Manuals/calci/KSTESTNORMAL

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

• $xRange$ is the array of x values.
• $ObservedFrequency$ is the frequency of values to test.
• $Mean$ is the mean of set of values.
• $Stdev$ is the standard deviation of the set of values.
• $Test$ is the type of the test.
• $Logicalvalue$ 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 $x_{i}$, where i=1,...,n.
• 2.Smallest empirical estimate of fraction of points falling below $x_{i}$, and computed as ${\frac {(i-1)}{n}}$ for i=1,...,n.
• 3.Largest empirical estimate of fraction of points falling below $x_{i}$ and computed as ${\frac {i}{n}}$ for i=1,...,n.
• 4.Theoretical estimate of fraction of points falling below $x_{i}$ and computed as $F(x_{i})$, 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 statistic for the cumulative distribution F(x) is:$D_{n}=Sup_{x}|F_{n}(x)-F(x)|$,where $sup_{x}$ is the supremum of the set of distances.
• $F_{n}(x)$ is the empirical distribution function for n,with the observations $X_{i}$ is defined as:$F_{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}I_{X_{i}\leq x}$,where $I_{X_{i}\leq x}$ is the indicator function, equal to 1 if $X_{i}\leq x$ and equal to 0 otherwise.