The contents of the paper is concerned with the two-sample problem where $F_m\left(x\right)$ and $G_n\left(x\right)$ are two empirical distribution functions. The difference $F_m\left(x\right)-G_n\left(x\right)$ changes only at an $x_i,i=1,2,...,m+n$, corresponding to one of the observations. Let $R_{mn}^{+}\left(j\right)$ denote the subscript $i$ for which $F_m\left(x_i\right)-G_n\left(x_i\right)$ achieves its maximum value $D_{mn}^{+}$ for the $j$th time $(j=1,2,...)$. The paper deals with the probabilities for $R_{mn}^{+}\left(j\right)$ and for the vector $(D_{mn}^{+},R_{mn}^{+}\left(j\right))$ under $H_0:F=G$, thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model.

An asymptotic local power of Pearson chi-squared tests is considered, based on convex mixtures of the null densities with fixed alternative densities when the mixtures tend to the null densities for sample sizes $n\to \infty .$ This local power is used to compare the tests with fixed partitions $𝒫$ of the observation space of small partition sizes $\left|𝒫\right|$ with the tests whose partitions $𝒫={𝒫}_n$ depend on $n$ and the partition sizes $|{𝒫}_n|$ tend to infinity for $n\to \infty$. New conditions are presented under which it is asymptotically optimal...

For the Bickel-Rosenblatt goodness-of-fit test with fixed bandwidth studied by Fan (1998) we derive its Bahadur exact slopes in a neighbourhood of a simple hypothesis f = f0 and we use them to get a better understanding on the role played by the smoothing parameter in the detection of departures from the null hypothesis. When f0 is an univariate normal distribution and we take for kernel the standard normal density function, we compute these slopes for a set of Edgeworth alternatives which give...

We discuss two families of tests for normality based on characterizations of continuous distributions via order statistics and record values. Simulations of their powers show that they are competitive to widely recommended tests in the literature.

This paper is concerned with the properties of two statistics based on the logarithms of disjoint m-spacings. The asymptotic normality is established in an elementary way and exact and asymptotic means and variances are computed in the case of uniform distribution on the interval [0,1]. This result is generalized to the case when the sample is drawn from a distribution with positive step density on [0,1]. Bahadur approximate efficiency of tests based on those statistics is found for such alternatives....

In this paper we deepen the study of the nonlinear principal components introduced by Salinelli in 1998, referring to a real random variable. New insights on their probabilistic and statistical meaning are given with some properties. An estimation procedure based on spline functions, adapting to a statistical framework the classical Rayleigh–Ritz method, is introduced. Asymptotic properties of the estimator are proved, providing an upper bound for the rate of convergence under suitable mild conditions....