The paper deals with the asymptotic distribution of the least squares estimator of a change point in a regression model where the regression function has two phases --- the first linear and the second quadratic. In the case when the linear coefficient after change is non-zero the limit distribution of the change point estimator is normal whereas it is non-normal if the linear coefficient is zero.

Recently Hušková (1998) has studied the least squares estimator of a change-point in gradually changing sequence supposing that the sequence increases (or decreases) linearly after the change-point. The present paper shows that the limit behavior of the change-point estimator for more complicated gradual changes is similar. The limit variance of the estimator can be easily calculated from the covariance function of a limit process.

In this paper, we study the admissibility of linear estimator of regression coefficient in linear model under the extended balanced loss function (EBLF). The sufficient and necessary condition for linear estimators to be admissible are obtained respectively in homogeneous and non-homogeneous classes. Furthermore, we show that admissible linear estimator under the EBLF is a convex combination of the admissible linear estimator under the sum of square residuals and quadratic loss function.

The aim of this paper is to characterize the Multivariate Gauss-Markoff model $\left(MGM\right)$ as in () with singular covariance matrix and missing values. $MGMDP2$ model and completed $MGMDP2Q$ model are obtained by three transformations $D$, $P$ and $Q$ (cf. ()) of $MGM$. The unified theory of estimation (Rao, 1973) which is of interest with respect to $MGM$ has been used. The characterization is reached by estimation of parameters: scalar ${\sigma }^2$ and linear combination ${\lambda }^{\text{'}}\overline{B}$ ( $\overline{B}=vecB)$ as in (), (), () as well as by the model of the form () (cf. Th. )....

For any orthogonal multi-way classification, the sums of squares appearing in the analysis of variance may be expressed by the standard quadratic forms involving only squares of the marginal and total sums of observations. In this case the forms are independent and nonnegative definite. We characterize all two-way classifications preserving these properties for some and for all of the standard quadratic forms.

In this paper we give the expression of the multiple correlation coefficient in a linear model according to the coefficients of correlation. This expression makes it possible to analyze from a numerical point of view the instability of estimates in the case of collinear explanatory variables in the linear model or in the autoregressive model. This numerical approach, that we show on two examples, thus supplements the usual approach of the quasi colinearity, founded on the statistical properties...

Employing recently derived asymptotic representation of the least trimmed squares estimator, the combinations of the forecasts with constraints are studied. Under assumption of unbiasedness of individual forecasts it is shown that the combination without intercept and with constraint imposed on the estimate of regression coefficients that they sum to one, is better than others. A numerical example is included to support theoretical conclusions.

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

A formula for evaluation of the distribution of a linear combination of independent inverted gamma random variables by one-dimensional numerical integration is presented. The formula is direct application of the inversion formula given by Gil–Pelaez [gil-pelaez]. This method is applied to computation of the generalized $p$-values used for exact significance testing and interval estimation of the parameter of interest in the Behrens–Fisher problem and for variance components in balanced mixed linear...