of testing for independence in time series in the presence of a nonparametric marginal distribution under the null. The critical region of the tests is the upper tail of the distribution of an estimate of the Kullback-Leibler information criterion, whose desirable properties make it convenient to describe in a reasonably comprehensible fashion some of the consistent directions. A test is consistent against one direction of departure from the null hypothesis if the probability of rejection approaches one no matter how small the departure in that direction. For continuous distributions, two random variables Y and Z are independent if and only if their joint probability density f(y, z) equals the product of the marginal densities g(y) and h(z) for all y, z; our test for independence is consistent wherever f(y, z) deviates from g(y)h(z) by even small amounts on a set of arbitrarily small non-zero measure, providing some regularity conditions hold. It will be helpful to briefly introduce the Kullback-Leibler information criterion and describe some of its properties. Following definitions of the entropy of a distribution by Shannon (1948), Wiener (1948), a measure of information for discriminating between two hypotheses was proposed by Kullback and Leibler (1951). Let X be a p-vector-valued random variable with absolutely continuous distribution function. Consider the hypotheses HI: pdf(X) = f(x) H2: pdf(X) = g(x). The mean information for discrimination between HI and H2 per observation from f is
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Citations
Nonparametric entropy estimation. An overview
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