The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO estimator, but it is also well-known that the asymptotic results can give a wrong picture of the LASSO estimator's actual finite-sample behaviour. The finite sample distribution of the LASSO estimator has been previously studied for the special case of orthogonal models....

The consistency of the least trimmed squares estimator (see Rousseeuw [Rous] or Hampel et al. [HamRonRouSta]) is proved under general conditions. The assumptions employed in paper are discussed in details to clarify the consequences for the applications.

$\sqrt{n}$-consistency of the least trimmed squares estimator is proved under general conditions. The proof is based on deriving the asymptotic linearity of normal equations.

Asymptotic normality of the least trimmed squares estimator is proved under general conditions. At the end of paper a discussion of applicability of the estimator (including the discussion of algorithm for its evaluation) is offered.

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...