Let ${\theta }^{*}$ be a biased estimate of the parameter $\vartheta$ based on all observations $x_1$, $\cdots$, $x_n$ and let ${\theta }_{-i}^{*}$ ($i=1,2,\cdots ,n$) be the same estimate of the parameter $\vartheta$ obtained after deletion of the $i$-th observation. If the expectation of the estimators ${\theta }^{*}$ and ${\theta }_{-i}^{*}$ are expressed as $$\begin{array}{cc}\hfill \mathrm{E}\left({\theta }^{*}\right)& =\vartheta +a\left(n\right)b\left(\vartheta \right)\hfill \\ \hfill \mathrm{E}\left({\theta }_{-i}^{*}\right)& =\vartheta +a(n-1)b\left(\vartheta \right)i=1,2,\cdots ,n,\hfill \end{array}$$ where $a\left(n\right)$ is a known sequence of real numbers and $b\left(\vartheta \right)$ is a function of $\vartheta$, then this system of equations can be regarded as a linear model. The least squares method gives the generalized jackknife estimator. Using this method, it is possible to obtain the unbiased...

The two-sample Lepage test, devised for testing equality of the location and scale parameters against the alternative that at least for one of the parameters the equality does not hold, is extended to the general case of $k>1$ sampled populations. It is shown that its limiting distribution is the chi-square distribution with $2(k-1)$ degrees of freedom. This $k$-sample statistic is shown to yield consistent test and a formula for its noncentrality parameter under Pitman alternatives is derived. For some particular...

We present a two-dimensional linear regression model where both variables are subject to error. We discuss a model where one variable of each pair of observables is repeated. We suggest two methods to construct consistent estimators: the maximum likelihood method and the method which applies variance components theory. We study asymptotic properties of these estimators. We prove that the asymptotic variances of the estimators of regression slopes for both methods are comparable.

If in the model of measurement except useful parameters, which are to be determined, other auxiliary parameters occur as well, which were estimated from another experiment, then the type A and B uncertainties of measurement results must be taken into account. The type A uncertainty is caused by the new experiment and the type B uncertainty characterizes an accuracy of the parameters which must be used in estimation of useful parameters. The problem is to estimate of the type A uncertainty in the...

Nonlinear dynamic processes with time-varying time delays can often be encountered in industry. Time-delay estimation for nonlinear dynamic systems with time-varying time delays is an important issue for system identification. In order to estimate the dynamics of a process, a dynamic neural network with an external recurrent structure is applied in the modeling procedure. In the case where a delay is time varying, a useful way is to develop on-line time-delay estimation mechanisms to track the time-delay...

The problem of testability has been undertaken many times in the context of linear hypotheses. Almost all these considerations restricted to some algebraical conditions without reaching the nature of the problem. Therefore, a general and commonly acceptable notion of testability is still wanted. Our notion is based on a simple and natural decision theoretic requirement and is characterized in terms of the families of distributions corresponding to the null and the alternative hypothesis. Its consequences...

From the practical point of view the regression analysis and its Least Squares method is clearly one of the most used techniques of statistics. Unfortunately, if there is some problem present in the data (for example contamination), classical methods are not longer suitable. A lot of methods have been proposed to overcome these problematic situations. In this contribution we focus on special kind of methods based on trimming. There exist several approaches which use trimming off part of the observations,...

The aim of this paper is to develop two different methods for an executing of the deformation measurement and to prove that these two methods are equivalent which is a advantage for a conclusive verification of the results of the experiment in a practice.

Numerical results for a simple linear regression indicate that the non-simultaneous two-sided tolerance intervals nearly satisfy the condition of multiple-use confidence intervals, see Lee and Mathew (2002), but the numerical computation of the limits of the multiple-use confidence intervals is needed. We modified the Lieberman–Miller method (1963) for computing the simultaneous two-sided tolerance intervals in a simple linear regression with independent normally distributed errors. The suggested...