Double asymptotics for the chi-square statistic

Rempała, Grzegorz A.; Wesołowski, Jacek

doi:10.1016/j.spl.2016.09.004

Cited by 12 publications

(18 citation statements)

References 20 publications

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“…The asymptotic behavior of Pearson's chi-squared statistic has been well studied in the literature (Read and Cressie, 2012, for a review). In the high-dimensional case where the dimension (the number of bins) increases with the sample size, Rempała and Wesołowski (2016) investigate both the Poisson and Gaussian approximations for χ 2 n . Specifically, when π 0 is uniformly distributed, they show that χ 2 n is asymptotically Poisson when n/ √ d → c ∈ (0, ∞) and asymptotically Gaussian when n/ √ d → ∞.…”

Section: High-dimensional Asymptoticsmentioning

confidence: 99%

Multinomial goodness-of-fit based on U-statistics: High-dimensional asymptotic and minimax optimality

Kim

2020

Journal of Statistical Planning and Inference

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We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U -statistic for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U -statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic regime. Furthermore, we introduce a class of weights for the U -statistic that results in minimax rate optimal tests.Main results. The main results of this paper are as follows:1. Poissonian Asymptotic for the U -statistic (Section 3.1): We establish conditions under which the U -statistic has a Poisson limiting distribution.2. Gaussian Asymptotic for the U -statistic (Section 3.2): We also establish conditions under which the U -statistic has a Gaussian limiting distribution.

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Section: High-dimensional Asymptoticsmentioning

confidence: 99%

Multinomial goodness-of-fit based on U-statistics: High-dimensional asymptotic and minimax optimality

Kim

2020

Journal of Statistical Planning and Inference

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“…By using estimators a and b, the value of each parameter can be seen on table 2 below: --The use of mathematical model to predict the condition of the unit in the future certainly must involve suitability evaluation of the model (goodness of fit). Statistical method that often used to evaluate suitability of the model is Pearson's Chi-squared Test [7]. This method compares the amount of actual failure at a predetermined time interval, in this research is 6 months, with the expected amount of failure obtained from mathematical model that has been made.…”

Section: Unitmentioning

confidence: 99%

Reliability Analysis of Diesel Engine at LNG Plant using Counting Process

Darmanto

Atmanto

Permana

2019

Journal of Vocational Studies on Applied Research

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The purpose of this research is to analyze the reliability of diesel engine as driver for fire water pump. To determine the reliability level of the diesel engine, this research will apply counting process so that the rate of failure of the diesel engine can be known. The data used as basis for calculation is failure data gained from maintenance work order databases from 2012 to 2017. The data obtained will be processed using counting process method to produce mathematical modeling to predict the amount of failure to diesel engines in the future. From 4 diesel engines, only 3 parametric failure rate (l) that could be generated, for 33-GE-5A, for 33-GE-5B, and for 33-GE-5C, since 33-GE-5D was severely damaged in September 2015. The mathematical modeling will be verified using the Pearson's Chi-squared Test method to ensure the validity of the mathematical model can be guaranteed. The result of the goodness of fit test shows that only parametric failure rate (l) for 33-GE-5B and 33-GE-5C that could be accepted. The outcome mathematical model will be used to predict future behavior and failure of the unit so effective and efficient maintenance strategy for 33-GE-5B and 33-GE-5C could be applied

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“…In this paper, we are interested in the goodness-of-fit test when both n and k go to infinity, that is, we allow that k = k n changes with n and k n can be even much larger than n. We note that asymptotic normality of Pearson's chi-squared test statistics has been obtained by Tumanyan [18] and Holst [7] when n/k n → a ∈ (0, ∞) and some restrictive conditions are held. A recent work by Rempa la and Weso lowski [17] extended this scope by imposing conditions on the following decomposition of Pearson By assuming that S n2 is negligible, Rempa la and Weso lowski [17] showed that X 2 n is asymptotically normal if n 2 /k n → ∞ as n → ∞. The conditions imposed in Rempa la and Weso lowski [17] will be discussed further in Section 2.…”

mentioning

confidence: 99%

“…A recent work by Rempa la and Weso lowski [17] extended this scope by imposing conditions on the following decomposition of Pearson By assuming that S n2 is negligible, Rempa la and Weso lowski [17] showed that X 2 n is asymptotically normal if n 2 /k n → ∞ as n → ∞. The conditions imposed in Rempa la and Weso lowski [17] will be discussed further in Section 2. Since the negligibility condition is trivially true for equiprobable cells, that is, p 1 = • • • = p kn , X 2 n has a normal limit, and furthermore, Rempa la and Weso lowski [17] showed in this case that X 2 n , after properly normalized, converges in distribution to a Poisson distribution if n 2 /k n → λ ∈ (0, ∞).…”

mentioning

confidence: 99%

“…The conditions imposed in Rempa la and Weso lowski [17] will be discussed further in Section 2. Since the negligibility condition is trivially true for equiprobable cells, that is, p 1 = • • • = p kn , X 2 n has a normal limit, and furthermore, Rempa la and Weso lowski [17] showed in this case that X 2 n , after properly normalized, converges in distribution to a Poisson distribution if n 2 /k n → λ ∈ (0, ∞).…”

mentioning

confidence: 99%

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Pearson's goodness-of-fit tests for sparse distributions

Chang¹,

Li²,

Qi³

2021

Preprint

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Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say k, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with k − 1 degrees of freedom when the sample size n goes to infinity. In real applications, the number k often changes with n and may be even much larger than n. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions.We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.

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Double asymptotics for the chi-square statistic

Abstract: Consider distributional limit of the Pearson chi-square statistic when the number of classes mn increases with the sample size n and n/mn→λ. Under mild moment conditions, the limit is Gaussian for λ = ∞, Poisson for finite λ > 0, and degenerate for λ = 0.

Cited by 12 publications

References 20 publications

Multinomial goodness-of-fit based on U-statistics: High-dimensional asymptotic and minimax optimality

Multinomial goodness-of-fit based on U-statistics: High-dimensional asymptotic and minimax optimality

Reliability Analysis of Diesel Engine at LNG Plant using Counting Process

Pearson's goodness-of-fit tests for sparse distributions

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