Some limit behavior for linear combinations of order statistics

Yu Miao; Mengyao Ma

Displaying similar documents to “Some limit behavior for linear combinations of order statistics”

Stress-strength based on $m$ -generalized order statistics and concomitant for dependent families

Filippo Domma, Abbas Eftekharian, Mostafa Razmkhah (2019)

Applications of Mathematics

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The stress-strength model is proposed based on the $m$ -generalized order statistics and the corresponding concomitant. For the dependency between $m$ -generalized order statistics and its concomitant, a bivariate copula expansion is considered and the stress-strength model is obtained for two special cases of order statistics and upper record values. In the particular case of copula function, the generalized Farlie-Gumbel-Morgenstern bivariate distribution function is considered with proportional...

On the distribution of a nonnegative difference between two $χ^{2}$ -distributed second degree polynomial statistics

J. K. Baksalary, R. Kala (1983)

Applicationes Mathematicae

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Asymptotic rate of convergence in the degenerate U-statistics of second order

Olga Yanushkevichiene (2010)

Banach Center Publications

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Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T (X ₁, . . ., X ₙ) = n^{- 1} \sum_{1 \leq i L e t q_{i} (i \geq 1) b e e i g e n v a l u e s o f t h e H i l b e r t - S c h m i d t o p e r a t o r a s s o c i a t e d w i t h t h e k e r n e l h (x, y), a n d q ₁ b e t h e l a r g e s t i n a b s o l u t e v a l u e o n e . W e p r o v e t h a t}$ Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12) $,$ where $G_{i}$ , 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β^{'} : = E | h (X, X ₁) | ³ + {E | h (X, X ₁) |}^{18 / 5} + E | g (X) | ³ + {E | g (X) |}^{18 / 5} + 1 < \infty$ .

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...

A Deformed Quon Algebra

Hery Randriamaro (2019)

Communications in Mathematics

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The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators $a_{i, k}$ , $(i, k) \in ℕ^{*} \times [m]$ , on an infinite dimensional vector space satisfying...

The Trends in the Usage of TeX for the Preparation of Theses and Dissertations at the Masaryk University in Brno

Vít Novotný (2015)

Zpravodaj Československého sdružení uživatelů TeXu

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This article is a statistical analysis of theses and dissertations written and defended during the period 2010-2015 at the Masaryk University in Brno (MU). The author assesses the trends in the usage of TeX and tests the hypothesis that theses and dissertations written using TeX received significantly better rating than those not written using TeX.

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

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We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$ . Restrict it to some large cube $Λ$ . Consider now $I_{Λ}$ , a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Λ$ grows to infinity. We prove that, with probability one in the...

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

Andrzej Kornacki (1997)

Applications of Mathematics

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Necessary and sufficient conditions are derived for the inclusions $J_{0} \subset J$ and $J_{0}^{*} \subset J^{*}$ to be fulfilled where $J_{0}$ , $J_{0}^{*}$ and $J$ , $J^{*}$ are some classes of invariant linearly sufficient statistics (Oktaba, Kornacki, Wawrzosek (1988)) corresponding to the Gauss-Markov models $G M_{0} = (y, X_{0} β_{0}, σ_{0}^{2} V_{0})$ and $G M = (y, X β, σ^{2} V)$ , respectively.

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let ${X_{i j}}$ , $i, j = \dots$ , be a double array of independent and identically distributed (i.i.d.) real random variables with $E X_{11} = μ$ , $E | X_{11} {- μ |}^{2} = 1$ and $E | X_{11} |^{4} l t; \infty$ . Consider sample covariance matrices (with/without empirical centering) $𝒮 = \frac{1}{n} \sum_{j = 1}^{n} (𝐬_{j} - \bar{𝐬}) {(𝐬_{j} - \bar{𝐬})}^{T}$ and $𝐒 = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j} 𝐬_{j}^{T}$ , where $\bar{𝐬} = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j}$ and $𝐬_{j} = 𝐓_{n}^{1 / 2} {(X_{1 j}, ..., X_{p j})}^{T}$ with ${(𝐓_{n}^{1 / 2})}^{2} = 𝐓_{n}$ , 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...

Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $λ ₁ \geq \dots \geq λ_{k}$ may be enumerated according to descents $λ_{i} > λ_{i + 1}$ while tracking the individual parities of $λ_{i}$ and $λ_{i + 1}$ . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations...

Моментные неравенства для обобщенных $L$ -статистик

И.С. Борисов, Е.А. Бакланов (1998)

Sibirskij matematiceskij zurnal

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Spectra of elements in the group ring of SU(2)

Alex Gamburd, Dmitry Jakobson, Peter Sarnak (1999)

Journal of the European Mathematical Society

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We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU (2)$ , providing an elementary solution of Ruziewicz problem on $S^{2}$ as well as giving many new examples of finitely generated subgroups of $SU (2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $𝐑 [SU (2)]$ in the $N$ -th irreducible representation of $SU (2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $𝐑 [SU (2)]$ , the “unfolded” consecutive...

The TeX Live Guide, version 2

Sebastian Rahtz, Michel Goossens (1997)

Zpravodaj Československého sdružení uživatelů TeXu

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A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space $𝒬$ of quadratic forms in $ℝ^{n}$ is splitted in a direct sum $𝒬_{1} \oplus ... \oplus 𝒬_{k}$ and if $X$ and $Y$ are independent random variables of $ℝ^{n}$ , assume that there exist a real number $a$ such that $E (X | X + Y) = a (X + Y)$ and real distinct numbers $b_{1}, . . ., b_{k}$ such that $E (q (X) | X + Y) = b_{i} q (X + Y)$ for any $q$ in $𝒬_{i} .$ We prove that this happens only when $k = 2$ , when $ℝ^{n}$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

A depth-based modification of the k-nearest neighbour method

Ondřej Vencálek, Daniel Hlubinka (2021)

Kybernetika

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We propose a new nonparametric procedure to solve the problem of classifying objects represented by $d$ -dimensional vectors into $K \geq 2$ groups. The newly proposed classifier was inspired by the $k$ nearest neighbour (kNN) method. It is based on the idea of a depth-based distributional neighbourhood and is called $k$ nearest depth neighbours (kNDN) classifier. The kNDN classifier has several desirable properties: in contrast to the classical kNN, it can utilize global properties of the considered...