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.

Zbyněk Šidák, the chief editor of the Applications of Mathematics, an outstanding Czech statistician and probabilist, died on November 12, 1999, aged 66 years. This article is devoted to memory of him and outlines his life and scientific work.

This paper deals with the likelihood ratio test (LRT) for testing hypotheses on the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRT statistics under some conditions that are proved to be almost necessary. A detailed solution is given for two testing problems: the test of a single distribution against any mixture, with application to Gaussian, Poisson and binomial distributions; the test of the number of populations...

We study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime. We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

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...

In this paper, we consider a repair-cost limit replacement problem with imperfect repair and develop a graphical method to determine the optimal repair-cost limit which minimizes the expected cost per unit time in the steady-state, using the Lorenz transform of the underlying repair-cost distribution function. The method proposed can be applied to an estimation problem of the optimal repair-cost limit from empirical repair-cost data. Numerical examples are devoted to examine asymptotic properties...

In this paper, we consider a repair-cost limit replacement problem with imperfect repair and develop a graphical method to determine the optimal repair-cost limit which minimizes the expected cost per unit time in the steady-state, using the Lorenz transform of the underlying repair-cost distribution function. The method proposed can be applied to an estimation problem of the optimal repair-cost limit from empirical repair-cost data. Numerical examples are devoted to examine asymptotic properties...

We prove the Lukacs characterization of the Wishart distribution on non-octonion symmetric cones of rank greater than 2. We weaken the smoothness assumptions in the version of the Lukacs theorem of [Bobecka-Wesołowski, Studia Math. 152 (2002), 147-160]. The main tool is a new solution of the Olkin-Baker functional equation on symmetric cones, under the assumption of continuity of respective functions. It was possible thanks to the use of Gleason's theorem.

The Lukacs theorem is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on (0,∞). In 1962 Olkin and Rubin extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason...

The present paper deals with the extension of the likelihood estimation to the situation where the experimentation does not provide exact information but rather vague information.The extension process tries to achieve three fundamental objectives: the new method must be an extension of the maximum likelihood method, it has to be very simple to apply and it must allow for an interesting interpretation.These objectives are achieved herein by using the following concepts: the fuzzy information (introduced...