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.

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.

A new method for process control based on goodness of fit tests

2015 ◽  
Vol 32 (2) ◽  
pp. 132-143
Author(s):  
Mohammad Saleh Owlia ◽  
Mohammad Saber Fallah Nezhad ◽  
Mohesn Sheikh Sajadieh
Keyword(s):  
Process Control ◽  
Design Methodology ◽  
Goodness Of Fit ◽  
New Method ◽  
Content Type ◽  
Goodness Of Fit Tests ◽  
Kolmogorov Smirnov ◽  
Shift Detection ◽  
The Subject ◽  
Smirnov Test

Purpose – The purpose of this paper is to propose a new method based on goodness of fit tests for shift detection problems. Design/methodology/approach – In this method, although the distribution of gathered data from the process is the subject of control, but any out-of-control signal could also be generalized to the overall state of the process including the parameters of the distribution. Findings – Results of simulation study denote that among goodness of fit tests, the χ2 test has a better performance than the Kolmogorov-Smirnov test in detecting shifts of process. Also comparison of proposed method with traditional methods denotes that, proposed method generally has smaller probabilities of first and second type errors. Originality/value – To the best of author’s knowledge, no attention has previously been paid to application of goodness of fit tests in process control.

scholarly journals Um teste de ajuste com foco no valor em risco condicionado

2008 ◽  
Vol 6 (2) ◽  
pp. 139
Author(s):  
José Santiago Fajardo Barbachan ◽  
Aquiles Rocha de Farias ◽  
José Renato Haas Ornelas
Keyword(s):  
Value At Risk ◽  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Hypothesis Test ◽  
Risk Measure ◽  
Conditional Value At Risk ◽  
Small Samples ◽  
Risk Management Framework ◽  
Kolmogorov Smirnov ◽  
Goodness Of Fit Hypothesis

To verify whether an empirical distribution has a specific theoretical distribution, several tests have been used like the Kolmogorov-Smirnov and the Kuiper tests. These tests try to analyze if all parts of the empirical distribution has a specific theoretical shape. But, in a Risk Management framework, the focus of analysis should be on the tails of the distributions, since we are interested on the extreme returns of financial assets. This paper proposes a new goodness-of-fit hypothesis test with focus on the tails of the distribution. The new test is based on the Conditional Value at Risk measure. Then we use Monte Carlo Simulations to assess the power of the new test with different sample sizes, and then compare with the Crnkovic and Drachman, Kolmogorov-Smirnov and the Kuiper tests. Results showed that the new distance has a better performance than the other distances on small samples. We also performed hypothesis tests using financial data. We have tested the hypothesis that the empirical distribution has a Normal, Scaled Student-t, Generalized Hyperbolic, Normal Inverse Gaussian and Hyperbolic distributions, based on the new distance proposed on this paper.

Curve Fitting

1935 ◽  
Vol 4 (04) ◽  
pp. 222-233
Author(s):  
F. M. Redington
Keyword(s):  
Curve Fitting ◽  
Special Interest ◽  
Goodness Of Fit ◽  
Summation Methods ◽  
Starting Point ◽  
The Subject

For the purposes of this paper the term “curve-fitting” is not synonymous with “graduation.” It is restricted to graduation by means of an explicit algebraic or trigonometrical formula: graphic and summation methods are excluded. Even so the subject is unlimited, and this paper cannot provide more than an explanation of a few general principles, and a brief exposition of some of the problems of special interest to the actuary.Our starting point is this: we have decided to fit a curve (vx) to a series of crude figures (ux) and the problems that confront us are:—(a) What formula are we to ascribe to vx?(b) How are we to choose the constants in vx in the best possible way?(c) How are we to test the goodness of fit and to compare the results of different graduations?

scholarly journals Evaluation of statistical distributions to analyze the pollution of Cd and Pb in urban runoff

2017 ◽  
Vol 75 (9) ◽  
pp. 2072-2082 ◽  
Author(s):  
Amin Toranjian ◽  
Safar Marofi
Keyword(s):  
Goodness Of Fit ◽  
Damage Identification ◽  
Urban Runoff ◽  
Distribution Functions ◽  
Environmental Damage ◽  
Value Functions ◽  
Event Mean Concentration ◽  
Kolmogorov Smirnov ◽  
Anderson Darling ◽  
Instantaneous Concentration

Heavy metal pollution in urban runoff causes severe environmental damage. Identification of these pollutants and their statistical analysis is necessary to provide management guidelines. In this study, 45 continuous probability distribution functions were selected to fit the Cd and Pb data in the runoff events of an urban area during October 2014–May 2015. The sampling was conducted from the outlet of the city basin during seven precipitation events. For evaluation and ranking of the functions, we used the goodness of fit Kolmogorov–Smirnov and Anderson–Darling tests. The results of Cd analysis showed that Hyperbolic Secant, Wakeby and Log-Pearson 3 are suitable for frequency analysis of the event mean concentration (EMC), the instantaneous concentration series (ICS) and instantaneous concentration of each event (ICEE), respectively. In addition, the LP3, Wakeby and Generalized Extreme Value functions were chosen for the EMC, ICS and ICEE related to Pb contamination.

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