Let ${𝕖}_t={(e_{t1},\cdots ,e_{tp})}^{\text{'}}$ be a $p$-dimensional nonnegative strict white noise with finite second moments. Let $h_{ij}\left(x\right)$ be nondecreasing functions from $[0,\infty )$ onto $[0,\infty )$ such that $h_{ij}\left(x\right)\le x$ for $i,j=1,\cdots ,p$. Let $𝕌=\left(u_{ij}\right)$ be a $p\times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process ${𝕏}_t={(X_{t1},\cdots ,X_{tp})}^{\text{'}}$ by $$X_{tj}=u_{j1}{h}_{1j}\left(X_{t-1,1}\right)+\cdots +u_{jp}{h}_{pj}\left(X_{t-1,p}\right)+e_{tj}$$ for $j=1,\cdots ,p$. A method for estimating $𝕌$ from a realization ${𝕏}_1,\cdots ,{𝕏}_n$ is proposed. It is proved that the estimators are strongly consistent.

We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance ${\sigma }^2$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $𝔽$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $𝔽$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown....

The paper deals with sufficient conditions for the existence of general approximate minimum distance estimator (AMDE) of a probability density function $f_0$ on the real line. It shows that the AMDE always exists when the bounded $\phi$-divergence, Kolmogorov, Lévy, Cramér, or discrepancy distance is used. Consequently, $n^{-1/2}$ consistency rate in any bounded $\phi$-divergence is established for Kolmogorov, Lévy, and discrepancy estimators under the condition that the degree of variations of the corresponding...

In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when...

Real valued $M$-estimators ${\widehat{\theta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(\theta \right))$ in a statistical model with observations $Y_i\sim F_{{\theta }_0}$ are replaced by ${ℝ}^p$-valued $M$-estimators ${\widehat{\beta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(u\left(z_i^T\beta \right)\right))$ in a new model with observations $Y_i\sim F_{u\left(z_i^t{\beta }_0\right)}$, where $z_i\in {ℝ}^p$ are regressors, ${\beta }_0\in {ℝ}^p$ is a structural parameter and $u:ℝ\to ℝ$ a structural function of the new model. Sufficient conditions for the consistency of ${\widehat{\beta }}_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” ${\widehat{\theta }}_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical...

The model $y={\sum }_{j=1}^w{X}_j\beta {̲}_j+e̲$ is generalized orthogonal if the orthogonal projection matrices on the range spaces of matrices $X_j$, j = 1, ..., w, commute. Unbiased estimators are obtained for the variance components of such models with cross-nesting.

This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_i=f\left(X_i\right)+{\eta }_i$, i=1,...,n, where $X_i\in A\subset {ℝ}^d$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and ${\eta }_i$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{-1}{\sum }_{i=1}^n(f\left(X_i\right)-f{̂}_N\left(X_i\right))²$ for the estimator $f{̂}_N\left(x\right)={\sum }_{k=1}^{N}c{̂}_k{e}_k\left(x\right)$, constructed using an orthonormal system...

The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_k$, k = 0,±1,..., for the observation model $y_j=f\left(u_j\right)+{\eta }_j$, j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_j$ are independent random variables uniformly distributed in the observation interval [-T,T], ${\eta }_j$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions...