scholarly journals Strong consistencies of the bootstrap moments

1991 ◽  
Vol 14 (4) ◽  
pp. 797-802 ◽  
Author(s):  
Tien-Chung Hu
Keyword(s):  
Positive Integer ◽  
Sample Size ◽  
Real Number ◽  
Random Variables ◽  
Random Variable ◽  
Bootstrap Sample ◽  
Data Set ◽  
Sample Moment ◽  
Almost All ◽  
Independent Identically Distributed

LetXbe a real valued random variable withE|X|r+δ<∞for some positive integerrand real number,δ,0<δ≤r, and let{X,X1,X2,…}be a sequence of independent, identically distributed random variables. In this note, we prove that, for almost allw∈Ω,μr;n*(w)→μrwith probability1. iflimn→∞infm(n)n−β>0for someβ>r−δr+δ, whereμr;n*is the bootstraprthsample moment of the bootstrap sample some with sample sizem(n)from the data set{X,X1,…,Xn}andμris therthmoment ofX. The results obtained here not only improve on those of Athreya [3] but also the proof is more elementary.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (02) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, Xn :n. If the Xi 's have a geometric distribution then the conditional distribution of Xk +1:n – Xk :n given Xk+ 1:n – Xk :n &gt; 0 is the same as the distribution of X 1:n–k . Also the random variable X 2:n – X 1:n is independent of the event [X 1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

A characterization of the geometric distribution

1983 ◽  
Vol 20 (01) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, X n:n . We prove that if the random variable X2:n – X 1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k &gt; m &gt; 1, then the Xi 's are geometric. This is related to a characterization problem raised by Arnold (1980).

A characterization of the geometric distribution

1983 ◽  
Vol 20 (1) ◽  
pp. 209-212 ◽  
Author(s):  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Characterization Problem ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (2) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. If the Xi's have a geometric distribution then the conditional distribution of Xk+1:n – Xk:n given Xk+1:n – Xk:n > 0 is the same as the distribution of X1:n–k. Also the random variable X2:n – X1:n is independent of the event [X1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

1987 ◽  
Vol 24 (04) ◽  
pp. 838-851 ◽  
Author(s):  
W. J. Voorn
Keyword(s):  
Positive Integer ◽  
Sample Size ◽  
Random Variable ◽  
Logistic Distribution ◽  
Size Distributions ◽  
Random Sample Size ◽  
Maximum Value ◽  
Maximum Stability ◽  
Independent Observations

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

scholarly journals Improving the Asmussen–Kroese-Type Simulation Estimators

2012 ◽  
Vol 49 (4) ◽  
pp. 1188-1193 ◽  
Author(s):  
Samim Ghamami ◽  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Nonnegative Integer ◽  
Rare Event ◽  
Random Variables ◽  
Random Variable ◽  
Heavy Tailed ◽  
Independent Identically Distributed

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Perpetuities with thin tails

1996 ◽  
Vol 28 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Charles M. Goldie ◽  
Rudolf Grübel
Keyword(s):  
Difference Equations ◽  
Random Variables ◽  
Random Variable ◽  
Probabilistic Algorithms ◽  
Stochastic Difference Equations ◽  
Tail Behaviour ◽  
Independent Identically Distributed

We investigate the behaviour of P(R ≧ r) and P(R ≦ −r) as r → ∞for the random variable where is an independent, identically distributed sequence with P(− 1 ≦ M ≦ 1) = 1. Random variables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic algorithms and elsewhere. Exponential and Poissonian tail behaviour can arise.

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