相关论文: The Probabilities of Large Deviations for the Chi-…
We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m} \to {\lambda}$. Under mild moment conditions, the limit is Gaussian for…
We consider whether the asymptotic distributions for the log-likelihood ratio test statistic are expected to be Gaussian or chi-squared. Two straightforward examples provide insight on the difference.
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…
If a discrete probability distribution in a model being tested for goodness-of-fit is not close to uniform, then forming the Pearson chi-square statistic can involve division by nearly zero. This often leads to serious trouble in practice…
Consider $k$ independent random samples from $p$-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of $k$ covariance…
In the present paper, we consider the Pearson chi-square statistic defined on a finite alphabet which is assumed to dynamically vary as the sample size increases, and establish its moderate deviation principle.
Consider the likelihood ratio test (LRT) statistics for the independence of sub-vectors from a $p$-variate normal random vector. We are devoted to deriving the limiting distributions of the LRT statistics based on a random sample of size…
For random samples of size n obtained from p-variate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the high-dimensional setting. These test statistics have been…
For testing goodness of fit it is very popular to use either the chi square statistic or G statistics (information divergence). Asymptotically both are chi square distributed so an obvious question is which of the two statistics that has a…
Thousands of experiments are analyzed and papers are published each year involving the statistical analysis of grouped data. While this area of statistics is often perceived -- somewhat naively -- as saturated, several misconceptions still…
In this paper we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the…
The likelihood ratio test is widely used in exploratory factor analysis to assess the model fit and determine the number of latent factors. Despite its popularity and clear statistical rationale, researchers have found that when the…
It is well-known that each statistic in the family of power divergence statistics, across $n$ trials and $r$ classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio and Freeman-Tukey statistics correspond…
In this paper new families of test statistics are introduced and studied for the problem of comparing two treatments in terms of the likelihood ratio order. The considered families are based on phi-divergence measures and arise as natural…
We study confidence regions and approximate chi-squared tests for variable groups in high-dimensional linear regression. When the size of the group is small, low-dimensional projection estimators for individual coefficients can be directly…
The cumulants and moments of the log of the non-central chi-square distribution are derived. For example, the expected log of a chi-square random variable with v degrees of freedom is log(2) + psi(v/2). Applications to modeling probability…
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel…
Consider the family of power divergence statistics based on $n$ trials, each leading to one of $r$ possible outcomes. This includes the log-likelihood ratio and Pearson's statistic as important special cases. It is known that in certain…
Two--sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence…
The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed…