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 Some Limit Theorems Involving the Empirical Distribution Function

1967 ◽  
Vol 10 (5) ◽  
pp. 739-741
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Mutually Independent

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that isFn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.Using a theorem of Manija [4], we proved among others the following statement in [1].

A New Proof of Some Results of Rényi and the Asymptotic Distribution of the Range of his Kolmogorov-Smirnov Type Random Variables

1967 ◽  
Vol 19 ◽  
pp. 550-558 ◽  
Author(s):  
Miklós Csörgö
Keyword(s):  
Distribution Function ◽  
Asymptotic Distribution ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Kolmogorov Smirnov ◽  
Mutually Independent

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.

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 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.

scholarly journals Limiting behavior of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables

1997 ◽  
Vol 10 (1) ◽  
pp. 3-20 ◽  
Author(s):  
Shan Sun ◽  
Ching-Yuan Chiang
Keyword(s):  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Limiting Behavior ◽  
U Statistics ◽  
Strongly Mixing ◽  
Almost Sure Representation ◽  
Mixing Sequences ◽  
Empirical Distribution Functions

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.

Records from a power model of independent observations

1995 ◽  
Vol 32 (4) ◽  
pp. 982-990 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Distribution Function ◽  
Wide Spectrum ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Power Model ◽  
Record Times ◽  
Independent Sequence ◽  
Independent Observations

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

Microdata Disclosure Control by Resampling - Effects on Regression Results

2005 ◽  
Vol 225 (5) ◽  
Author(s):  
Sandra Gottschalk
Keyword(s):  
Empirical Distribution ◽  
Continuous Distribution ◽  
Synthetic Data ◽  
Original Data ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Disclosure Control ◽  
Resampling Procedure ◽  
Original Survey ◽  
Empirical Distribution Functions

SummaryNonparametric resampling is a method for generating synthetic microdata and is introduced as a procedure for microdata disclosure limitation. Theoretically, re-identification of individuals or firms is not possible with synthetic data. The resampling procedure creates datasets - the resample - which nearly have the same empirical cumulative distribution functions as the original survey data and thus permit econometricians to calculate meaningful regression results. The idea of nonparametric resampling, especially, is to draw from univariate or multivariate empirical distribution functions without having to estimate these explicitly. Until now, the resampling procedure shown here has only been applicable to variables with continuous distribution functions. Monte Carlo simulations and applications with data from the Mannheim Innovation Panel show that results of linear and nonlinear regression analyses can be reproduced quite precisely by nonparametric resamples. A univariate and a multivariate resampling version are examined. The univariate version as well as the multivariate version which is using the correlation structure of the original data as a scaling instrument turn out to be able to retain the coefficients of model estimations. Furthermore, multivariate resampling best reproduces regression results if all variables are anonymised.

On the law of the iterated logarithm for inter-record times

1970 ◽  
Vol 7 (02) ◽  
pp. 432-439 ◽  
Author(s):  
William E. Strawderman ◽  
Paul T. Holmes
Keyword(s):  
Distribution Function ◽  
Probability Space ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Times ◽  
Iterated Logarithm ◽  
The Law ◽  
Image Position ◽  
Independent Identically Distributed

Let X 1, X2, X 3 , ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr } be defined as Also define The following theorem is due to Renyi [5].

scholarly journals On the number and sum of near-record observations

2005 ◽  
Vol 37 (03) ◽  
pp. 765-780 ◽  
Author(s):  
N. Balakrishnan ◽  
A.G. Pakes ◽  
A. Stepanov
Keyword(s):  
Distribution Function ◽  
Asymptotic Properties ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Record Time ◽  
Record Value ◽  
Independent And Identically Distributed

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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