Asymptotic Minimaxity of Chi-Square Tests

Ermakov, Mikhail

doi:10.1137/s0040585x97976441

Cited by 18 publications

(42 citation statements)

References 21 publications

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“…Note that the same situation takes place in the case of chi-squared tests as well (see Ermakov [11]). …”

Section: It Holdsmentioning

confidence: 79%

“…Thus it seems natural to consider the testing nonparametric hypotheses from the distance positions and to develop rigorous justification of this approach. From viewpoint of asymptotic minimaxity such an argumentation has been developed in Ermakov [10,11] in the case of standard goodness-of-fit tests. These results are based on the interpretation of test statistics of Kolmogorov, omega-square and chi-squared tests as the corresponding norms or seminorms (in the case of chi-squared tests) N n (F n − F 0 ) depending on a difference of empirical distribution functionF n of independent sample X 1 , .…”

Section: Dy (T) = S(t)dt + Q(t)dw(t)mentioning

confidence: 99%

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On Asymptotic Minimaxity of Kernel-based Tests

Ermakov

2003

ESAIM: PS

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Abstract. In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the L2-norms of signal smoothed by the kernels exceed some constants ρ > 0. The constant ρ depends on the power of noise and ρ → 0 as → 0. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended to the problems of testing nonparametric hypotheses on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained for the problems of testing parametric hypotheses versus nonparametric sets of alternatives.Mathematics Subject Classification. 62G10, 62G20.

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“…Note that the same situation takes place in the case of chi-squared tests as well (see Ermakov [11]). …”

Section: It Holdsmentioning

confidence: 79%

Section: Dy (T) = S(t)dt + Q(t)dw(t)mentioning

confidence: 99%

On Asymptotic Minimaxity of Kernel-based Tests

Ermakov

2003

ESAIM: PS

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“…For the case of growing alphabets, the existence of tests for the simple-versus-composite problem is studied by Barron [9], Paninski [10] and Ermakov [11]. The works [9], [10] also address the converse problem of determining the the smallest growth rate beyond which (respectively) uniformly exponentially consistent and consistent tests do not exist.…”

Section: B Related Workmentioning

confidence: 99%

Classification of Homogeneous Data With Large Alphabets

Kelly

Wagner

Tularak

et al. 2013

IEEE Trans. Inform. Theory

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Given training sequences generated by two distinct, but unknown, distributions sharing a common alphabet, we study the problem of determining whether a third test sequence was generated according to the first or second distribution using only the training data. To better model sources such as natural language, for which the underlying distributions are difficult to learn, we allow the alphabet size to grow and therefore the probability distributions to change with the blocklength. Our primary focus is the situation in which the underlying probabilities are all of the same order, and in this regime we give conditions on the alphabet growth rate and distributions guaranteeing the existence of universally consistent tests, i.e. tests having a probability of error tending to zero with the blocklength for any underlying distributions. We show that some commonly used statistical tests are universally consistent provided the alphabet is sub-linear but these tests are inconsistent for linear growth rates. We then propose a classifier that is universally consistent with up-to quadratic alphabet growth and that no classifier can handle the case in which the alphabet grows quadratically or faster. If the tester is given the underlying distributions in place of the training data, we prove that consistent testing is possible regardless of the growth of the underlying alphabet. Our results are then used to illuminate the problem of classifying arbitrary (i.e. non-homogeneous) distributions on growing alphabets.

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“…[4] Pearson's test statistic is asymptotically χ 2 -distributed; [5] When ε m −0.5 (CLT analysis), Pearson's chi-square test is asymptotically minimax; [1] GLRT has optimal error exponents for fixed m; [6] There exists a test with nonzero classical error exponents (see (2)) if and only if m = O(n).…”

Section: Related Workmentioning

confidence: 99%

“…2 ), and CLT analysis is applied, Pearson's chi-square test has been shown to be asymptotically minimax [5]. In the asymptotic minimaxity, two tests are compared using the absolute difference between probabilities of error; in generalized error exponents, a finer comparison using the ratio between probabilities of error is considered, and Pearson's chi-square test is not optimal:…”

Section: Pearson's Test Is Not Optimalmentioning

confidence: 99%

Error exponents for composite hypothesis testing with small samples

Huang

Meyn

2012

2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We consider the small sample composite hypothesis testing problem, where the number of samples n is smaller than the size of the alphabet m. A suitable model for analysis is the high-dimensional model in which both n and m tend to infinity, and n = o(m). We propose a new performance criterion based on large deviation analysis, which generalizes the classical error exponent applicable for large sample problems (in which m = O(n)). The results are:(i) The best achievable probability of error P e decays as − log(P e ) = (n 2 /m)(1 + o (1)

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Asymptotic Minimaxity of Chi-Square Tests

Cited by 18 publications

References 21 publications

On Asymptotic Minimaxity of Kernel-based Tests

On Asymptotic Minimaxity of Kernel-based Tests

Classification of Homogeneous Data With Large Alphabets

Error exponents for composite hypothesis testing with small samples

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