Advanced Search

The Minimum Norm Quadratic Unbiased Invariant Estimator of the estimable linear function of the unknown variance-covariance component parameter θ in the linear model with given linear restrictions of the type Rθ = c is derived in two special structures: replicated and growth-curve model.

The paper deals with the estimation of unknown vector parameter of mean and scalar parameters of variance as well in two-stage linear model, which is a special type of mixed linear model. The necessary and sufficient condition for the existence of uniformly best unbiased estimator of parameter of means is given. The explicite formulas for these estimators and for the estimators of the parameters of variance as well are derived.

The paper deals with a linear model with linear variance-covariance structure, where the linear function of the parameter of expectation is to be estimated. The two-stage estimator is based on the observation of the vector $Y$ and on the invariant quadratic estimator of the variance-covariance components. Under the assumption of symmetry of the distribution and existence of finite moments up to the tenth order, an approach to determining the upper bound for the difference in variances of the estimators...

The paper deals with the estimation of the unknown vector parameter of the mean and the parameters of the variance in the general $n$-stage linear model. Necessary and sufficient conditions for the existence of the uniformly minimum variance unbiased estimator (UMVUE) of the mean-parameter under the condition of normality are given. The commonly used least squares estimators are used to derive the expressions of UMVUE-s in a simple form.

Let $𝐘$ be an $n$-dimensional random vector which is $N_n(𝐀\mathbf{0},𝐊)$ distributed. A minimum variance unbiased estimator is given for $f\left(o\right)$ provided $f$ is an unbiasedly estimable functional of an unknown $k$-dimensional parameter $\mathbf{0}$.

The aim of the paper is to estimate a function $\gamma =tr\left(D\beta {\beta }^{\text{'}}\right)+tr(C\sum )$ (with $d,C$ known matrices) in a regression model $(Y,X\beta ,\sum )$ with an unknown parameter $\beta$ and covariance matrix $\sum$. Stochastically independent replications $Y_1,...,Y_m$ of the stochastic vector $Y$ are considered, where the estimators of $X\beta$ and $\sum$ are $\overline{Y}=\frac{1}{m}{\sum }_{i=1}^m{Y}_i$ and $\widehat{\sum }={(m-1)}^{-1}{\sum }_{i=1}^m(Y_i-\overline{Y}){(Y_i-\overline{Y})}^{\text{'}}$, respectively. Locally and uniformly best inbiased estimators of the function $\gamma$, based on $\overline{Y}$ and $\widehat{\sum }$, are given.

The paper deals with an optimal estimation of the quadratic function ${\beta }^{\text{'}}𝐃\beta$, where $\beta \in {ℛ}^k,𝐃$ is a known $k\times k$ matrix, in the model $𝐘,𝐗\beta ,{\sigma }^{\mathbf{2}}𝐈$. The distribution of $𝐘$ is assumed to be symmetric and to have a finite fourth moment. An explicit form of the best unbiased estimator is given for a special case of the matrix $𝐗$.

The MINQUE of the linear function ${\int }^{\text{'}}\vartheta$ of the unknown variance-components parameter $\vartheta$ in mixed linear model under linear restrictions of the type $𝐑\vartheta =c$ is defined and derived. As an illustration of this estimator the example of the one-way classification model with the restrictions ${\vartheta }_1=k{\vartheta }_2$, where $k\ge 0$, is given.