scholarly journals A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance

1952 ◽  
Vol 23 (2) ◽  
pp. 286-289 ◽  
Author(s):  
Sigeiti Moriguti
Keyword(s):  
Lower Bound ◽  
Continuous Distribution ◽  
Finite Variance ◽  
Absolutely Continuous

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (1) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (01) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (2) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

scholarly journals Bivariate Power-Skew-Elliptical Distribution

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1327
Author(s):  
Guillermo Martínez-Flórez ◽  
Roger Tovar-Falón ◽  
Héctor W. Gómez
Keyword(s):  
Continuous Distribution ◽  
Real Data ◽  
Bivariate Distribution ◽  
Asymptotic Covariance Matrix ◽  
Unknown Parameters ◽  
Bivariate Model ◽  
Absolutely Continuous ◽  
Sandwich Type ◽  
Ml Estimator ◽  
Special Case

In this article, we introduce a power-skew-elliptical (PSE) distribution in the bivariate setting. The new bivariate model arises in the context of conditionally specified distributions. The proposed bivariate model is an absolutely continuous distribution whose marginals are univariate PSE distributions. The special case of the bivariate power-skew-normal (BPSN) distribution is studied in details. General properties of the BPSN distribution are derived and the estimation of the unknown parameters by maximum pseudo-likelihood is discussed. Further, a sandwich type matrix, which is a consistent estimator for the asymptotic covariance matrix of the maximum likelihood (ML) estimator is determined. Two applications for real data of the proposed bivariate distribution is provided for illustrative purposes.

Absolutely continuous invariant measures for piecewise expanding C2 transformations in Rn on domains with cusps on the boundaries

1996 ◽  
Vol 16 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Kourosh Adl-Zarabi
Keyword(s):  
Lower Bound ◽  
Invariant Measure ◽  
Finite Number ◽  
Invariant Measures ◽  
Bounded Region ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Derivative Matrix ◽  
Absolutely Continuous Invariant

AbstractLet Ω be a bounded region in Rn and let be a partition of Ω into a finite number of subsets having piecewise C2 boundaries. The boundaries may contain cusps. Let τ: Ω → Ω be piecewise C2 on and expanding in the sense that there exists α > 1 such that for any i = 1, 2,…,m, where is the derivative matrix of and ‖·‖ is the euclidean matrix norm. The main result provides a lower bound on α which guarantees the existence of an absolutely continuous invariant measure for τ.

Completeness of Order Statistics

1954 ◽  
Vol 6 ◽  
pp. 42-45 ◽  
Author(s):  
D. A. S. Fraser
Keyword(s):  
Order Statistics ◽  
Continuous Distribution ◽  
Absolutely Continuous ◽  
Sufficient Statistic ◽  
Borel Sets ◽  
Uniform Distributions ◽  
General Distributions ◽  
Non Parametric

1. Introduction. Under the non-parametric assumption that a set of observations is a sample from an absolutely continuous distribution, the order statistics are known to form a complete sufficient statistic. It is proved in this note that it suffices to have the class of uniform distributions over finite numbers of intervals or the class of uniform distributions over sets of a ring which is a basis for the σ-algebra of Borel sets. This result is derived as a particular case of that of several samples from more general distributions.

scholarly journals Sharp bounds for exponential approximations under a hazard rate upper bound

2015 ◽  
Vol 52 (03) ◽  
pp. 841-850 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Exponential Distribution ◽  
Hazard Rate ◽  
Residual Life ◽  
Continuous Distribution ◽  
Rate Function ◽  
Mean Residual Life ◽  
Birth And Death Processes ◽  
Absolutely Continuous ◽  
Sharp Bounds ◽  
Life Distributions

Consider an absolutely continuous distribution on [0, ∞) with finite meanμand hazard rate functionh(t) ≤bfor allt. Forbμclose to 1, we would expectFto be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance betweenFand an exponential distribution with meanμ, as well as betweenFand an exponential distribution with failure rateb. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

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