Let $X_j,1\le j\le N$, be independent random $p$-vectors with respective continuous cumulative distribution functions $F_{j}1\le j\le N$. Define the $p$-vectors $R_j$ by setting $R_{ij}$ equal to the rank of $X_{ij}$ among $X_{ij},...,X_{iN},1\le i\le p,1\le j\le N$. Let $a^{\left(N\right)}(.)$ denote a multivariate score function in $R_p$, and put $S={\sum }_{j=1}^N{c}_j{a}^{\left(N\right)}\left(R_j\right)$, the $c_j$ being arbitrary regression constants. In this paper the asymptotic distribution of $S$ is investigated under various sets of conditions on the constants, the score functions, and the underlying distribution functions. In particular, asymptotic normality of $S$ is established...

We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins–Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard algorithms fail because the expected-value function grows too fast. In this work, we give a self contained proof of a central limit theorem for this algorithm under local assumptions on the expected-value function, which are fairly...

In this paper we consider a kernel estimator of a density in a convolution model and give a central limit theorem for its integrated square error (ISE). The kernel estimator is rather classical in minimax theory when the underlying density is recovered from noisy observations. The kernel is fixed and depends heavily on the distribution of the noise, supposed entirely known. The bandwidth is not fixed, the results hold for any sequence of bandwidths decreasing to 0. In particular the central limit...

We build a kernel estimator of the Markovian transition operator as an endomorphism on L¹ for some discrete time continuous states Markov processes which satisfy certain additional regularity conditions. The main result deals with the asymptotic normality of the kernel estimator constructed.

We study the limit behavior of certain classes of dependent random sequences (processes) which do not possess the Markov property. Assuming these processes depend on a control parameter we show that the optimization of the control can be reduced to a problem of nonlinear optimization. Under certain hypotheses we establish the stability of such optimization problems.

The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807–832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897–906]. First, we study the case of mixtures of autoregressive models (i.e. independent...

The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807–832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897–906]. First, we study the case of mixtures of autoregressive models (i.e. independent...

We consider a multivariate regression (growth curve) model of the form $Y=XBZ+\epsilon$, $E\epsilon =0$, $var(vec\epsilon )=W\otimes \Sigma$, where $W={\sum }_{i=1}^k{\theta }_i{V}_i$ and ${\theta }_i$’s are unknown scalar covariance components. In the case of replicated observations, we derive the explicit form of the locally best estimators of the covariance components under normality and asymptotic confidence ellipsoids for certain linear functions of the first order parameters $\left\{B_{ij}\right\}$ estimating simultaneously the first and the second order parameters.

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a step-wise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity...

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T(X₁,...,Xₙ)=n^{-1}{\sum }_{1\le iLet{q}_i\left(i\ge 1\right)beeigenvaluesoftheHilbert-Schmidtoperatorassociatedwiththekernelh(x,y),andq₁bethelargestinabsolutevalueone.Weprovethat}$Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12)$,$where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and ${\beta }^{\text{'}}:=E\left|h(X,X₁)\right|³+{E\left|h(X,X₁)\right|}^{18/5}+E\left|g\left(X\right)\right|³+{E\left|g\left(X\right)\right|}^{18/5}+1<\infty$.

The approach introduced in Janžura [Janzura 1997] is further developed and the asymptotic Rényi distances are studied mostly from the point of their monotonicity properties. The results are applied to the problems of statistical inference.