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.

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

1987 ◽  
Vol 24 (4) ◽  
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.

Stability of extremes with random sample size

1989 ◽  
Vol 26 (04) ◽  
pp. 734-743 ◽  
Author(s):  
W. J. Voorn
Keyword(s):  
Distribution Function ◽  
Sample Size ◽  
Random Sample ◽  
Distribution Functions ◽  
Random Sample Size ◽  
Maximum Value ◽  
Arbitrary Real Constant ◽  
Independent Observations ◽  
Degenerate Distribution ◽  
Image Position

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn ) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c 2, c 3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

Stability of extremes with random sample size

1989 ◽  
Vol 26 (4) ◽  
pp. 734-743 ◽  
Author(s):  
W. J. Voorn
Keyword(s):  
Distribution Function ◽  
Sample Size ◽  
Random Sample ◽  
Distribution Functions ◽  
Random Sample Size ◽  
Maximum Value ◽  
Arbitrary Real Constant ◽  
Independent Observations ◽  
Degenerate Distribution ◽  
Image Position

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c2, c3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

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).

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.

On characterization of certain probability distributions

1972 ◽  
Vol 71 (2) ◽  
pp. 347-352 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Distribution Function ◽  
Linear Regression ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Random Variable ◽  
The Other ◽  
Linear Statistic ◽  
Binomial Distributions ◽  
Independent Observations

Introduction: Let X1, X2, …, Xn be n (n ≤ 2) independent observations on a random variable X with distribution function F. Also let L = L (X1, X2, …, Xn) be a linear statistic and Q = Q (X1, X2, …, Xn) be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statistic Q on the other statistic L. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.

CHARACTERIZATION OF PORE SIZE DISTRIBUTIONS USING NON-NEWTONIAN FLUIDS

2020 ◽  
Author(s):  
Scott C. Hauswirth ◽  
◽  
Majdi Abou Najm ◽  
Christelle Basset
Keyword(s):  
Pore Size ◽  
Newtonian Fluids ◽  
Size Distributions ◽  
Pore Size Distributions

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.

scholarly journals Size-resolved characterization of organic aerosol in the North China Plain: new insights from high resolution spectral analysis

2021 ◽  
Author(s):  
Weiqi Xu ◽  
Chun Chen ◽  
Yanmei Qiu ◽  
Conghui Xie ◽  
Yunle Chen ◽  
...  
Keyword(s):  
Radiative Forcing ◽  
North China Plain ◽  
North China ◽  
Fine Particles ◽  
Large Fraction ◽  
Organic Aerosol ◽  
Size Distributions ◽  
The North ◽  
The Impact

Organic aerosol (OA), a large fraction of fine particles, has a large impact on climate radiative forcing and human health, and the impact depends strongly on size distributions. Here we...

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