Statistical Goodness-of-Fit Measure

A Statistical Goodness-of-Fit Measure is a statistic function that quantifies how well a posited theoretical distribution function fits an empirical distribution function (on observed data).

• Example(s):
  • a Kolmogorov–Smirnov test;
  • a Cramér–von Mises criterion;
  • an Anderson–Darling test;
  • a Chi-Squared Goodness-of-Fit Test, such as Pearson's chi-squared test.
  • a Z-Test.
  • …
• Counter-Example(s):
  • a Statistical Independence Test.
• See: Goodness of Fit, Hypothesis Testing, Independence Test, Lack-of-Fit Sum of Squares, Normality Test, Kolmogorov–Smirnov, Analysis of Variance.

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References

2016

• (Encyclopedia of Mathematics, 2016) ⇒ https://www.encyclopediaofmath.org/index.php/Goodness-of-fit_test
  • QUOTE: A statistical test for goodness of fit. The essence of such a test is the following. Let [math]\displaystyle{ X_1,\cdots,X_n }[/math] be independent identically-distributed random variables whose distribution function F is unknown. Then the problem of statistically testing the hypothesis [math]\displaystyle{ H0 }[/math] that [math]\displaystyle{ F≡F_0 }[/math] for some given distribution function [math]\displaystyle{ F0 }[/math] is called a problem of testing goodness of fit. For example, if [math]\displaystyle{ F0 }[/math] is a continuous distribution function, then as a goodness-of-fit test for testing [math]\displaystyle{ H_0 }[/math] one can use the Kolmogorov test.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/goodness_of_fit Retrieved:2015-2-28.
  • The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions (see Kolmogorov–Smirnov test), or whether outcome frequencies follow a specified distribution (see Pearson's chi-squared test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.

2012

• http://en.wikipedia.org/wiki/Goodness_of_fit#Fit_of_distributions
  • In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used:
    • Kolmogorov–Smirnov test;
    • Cramér–von Mises criterion;
    • Anderson–Darling test;
    • Chi Square Goodness of Fit Test;

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: Goodness-Of-Fit Test: A test of the fit of some model to a set of data. The most commonly used tests are the chi-squared test and the likelihood-ratio test. See also INDEX OF DISPERSION; KOLMOGOROV-SMIRNOV TEST.

2004

• http://www.quality-control-plan.com/StatGuide/sg_glos.htm#goodness-of-fit
  • QUOTE: Goodness-of-fit tests test the conformity of the observed data's empirical distribution function with a posited theoretical distribution function. The chi-square goodness-of-fit test does this by comparing observed and expected frequency counts. The Kolmogorov-Smirnov test does this by calculating the maximum vertical distance between the empirical and posited distribution functions.

1989

• (Rice, 1989) ⇒ William R. Rice. (1989). “Analyzing Tables of Statistical Tests.” In: Evolution, 43(1).

1976

• TURNBULBL., W. and WEISSL, . (1976). A likelihood ratio statistic for testing goodness of fit with randomly censored data. Technical Report No. 307, School of Operations Research, Cornell University.,

1900

• (Pearson, 1900) ⇒ Karl Pearson. (1900). “On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling.” In: Philosophical Magazine Series 5. 50 (302): 157–175. doi:10.1080/14786440009463897.

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