Glivenko-cantelli type theorems for distance functions based on the modified empirical distribution function of M. kac and for the empirical process with random sample size in general

1973 ◽  
pp. 149-164 ◽  
Author(s):  
Miklós Csörgő
Keyword(s):  
Distribution Function ◽  
Sample Size ◽  
Random Sample ◽  
Empirical Process ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Distance Functions ◽  
Random Sample Size

scholarly journals On weak convergence of empirical processes for random number of independent stochastic vectors

1973 ◽  
Vol 73 (1) ◽  
pp. 139-144 ◽  
Author(s):  
Pranab Kumar Sen
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Random Number ◽  
Empirical Process ◽  
Empirical Processes ◽  
Random Sample Size ◽  
Martingale Property

AbstractBy the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.

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.

scholarly journals Exact and Limiting Probability Distributions of Some Smirnov Type Statistics

1965 ◽  
Vol 8 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Random Sample ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Probability Distributions ◽  
Continuous Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Exact Distributions ◽  
Image Position

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variablesand and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.

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.

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