A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set $\Xi \subseteq {ℝ}^d$ with values in the extended convex ring is introduced. The method is based on the summary statistics – normalized intrinsic volumes densities of the $\epsilon$-parallel sets to $\Xi$. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation...

We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of an $\alpha$-mixing process $(Y,X)$, where $Y=f\left(X\right)+\xi$, $X$ represents the design and $\xi$ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that $\xi$ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework....

Explicit formulas for the non-centrality parameters of the limiting chi-square distribution of proposed multisample rank based test statistics, aimed at testing the hypothesis of the simultaneous equality of location and scale parameters of underlying populations, are obtained by means of a general assertion concerning the location-scale test statistics. The finite sample behaviour of the proposed tests is discussed and illustrated by simulation estimates of the rejection probabilities. A modification...

We consider a centered Gaussian random field X = X t : t ∈ T with values in a Banach space $$𝔹$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = convX t : t ∈ T n, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense...

We consider an estimate of the mode $\theta$ of a multivariate probability density $f$ with support in ${ℝ}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1,\cdots ,X_n$. The estimate ${\theta }_n$ is defined as any $x$ in $\{X_1,\cdots ,X_n\}$ such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that ${\theta }_n$ behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of ${\theta }_n$ follows without further work.

We consider an estimate of the mode θ of a multivariate probability density f with support in ${ℝ}^d$ using a kernel estimate fn drawn from a sample X1,...,Xn. The estimate θn is defined as any x in {X1,...,Xn} such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that θn behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of fn. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of θn follows without further work.

The equivalence of the symmetry of density of the distribution of observations and the oddness and evenness of the score-generating functions for the location and the scale problem, respectively, is established at first. Then, it is shown that the linear rank statistics with scores generated by these functions are asymptotically independent under the hypothesis of randomness as well as under contiguous alternatives in the last part of the paper. The linear and quadratic forms of these statistics...

In this paper we introduce several algorithms to generate all the vectors in the support of a multinomial distribution. Computational studies are carried out to analyze their efficiency with respect to the CPU time and to calculate their efficiency frontiers. The proposed algorithm is used to calculate exact distributions of power divergence test statistics under the hypothesis of uniformity. Finally, several exact power comparisons are done for different divergence statistics and families of alternatives...

In the article, we consider construction of prediction intervals for stationary time series using Bühlmann's [8], [9] sieve bootstrapapproach. Basic theoretical properties concerning consistency are proved. We extend the results obtained earlier by Stine [21], Masarotto and Grigoletto [13] for an autoregressive time series of finite order to the rich class of linear and invertible stationary models. Finite sample performance of the constructed intervals is investigated by computer simulations.

The aim of this paper is to establish theorems on the absolute continuity of translation as well as scale invariant statistics in general, from which the related results by Hodges-Lehmann and Puri-Sen follow. The continuity relations between the joint cdf of a random vector and its marginal cdf's are also considered.

This paper deals with the convergence in distribution of the maximum of n independent and identically distributed random variables under power normalization. We measure the difference between the actual and asymptotic distributions in terms of the double-log scale. The error committed when replacing the actual distribution of the maximum under power normalization by its asymptotic distribution is studied, assuming that the cumulative distribution function of the random variables is known. Finally,...

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.