Quadratic forms in order statistics used as goodness-of-fit criteria

Biometrika ◽  
1972 ◽  
Vol 59 (3) ◽  
pp. 605-611 ◽  
Author(s):  
H. O. HARTLEY ◽  
R. C. PFAFFENBERGER
Keyword(s):  
Order Statistics ◽  
Quadratic Forms ◽  
Goodness Of Fit

Order Statistics and Applications

2018 ◽  
pp. 1856-1868
Author(s):  
E. Jack Chen
Keyword(s):  
Distribution Function ◽  
Statistical Analysis ◽  
Order Statistics ◽  
Goodness Of Fit ◽  
Ranking And Selection ◽  
Quantile Estimation ◽  
Cumulative Distribution ◽  
Parametric Statistics ◽  
Goodness Of Fit Hypothesis ◽  
Tests Of Independence

Order statistics refer to the collection of sample observations sorted in ascending order and are among the most fundamental tools in non-parametric statistics and inference. Statistical inference established based on order statistics assumes nothing stronger than continuity of the cumulative distribution function of the population and is simple and broadly applicable. We discuss how order statistics are applied in statistical analysis, e.g., tests of independence, tests of goodness of fit, hypothesis tests of equivalence of means, ranking and selection, and quantile estimation. These order-statistics techniques are key components of many studies.

Pearson’s chi-squared statistics: approximation theory and beyond

Biometrika ◽  
2019 ◽  
Vol 106 (3) ◽  
pp. 716-723
Author(s):  
Mengyu Xu ◽  
Danna Zhang ◽  
Wei Biao Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Approximation Theory ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Goodness Of Fit ◽  
High Dimensional ◽  
Goodness Of Fit Test ◽  
Chi Squared

Summary We establish an approximation theory for Pearson’s chi-squared statistics in situations where the number of cells is large, by using a high-dimensional central limit theorem for quadratic forms of random vectors. Our high-dimensional central limit theorem is proved under Lyapunov-type conditions that involve a delicate interplay between the dimension, the sample size, and the moment conditions. We propose a modified chi-squared statistic and introduce an adjusted degrees of freedom. A simulation study shows that the modified statistic outperforms Pearson’s chi-squared statistic in terms of both size accuracy and power. Our procedure is applied to the construction of a goodness-of-fit test for Rutherford’s alpha-particle data.

scholarly journals Computing the exact distributions of some functions of the ordered multinomial counts: maximum, minimum, range and sums of order statistics

2019 ◽  
Vol 6 (10) ◽  
pp. 190198 ◽  
Author(s):  
Marco Bonetti ◽  
Pasquale Cirillo ◽  
Anton Ogay
Keyword(s):  
New York ◽  
Order Statistics ◽  
Goodness Of Fit ◽  
Multinomial Distribution ◽  
Exact Algorithms ◽  
Statistical Testing ◽  
New York University ◽  
Goodness Of Fit Tests ◽  
Computational Procedures ◽  
Phd Thesis

Starting from seminal neglected work by Rappeport (Rappeport 1968 Algorithms and computational procedures for the application of order statistics to queuing problems. PhD thesis, New York University), we revisit and expand on the exact algorithms to compute the distribution of the maximum, the minimum, the range and the sum of the J largest order statistics of a multinomial random vector under the hypothesis of equiprobability. Our exact results can be useful in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. We describe the algorithms, motivate their use in statistical testing and illustrate two applications. We also provide the codes and ready-to-use tables of critical values.

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