Stability of invariant linearly sufficient statistics in the general Gauss-Markov model

Andrzej Kornacki

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Displaying similar documents to “Stability of invariant linearly sufficient statistics in the general Gauss-Markov model”

Characterization of the multivariate Gauss-Markoff model with singular covariance matrix and missing values

Wiktor Oktaba (1998)

Applications of Mathematics

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The aim of this paper is to characterize the Multivariate Gauss-Markoff model $(M G M)$ as in () with singular covariance matrix and missing values. $M G M D P 2$ model and completed $M G M D P 2 Q$ model are obtained by three transformations $D$ , $P$ and $Q$ (cf. ()) of $M G M$ . The unified theory of estimation (Rao, 1973) which is of interest with respect to $M G M$ has been used. The characterization is reached by estimation of parameters: scalar $σ^{2}$ and linear combination $λ^{'} \bar{B}$ ( $\bar{B} = v e c B)$ as in (), (), () as well as by the model of the form ()...

A spatial individual-based contact model with age structure

Dominika Jasińska (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The Markov dynamics of an infinite continuum birth-and-death system of point particles with age is studied. Each particle is characterized by its location $x \in ℝ^{d}$ and age $a_{x} \geq 0$ . The birth and death rates of a particle are age dependent. The states of the system are described in terms of probability measures on the corresponding configuration space. The exact solution of the evolution equation for the correlation functions of first and second orders is found.

Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model

Wiktor Oktaba (1995)

Applications of Mathematics

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By using three theorems (Oktaba and Kieloch [3]) and Theorem 2.2 (Srivastava and Khatri [4]) three results are given in formulas (2.1), (2.8) and (2.11). They present asymptotically normal confidence intervals for the determinant $| σ^{2} \sum |$ in the MGM model $(U, X B, σ^{2} \sum \otimes V)$ , $\sum > 0$ , scalar $σ^{2} > 0$ , with a matrix $V \geq 0$ . A known $n \times p$ random matrix $U$ has the expected value $E (U) = X B$ , where the $n \times d$ matrix $X$ is a known matrix of an experimental design, $B$ is an unknown $d \times p$ matrix of parameters and $σ^{2} \sum \otimes V$ is the covariance matrix of $U, \otimes$ being the symbol...

Tangential Markov inequality in $L^{p}$ norms

Agnieszka Kowalska (2015)

Banach Center Publications

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In 1889 A. Markov proved that for every polynomial p in one variable the inequality $| | p^{'} {| |}_{[- 1, 1]} {\leq (d e g p) ² | | p | |}_{[- 1, 1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs...

Different kinds of sufficiency in the general Gauss-Markov model

Andrzej Kornacki (2007)

Mathematica Slovaca

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Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables

Jean-Louis Bon, Eugen Păltănea (2006)

ESAIM: Probability and Statistics

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The paper is motivated by the stochastic comparison of the reliability of non-repairable $k$ -out-of- $n$ systems. The lifetime of such a system with nonidentical components is compared with the lifetime of a system with identical components. Formally the problem is as follows. Let $U_{i}, i = 1, . . ., n,$ be positive independent random variables with common distribution $F$ . For $λ_{i} > 0$ and $μ > 0$ , let consider $X_{i} = U_{i} / λ_{i}$ and $Y_{i} = U_{i} / μ, i = 1, . . ., n$ . Remark that this is no more than a change of scale for each term. For $k \in {1, 2, . . ., n},$ let us define $X_{k : n}$ to be the $k$ th order...

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

Existentially closed II₁ factors

Ilijas Farah, Isaac Goldbring, Bradd Hart, David Sherman (2016)

Fundamenta Mathematicae

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We examine the properties of existentially closed ( $^{ω}$ -embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ( $^{ω}$ -embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th().

On invariant, dual invariant and absolute formulas

Andrzej Mostowski

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CONTENTS Introduction..............................................................................................................................................................3 1. Lemmas concerning first order formulas.....................................................................................................5 2. Representability of recursively enumerable sets........................................................................................9 3. Simple theory of types.......................................................................................................................................10...

Iterated Boolean random varieties and application to fracture statistics models

Dominique Jeulin (2016)

Applications of Mathematics

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Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in $ℝ^{2}$ and $ℝ^{3}$ and on random planes in $ℝ^{3}$ . The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set $K$ and the Choquet capacity $T (K)$ are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical...