Modified Asymptotic Formulas for Critical Values of the Kolmogorov Test Statistic

1980 ◽  
Vol 34 (2) ◽  
pp. 110 ◽  
Author(s):  
H. Leon Harter
Keyword(s):  
Critical Values ◽  
Asymptotic Formulas ◽  
Test Statistic ◽  
Kolmogorov Test

Extended tables of critical values for Wilcoxon's test statistic

Biometrika ◽  
1963 ◽  
Vol 50 (1-2) ◽  
pp. 177-186 ◽  
Author(s):  
L. R. VERDOOREN
Keyword(s):  
Critical Values ◽  
Test Statistic

scholarly journals Inference on Causal and Structural Parameters using Many Moment Inequalities

2018 ◽  
Vol 86 (5) ◽  
pp. 1867-1900 ◽  
Author(s):  
Victor Chernozhukov ◽  
Denis Chetverikov ◽  
Kengo Kato
Keyword(s):  
Structural Parameters ◽  
Type Inequality ◽  
Moderate Deviation ◽  
Critical Values ◽  
Moment Inequalities ◽  
Mixing Conditions ◽  
Test Statistic ◽  
Large Classes ◽  
Deviation Inequality ◽  
Selection Of

Abstract This article considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by $p$, is possibly much larger than the sample size $n$. There is a variety of economic applications where solving this problem allows to carry out inference on causal and structural parameters; a notable example is the market structure model of Ciliberto and Tamer (2009) where $p=2^{m+1}$ with $m$ being the number of firms that could possibly enter the market. We consider the test statistic given by the maximum of $p$ Studentized (or $t$-type) inequality-specific statistics, and analyse various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (1) the union bound combined with a moderate deviation inequality for self-normalized sums, (2) the multiplier and empirical bootstraps, and (3) two-step and three-step variants of (1) and (2) by incorporating the selection of uninformative inequalities that are far from being binding and a novel selection of weakly informative inequalities that are potentially binding but do not provide first-order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with the error in size decreasing polynomially in $n$ while allowing for $p$ being much larger than $n$; indeed $p$ can be of order $\exp (n^{c})$ for some $c > 0$. Importantly, all these results hold without any restriction on the correlation structure between $p$ Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, in the online supplement, we show validity of a test based on the block multiplier bootstrap in the case of dependent data under some general mixing conditions.

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