scholarly journals On the continuation of the limit distribution of central order statistics under power normalization

2016 ◽  
Vol 43 (2) ◽  
pp. 145-155
Author(s):  
H. M. Barakat ◽  
E. M. Nigm ◽  
E. O. Abo Zaid
Keyword(s):  
Order Statistics ◽  
Limit Distribution ◽  
Power Normalization

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (1) ◽  
pp. 219-222 ◽  
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

On the Asymptotic Properties of Hogg's Q Statistic

1986 ◽  
Vol 35 (1-2) ◽  
pp. 77-84 ◽  
Author(s):  
H.N. Nagaraja
Keyword(s):  
Order Statistics ◽  
Limit Distribution ◽  
Asymptotic Theory ◽  
Asymptotic Properties ◽  
Linear Functions ◽  
Close Approximation ◽  
Optimum Proportion ◽  
Sample Data ◽  
Q Statistic ◽  
Family Of Distributions

The asymptotic properties of Qn, a measure of tail thickness introduced by Hogg, are investigated. The nondegenerate limit distribution of Qn is obtained as an application of a bivariate extension of a result on smooth linear functions of order statistics, due to Stigler. The limiting result is helpful in indicating the optimum proportion of the sample data to be used in the construction of Qn for discriminating between members of a symmetric family of distributions. The asymptotic theory is also used to find a close approximation to the expected value of Qn. This approximation works well for moderate sample sizes for distributions in the symmetric family.

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (01) ◽  
pp. 219-222
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A 1, A 2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n ), as n→ + ∞, in terms of limits of the Sk (n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X 1, X 2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

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