In this paper, we investigate the problem of the conditional cumulative of a scalar response variable given a random variable taking values in a semi-metric space. The uniform almost complete consistency of this estimate is stated under some conditions. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional quantile.

In this work, we introduce a local linear estimator of the conditional mode for a random real response variable which is subject to left-truncation by another random variable where the covariate takes values in an infinite dimensional space. We first establish both of pointwise and uniform almost sure convergences, with rates, of the conditional density estimator. Then, we deduce the strong consistency of the obtained conditional mode estimator. We finally illustrate the outperformance of our method...

We study the estimation of the mean function of a continuous-time stochastic process and its derivatives. The covariance function of the process is assumed to be nonparametric and to satisfy mild smoothness conditions. Assuming that n independent realizations of the process are observed at a sampling design of size N generated by a positive density, we derive the asymptotic bias and variance of the local polynomial estimator as n,N increase to infinity. We deduce optimal sampling densities, optimal...

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd-valued processes and to N-valued processes. In the particular case where square-integrable local time does exist, it is shown that our estimator is strictly better than the local...

Let $X_i$, $1\le i\le N$, be $N$ independent random variables (i.r.v.) with distribution functions (d.f.) $F_i(x,\Theta )$, $1\le i\le N$, respectively, where $\Theta$ is a real parameter. Assume furthermore that $F_i(·,0)=F(·)$ for $1\le i\le N$. Let $R=(R_1,...,R_N)$ and $R^{+}=(R_1^{+},...,R_N^{+})$ be the rank vectors of $X=(X_1,...,X_N)$ and $\left|X\right|=\left(\right|X_1|,...,|X_N\left|\right)$, respectively, and let $V=(V_1,...,V_N)$ be the sign vector of $X$. The locally most powerful rank tests (LMPRT) $S=S\left(R\right)$ and the locally most powerful signed rank tests (LMPSRT) $S=S(R^{+},V)$ will be found for testing $\Theta =0$ against $\Theta >0$ or $\Theta <0$ with $F$ being arbitrary and with $F$ symmetric, respectively.

We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on ${[0,1]}^d$ there exists a point set $x_1,...,x_N\in {[0,1]}^d$ whose star-discrepancy with respect to μ is of order $D_N*(x_1,...,x_N;\mu )\ll \left({\left(logN\right)}^{(3d+1)/2}\right)/N$. For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy...

A solution to the marginal problem is obtained in a form of parametric exponential (Gibbs–Markov) distribution, where the unknown parameters are obtained by an optimization procedure that agrees with the maximum likelihood (ML) estimate. With respect to a difficult performance of the method we propose also an alternative approach, providing the original basis of marginals can be appropriately extended. Then the (numerically feasible) solution can be obtained either by the maximum pseudo-likelihood...

This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate the conditional hazard function. An asymptotic formula for the mean square error of this estimator is calculated considering as usual the bias and variance.

We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii,...