Separable Statistics and Hypothesis Testing. The Case of Small Samples

Ivchenko, G. I.; Medvedev, Yu. I.

doi:10.1137/1123092

Cited by 32 publications

(41 citation statements)

References 4 publications

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“…Steck (1957), Moms (1975), and Holst (1972and Holst ( , 1976 established the asymptotic normality of goodness-of-fit statistics for multinomial models for which no parameters are estimated. Some deficiencies in Holst's discussion of asymptotic local power properties have been corrected by Ivchenko and Medvedev (1978). Read (1984) considered small sample properties of normal and adjusted chi-squared approximations for tests of the equiprobable multinomial model.…”

Section: Introductionmentioning

confidence: 99%

Goodness-of-Fit Tests for Log-Linear Models in Sparse Contingency Tables

Koehler

1986

Journal of the American Statistical Association

104

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The asymptotic normality of the likelihood ratio goodness-offit statistic is demonstrated for testing the fit of log-linear models with closed form maximum likelihood estimates in sparse contingency tables. Unlike the traditional chi-squared theory, the number of categories in the table increases as the sample size increases, but not all of the expected frequencies are required to become large. Some results of a small Monte Car10 study are presented. The traditional chi-squared approximation is reasonably accurate for the Pearson statistic for many sparse tables, but cases are presented for which it fails. The normal approximation can be much more accurate than the chi-squared approximation for the likelihood ratio statistic, but the bias of estimated moments is a potential problem for very sparse tables.

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Section: Introductionmentioning

confidence: 99%

Goodness-of-Fit Tests for Log-Linear Models in Sparse Contingency Tables

Koehler

1986

Journal of the American Statistical Association

104

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“…By studying word counts in large documents, Baayen [4] argues that the LNRE model is well suited to natural language sources. Detection and estimation problems involving LNRE sources have been considered [15], [16], but these have not addressed the estimation of important information-theoretic quantities such as sequence probabilities, entropies, and divergences. Paninski [17] proves the existence of a consistent entropy estimator in a similar regime.…”

Section: Connections To the Literaturementioning

confidence: 99%

Probability Estimation in the Rare-Events Regime

Wagner

Viswanath

Kulkarni

2011

IEEE Trans. Inform. Theory

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We address the problem of estimating the probability of an observed string that is drawn i.i.d. from an unknown distribution. Motivated by models of natural language, we consider the regime in which the length of the observed string and the size of the underlying alphabet are comparably large. In this regime, the maximum likelihood distribution tends to overestimate the probability of the observed letters, so the Good-Turing probability estimator is typically used instead. We show that when used to estimate the sequence probability, the Good-Turing estimator is not consistent in this regime. We then introduce a novel sequence probability estimator that is consistent. This estimator also yields consistent estimators for other quantities of interest and a consistent universal classifier.

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“…The complementary hypothesis (p(y)q(y)) will be denoted by H1. [4]- [7]. From the results of [5] it follows that in the case of partition intervals which 2 are equiprobable under H the power of the , test becomes degenerate with increasing N (tends to the significance level of the test), under alternatives which approach as n --> oo the basic hypothesis with speed n -1/2, i.e., such close alternatives in the X 2 test are asymptotically indistinguishable from the basic hypothesis.…”

Section: Introductionmentioning

confidence: 99%

“…In [4] and [7] it is shown that as n, N-and n/N a > 0 the test ,,2, being asymptotically optimal in the class of all symmetric tests, distinguishes H from the alternatives approaching it only with speed n -/4. Let us stress that in the given situation under the hypothesis H the vector u has equiprobable polynomial distribution, while the X 2 statistic is symmetric.…”

Section: Introductionmentioning

confidence: 99%