Somek-sample Kolmogorov-Smirnov-Rényi type theorems for empirical distribution functions

1966 ◽  
Vol 17 (3-4) ◽  
pp. 325-334 ◽  
Author(s):  
M. Csörgő
Keyword(s):  
Empirical Distribution ◽  
Distribution Functions ◽  
Kolmogorov Smirnov ◽  
Empirical Distribution Functions

Beyond the heuristic approach to Kolmogorov-Smirnov theorems

1982 ◽  
Vol 19 (A) ◽  
pp. 359-365 ◽  
Author(s):  
David Pollard
Keyword(s):  
Weak Convergence ◽  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Heuristic Approach ◽  
Convergence Theory ◽  
Simple Method ◽  
Starting Point ◽  
Kolmogorov Smirnov ◽  
The Subject

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

scholarly journals The use of personal weather station observations to improve precipitation estimation and interpolation

2021 ◽  
Vol 25 (2) ◽  
pp. 583-601
Author(s):  
András Bárdossy ◽  
Jochen Seidel ◽  
Abbas El Hachem
Keyword(s):  
Spatial Interpolation ◽  
Empirical Distribution ◽  
Precipitation Amount ◽  
Distribution Functions ◽  
Quantile Mapping ◽  
Novel Approach ◽  
Precipitation Estimation ◽  
Weather Stations ◽  
High Precipitation ◽  
Empirical Distribution Functions

Abstract. The number of personal weather stations (PWSs) with data available through the internet is increasing gradually in many parts of the world. The purpose of this study is to investigate the applicability of these data for the spatial interpolation of precipitation using a novel approach based on indicator correlations and rank statistics. Due to unknown errors and biases of the observations, rainfall amounts from the PWS network are not considered directly. Instead, it is assumed that the temporal order of the ranking of these data is correct. The crucial step is to find the stations which fulfil this condition. This is done in two steps – first, by selecting the locations using the time series of indicators of high precipitation amounts. Then, the remaining stations are then checked for whether they fit into the spatial pattern of the other stations. Thus, it is assumed that the quantiles of the empirical distribution functions are accurate. These quantiles are then transformed to precipitation amounts by a quantile mapping using the distribution functions which were interpolated from the information from the German National Weather Service (Deutscher Wetterdienst – DWD) data only. The suggested procedure was tested for the state of Baden-Württemberg in Germany. A detailed cross validation of the interpolation was carried out for aggregated precipitation amount of 1, 3, 6, 12 and 24 h. For each of these temporal aggregations, nearly 200 intense events were evaluated, and the improvement of the interpolation was quantified. The results show that the filtering of observations from PWSs is necessary as the interpolation error after the filtering and data transformation decreases significantly. The biggest improvement is achieved for the shortest temporal aggregations.

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.

scholarly journals The Reliability of Numerical Modeling in Remote Sensing Data Analysis

2020 ◽  
Vol 149 ◽  
pp. 02012
Author(s):  
Boris Dobronets ◽  
Olga Popova
Keyword(s):  
Empirical Distribution ◽  
Remote Sensing Data ◽  
Small Sample ◽  
Probability Density Functions ◽  
Distribution Functions ◽  
Data Uncertainty ◽  
Computational Statistics ◽  
New Approach ◽  
Interpolation Polynomials ◽  
Empirical Distribution Functions

The article deals with the problem of calculating reliable estimates of empirical distribution functions under conditions of small sample and data uncertainty. To study these issues, we develope computational probabilistic analysis as a new area in computational statistics. We propose a new approach based on random interpolation polynomials and order statistics. Arithmetic operations on probability density functions and procedures for constructing the probabilistic extensions are used.

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.

Sign in / Sign up

Export Citation Format

Share Document