### ABSTRACT - Some Tests on Exponential Populations.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

A test statistic for testing goodness-of-fit of the Cauchy distribution is presented. It is a quadratic form of the first and of the last order statistic and its matrix is the inverse of the asymptotic covariance matrix of the quantile difference statistic. The distribution of the presented test statistic does not depend on the parameter of the sampled Cauchy distribution. The paper contains critical constants for this test statistic, obtained from $50\phantom{\rule{0.166667em}{0ex}}000$ simulations for each sample size considered....

The correction sonsists of deriving correct explicit formulas for MLE of parameters $\mu ,\sigma $ of the normal distribution under the hypothesis $\mu +c\sigma \le m+\delta ,\mu -c\sigma \ge m-\delta $.

In the paper a test of the hypothesis $\mu +c\sigma \le M$, $\mu -c\sigma \ge m$ on parameters of the normal distribution is presented, and explicit formulas for critical regions are derived for finite sample sizes. Asymptotic null distribution of the test statistic is investigated under the assumption, that the true distribution possesses the fourth moment.

Let the random variable $X$ have the normal distribution $N(\mu ,{\sigma}^{2})$. Explicit formulas for maximum likelihood estimator of $\mu ,\sigma $ are derived under the hypotheses $\mu +c\sigma \le m+\delta ,\mu -c\sigma \ge m-\delta $, where $c,m,\delta $ are arbitrary fixed numbers. Asymptotic distribution of the likelihood ratio statistic for testing this hypothesis is derived and some of its quantiles are presented.

In the paper it is shown that exponential families of probabilities have the quadratic derivative of the likelihood ratio, and explicit formulas for this derivative are derived.

Test statistics for testing some hypotheses on characteristic roots of covariance matrices are presented, their asymptotic distribution is derived and a confidence interval for the proportional sum of the characteristic roots is constructed. The resulting procedures are robust against violation of the normality assumptions in the sense that they asymptotically possess chosen significance level provided that the population characteristic roots are distinct and the covariance matrices of certain quadratic...

**Page 1**
Next