83,197 research outputs found
Normal versus Noncentral Chi-square Asymptotics of Misspecified Models
The noncentral chi-square approximation
of the distribution of the likelihood ratio (LR) test statistic is a critical part of the methodology in structural equations modeling (SEM). Recently, it was argued by some authors that in certain situations normal distributions may give a better approximation of the distribution of the LR test statistic. The main goal of this paper is to evaluate the validity of employing
these distributions in practice. Monte Carlo simulation results indicate that the noncentral chi-square distribution describes behavior of the LR test statistic well under small, moderate and even severe misspecifications regardless of the sample size (as
long as it is sufficiently large), while the normal
distribution, with a bias correction, gives a slightly better approximation for extremely severe misspecifications. However, neither the noncentral chi-square distribution nor the theoretical normal distributions give a reasonable approximation of the LR
test statistics under extremely severe misspecifications. Of
course, extremely misspecified models are not of much practical interest
Empirical Likelihood Inference for the Area Under the ROC Curve
For a continuous-scale diagnostic test, the most commonly used summary index of the receiver operating characteristic (ROC) curve is the area under the curve (AUC) that measures the accuracy of the diagnostic test. In this paper we propose an empirical likelihood approach for the inference of AUC. We first define an empirical likelihood ratio for AUC and show that its limiting distribution is a scaled chi-square distribution. We then obtain an empirical likelihood based confidence interval for AUC using the scaled chi-square distribution. This empirical likelihood inference for AUC can be extended to stratified samples and the resulting limiting distribution is a weighted sum of independent chi-square distributions. We also conduct simulation studies to compare the relative performance of the proposed empirical likelihood based interval with the existing normal approximation based intervals and bootstrap intervals for AUC
Empirical likelihood for median regression model with designed censoring variables
AbstractWe propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies
Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion
The Negative Binomial distribution becomes highly skewed under extreme
dispersion. Even at moderately large sample sizes, the sample mean exhibits a
heavy right tail. The standard Normal approximation often does not provide
adequate inferences about the data's mean in this setting. In previous work, we
have examined alternative methods of generating confidence intervals for the
expected value. These methods were based upon Gamma and Chi Square
approximations or tail probability bounds such as Bernstein's Inequality. We
now propose growth estimators of the Negative Binomial mean. Under high
dispersion, zero values are likely to be overrepresented in the data. A growth
estimator constructs a Normal-style confidence interval by effectively removing
a small, pre--determined number of zeros from the data. We propose growth
estimators based upon multiplicative adjustments of the sample mean and direct
removal of zeros from the sample. These methods do not require estimating the
nuisance dispersion parameter. We will demonstrate that the growth estimators'
confidence intervals provide improved coverage over a wide range of parameter
values and asymptotically converge to the sample mean. Interestingly, the
proposed methods succeed despite adding both bias and variance to the Normal
approximation
Two-sample Behrens--Fisher problems for high-dimensional data: a normal reference F-type test
The problem of testing the equality of mean vectors for high-dimensional data
has been intensively investigated in the literature. However, most of the
existing tests impose strong assumptions on the underlying group covariance
matrices which may not be satisfied or hardly be checked in practice. In this
article, an F-type test for two-sample Behrens--Fisher problems for
high-dimensional data is proposed and studied. When the two samples are
normally distributed and when the null hypothesis is valid, the proposed F-type
test statistic is shown to be an F-type mixture, a ratio of two independent
chi-square-type mixtures. Under some regularity conditions and the null
hypothesis, it is shown that the proposed F-type test statistic and the above
F-type mixture have the same normal and non-normal limits. It is then justified
to approximate the null distribution of the proposed F-type test statistic by
that of the F-type mixture, resulting in the so-called normal reference F-type
test. Since the F-type mixture is a ratio of two independent chi-square-type
mixtures, we employ the Welch--Satterthwaite chi-square-approximation to the
distributions of the numerator and the denominator of the F-type mixture
respectively, resulting in an approximation F-distribution whose degrees of
freedom can be consistently estimated from the data. The asymptotic power of
the proposed F-type test is established. Two simulation studies are conducted
and they show that in terms of size control, the proposed F-type test
outperforms two existing competitors. The proposed F-type test is also
illustrated by a real data example
POWER APPROXIMATIONS FOR TEST STATISTICS WITH DOMINANT COMPONENTS
Abstract: We consider approximating the power functions of some tests for several hypothesis testing problems in time series. The test statistics of interest are ratios of quadratic forms in normal variables and their power is related to the distributions of weighted sums of Chi-square random variables. Conventionally, power functions are evaluated from these distributions at each alternative, numerically, by Pearson's moment approximation, Imhof's procedure, Edgeworth-type expansion or the Monte Carlo method. In this study, we propose analytic approximations to the power functions when part of the weighted sum of Chi-square random variables can be well-approximated by a scaled Chi-square variable in distribution. In applications, the proposed analytic approximation may be obtained easily by evaluating the power only at a few alternative values. Several illustrative examples are presented and they show excellent agreement with the true power functions
Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions
In this paper, we develop a non-asymptotic local normal approximation for
multinomial probabilities. First, we use it to find non-asymptotic total
variation bounds between the measures induced by uniformly jittered
multinomials and the multivariate normals with the same means and covariances.
From the total variation bounds, we also derive a comparison of the cumulative
distribution functions and quantile coupling inequalities between Pearson's
chi-square statistic (written as the normalized quadratic form of a multinomial
vector) and its multivariate normal analogue. We apply our results to find
confidence intervals for the negative entropy of discrete distributions. Our
method can be applied more generally to find confidence intervals for strictly
convex functions of the weights of discrete distributions.Comment: 20 pages, 3 figure
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