Convergence rate of perturbed empirical distribution functions

1979 ◽  
Vol 16 (01) ◽  
pp. 163-173 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Distribution Function ◽  
Convergence Rate ◽  
Uniform Distribution ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Unit Mass ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Empirical Distribution Functions ◽  
Non Parametric

Given an i.i.d. sequenceX1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator ofFis the e.d.f.Fn;whereUois the d.f. of the unit mass at zero. Anadmissible perturbation of the e.d.f., say, is obtained ifUois replaced by a d.f., whereis a sequence of d.f.'s converging weakly toUo.Suchperturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as. It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then‘has the Chung–Smirnov property'; i.e., with probability one,1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution andHnis the d.f. of the unit mass atn–αthen the above lim sup is ≦ 1 or = ∞, depending on whetheror

Convergence rate of perturbed empirical distribution functions

1979 ◽  
Vol 16 (1) ◽  
pp. 163-173 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Distribution Function ◽  
Convergence Rate ◽  
Uniform Distribution ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Unit Mass ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Empirical Distribution Functions ◽  
Non Parametric

Given an i.i.d. sequence X1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or

scholarly journals Bounds for distribution functions of sums of squares and radial errors

1991 ◽  
Vol 14 (3) ◽  
pp. 561-569 ◽  
Author(s):  
Roger B. Nelsen ◽  
Berthold Schweizer
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Sum Of Squares ◽  
Distribution Functions ◽  
Probable Error ◽  
Important Case ◽  
Sums Of Squares ◽  
Radial Error ◽  
Common Distribution ◽  
Common Distribution Function

Bounds are found for the distribution function of the sum of squaresX2+Y2whereXandYare arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible whenXandYare symmetric about0and yield expressions which can be evaluated explicitly whenXandYhave a common distribution function which is concave on(0,∞). Similar results are obtained for the radial error(X2+Y2)½. The important case whereXandYare normally distributed is discussed, and here best-possible bounds on the circular probable error are also obtained.

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

scholarly journals On a probability problem connected with Railway traffic

1991 ◽  
Vol 4 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Railway Traffic ◽  
Empirical Distribution Functions

Let Fn(x) and Gn(x) be the empirical distribution functions of two independent samples, each of size n, in the case where the elements of the samples are independent random variables, each having the same continuous distribution function V(x) over the interval (0,1). Define a statistic θn by θn/n=∫01[Fn(x)−Gn(x)]dV(x)−min0≤x≤1[Fn(x)−Gn(x)]. In this paper the limits of E{(θn/2n)r}(r=0,1,2,…) and P{θn/2n≤x} are determined for n→∞. The problem of finding the asymptotic behavior of the moments and the distribution of θn as n→∞ has arisen in a study of the fluctuations of the inventory of locomotives in a randomly chosen railway depot.

On probability properties of measures of random sets and the asymptotic behavior of empirical distribution functions

1968 ◽  
Vol 5 (1) ◽  
pp. 196-202 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Empirical Distribution ◽  
Asymptotic Properties ◽  
Practical Importance ◽  
Distribution Functions ◽  
Random Sets ◽  
Derived Properties ◽  
Asymptotic Distribution Function ◽  
Empirical Distribution Functions

Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.

On the comparison of a theoretical and an empirical distribution function

1971 ◽  
Vol 8 (2) ◽  
pp. 321-330 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Mutually Independent

Let ξ1, ξ2, ···, ξm be mutually independent random variables having a common distribution function P{ξr≦x} = F(x)(r = 1, 2, ···, m). Let Fm(x) be the empirical distribution function of the sample (ξ1, ξ2, ···, ξm), that is, Fm(x) is defined as the number of variables ≦x divided by m.

On the comparison of a theoretical and an empirical distribution function

1971 ◽  
Vol 8 (02) ◽  
pp. 321-330 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Mutually Independent

Let ξ 1 , ξ2, ···, ξm be mutually independent random variables having a common distribution function P {ξ r ≦x} = F(x)(r = 1, 2, ···, m). Let Fm (x) be the empirical distribution function of the sample (ξ 1, ξ 2 , ···, ξm), that is, Fm (x) is defined as the number of variables ≦x divided by m.

On probability properties of measures of random sets and the asymptotic behavior of empirical distribution functions

1968 ◽  
Vol 5 (01) ◽  
pp. 196-202 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Empirical Distribution ◽  
Asymptotic Properties ◽  
Practical Importance ◽  
Distribution Functions ◽  
Random Sets ◽  
Derived Properties ◽  
Asymptotic Distribution Function ◽  
Empirical Distribution Functions

Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

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