Difference between revisions of "Manuals/calci/KSTESTNORMAL"

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==Feature==
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<div style="font-size:25px">'''KSTESTNORMAL(XRange,ObservedFrequency,Mean,Stdev,Test,Logicalvalue)'''</div><br/>
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*<math>xRange</math> is the array of x values.
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*<math>ObservedFrequency</math> is the frequency of values to test.
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*<math>Mean</math> is the mean of set of values.
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*<math>Stdev</math> is the standard deviation of the set of values.
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*<math>Test</math> is the type of the test.
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*<math>Logicalvalue</math> is either TRUE or FALSE.
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==Description==
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*This function gives the test statistic of the K-S test.
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*K-S test is indicating the Kolmogorov-Smirnov test.
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*It is one of the non parametric test.
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*This test is the equality of continuous one dimensional probability distribution.
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*It can be used to compare sample with a reference probability distribution or to compare two samples.
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*This test statistic measures a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples.
 +
*The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples.
 +
*It is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.
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*This test can be modified to serve as a goodness of fit test.
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*The assumption of the KS test is:
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*Null Hypothesis(H0):The sampled population is normally distributed.
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*Alternative hypothesis(Ha):The sampled population is not  normally distributed.
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*The Kolmogorov-Smirnov test to compare a data set to a given theoretical distribution is as follows:
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*1.Data set sorted into increasing order and denoted as <math>x_i</math>, where i=1,...,n.
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*2.Smallest empirical estimate of fraction of points falling below <math>x_i</math>, and computed as <math>\frac{(i-1)}{n}</math> for i=1,...,n.
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*3.Largest empirical estimate of fraction of points falling below <math>x_i</math> and computed as <math>\frac{i}{n}</math> for i=1,...,n.
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*4.Theoretical estimate of fraction of points falling below <math>x_i</math> and computed as <math>F(x_i)</math>, where    F(x) is the theoretical distribution function being tested.
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5.Find the absolute value of difference of Smallest and largest empirical value  with the theoretical estimation of points.
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*This is a measure of "error" for this data point.
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*6.From the largest error, we can compute the test statistic.
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*The Kolmogorov-Smirnov test statitic for the cumulative distribution F(x) is:<math> D_n=Sup_x|F_n(x)-F(x)|</math>,where <math>sup_x</math> is the supremum of the set of distances.
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*<math>F_n(x)</math> is the empirical distribution function for n,with the observations <math>X_i</math> is defined as:<math>F_n(x)=\frac{1}{n}\sum_{i=1}^n I_{X_i\le x},where <math>I_{Xi\le x}</math> is the indicator function, equal to 1 if <math>X_i\le x</math> and equal to 0 otherwise.

Revision as of 04:17, 22 May 2014

KSTESTNORMAL(XRange,ObservedFrequency,Mean,Stdev,Test,Logicalvalue)


  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xRange} is the array of x values.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ObservedFrequency} is the frequency of values to test.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Mean} is the mean of set of values.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Stdev} is the standard deviation of the set of values.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Test} is the type of the test.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Logicalvalue} is either TRUE or FALSE.

Description

  • This function gives the test statistic of the K-S test.
  • K-S test is indicating the Kolmogorov-Smirnov test.
  • It is one of the non parametric test.
  • This test is the equality of continuous one dimensional probability distribution.
  • It can be used to compare sample with a reference probability distribution or to compare two samples.
  • This test statistic measures a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples.
  • The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples.
  • It is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.
  • This test can be modified to serve as a goodness of fit test.
  • The assumption of the KS test is:
  • Null Hypothesis(H0):The sampled population is normally distributed.
  • Alternative hypothesis(Ha):The sampled population is not normally distributed.
  • The Kolmogorov-Smirnov test to compare a data set to a given theoretical distribution is as follows:
  • 1.Data set sorted into increasing order and denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} , where i=1,...,n.
  • 2.Smallest empirical estimate of fraction of points falling below Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} , and computed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(i-1)}{n}} for i=1,...,n.
  • 3.Largest empirical estimate of fraction of points falling below Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} and computed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{i}{n}} for i=1,...,n.
  • 4.Theoretical estimate of fraction of points falling below Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} and computed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x_i)} , where F(x) is the theoretical distribution function being tested.

5.Find the absolute value of difference of Smallest and largest empirical value with the theoretical estimation of points.

  • This is a measure of "error" for this data point.
  • 6.From the largest error, we can compute the test statistic.
  • The Kolmogorov-Smirnov test statitic for the cumulative distribution F(x) is:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_n=Sup_x|F_n(x)-F(x)|} ,where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sup_x} is the supremum of the set of distances.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n(x)} is the empirical distribution function for n,with the observations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i} is defined as:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_n(x)=\frac{1}{n}\sum_{i=1}^n I_{X_i\le x},where <math>I_{Xi\le x}} is the indicator function, equal to 1 if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i\le x} and equal to 0 otherwise.
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