Process capability index Cpk has been widely used in the manufacturing industry as a process performance measure. In this paper, we investigate the natural estimator of the index Cpk, and show that under the assumption of normality its distribution can be expressed as a mixture of the chi-square and the normal distributions. We also implement the theory of hypothesis testing using the natural e...

With the availability of high-throughput microarray technologies, investigators can simultaneously measure the expression levels of many thousands of genes in a short period. Although there are rich statistical methods for analyzing microarray data in the literature, limited work has been done in mapping expression quantitative trait loci (eQTL) that influence the variation in levels of gene ex...

The standard variance components method for mapping quantitative trait loci is derived on the assumption of normality. Unsurprisingly, statistical tests based on this method do not perform so well if this assumption is not satisfied. We use the statistical concept of copulas to relax the assumption of normality and derive a test that can perform well under any distribution of the continuous tra...

The problem, whether every topological space has a weak compact reflection, was answered by M. Hušek in the negative. Assuming normality, M. Hušek fully characterized the spaces having a weak reflection in compact spaces as the spaces with the finite Wallman remainder. In this paper we prove that the assumption of normality may be omitted. On the other hand, we show that some covering propertie...

The paper discusses the contributions Student (W. S. Gosset) made to the three stages in which small-sample methodology was established in the period 1908–1033: (i) the distributions of the test-statistics under the assumption of normality; (ii) the robustness of these distributions against non-normality; (iii) the optimal choice of test statistics. The conclusions are based on a careful readin...

The exact confidence interval for is hypersensitive tominor violations of the normality assumption and its performance does not improvewith increasing sample size.An approximate confidence interval for is proposed and is shown to be nearly exact under normality with excellent small-sample properties under moderate nonnormality. The small-sample performance of the proposed interval may be furthe...

We propose U -statistics-based tests for null variance components in linear mixed models. We obtain their asymptotic distribution (for increasing number of units) under mild regularity conditions that include only the existence of the second moment for the distribution of the random effects and of the fourth moment for the distribution of the conditional errors. We also derive their nonnull dis...