We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal...

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal...

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.

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...