Solved – Is Kolmogorov-Smirnov test valid with discrete distributions

discrete datahypothesis testingkolmogorov-smirnov test

I'm comparing a sample and checking whether it distributes as some, discrete, distribution. However, I'm not enterily sure that Kolmogorov-Smirnov applies. Wikipedia seems to imply it does not. If it does not, how can I test the sample's distribution?

Best Answer

It does not apply to discrete distributions. See http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm for example.

Is there any reason you can't use a chi-square goodness of fit test? see http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm for more info.

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Solved – Explanation of the Kolmogorov-Smirnov Test with applications in Java

import java.util.Scanner;
class ABC
{
 public static void main(String[]args)
{
    Scanner s=new Scanner(System.in);
    int n;
    System.out.println("Enter Count of numbers:");
    n=s.nextInt();
    double d[]=new double[n];
    double dp[]=new double[n];
    double dm[]=new double[n];
    double dplus,dmin,dmax;
    System.out.println("Enter "+n+"Numbers:");
    for(int i=0;i<n;i++)
     {
         d[i]=s.nextDouble();
     }
    for(int i=0;i<n;i++)
    {
        for(int j=i;j<n;j++)
        {
            if(d[i]>d[j])
            {
                 double temp=d[i];
                 d[i]=d[j];
                 d[j]=temp;
            }
        }
    }
    for(int i=0;i<n;i++)
    {
        dp[i]=((double)(i+1)/n)-d[i];
    }
    for(int i=0;i<n;i++)
    {
        dm[i]=d[i]-((double)i/n);
    }
    System.out.println("i\tRi\t(i/N-Ri)\tRi-(i-1/N)");
    System.out.println("-------------------------------------------");
    for(int i=1;i<n;i++)
        {
            System.out.println((i+1)+"\t"+d[i]+"\t"+dp[i]+"\t"+dm[i]);
        }
    dplus=dp[0];
    dmin=dm[0];
    for(int i=0;i<n;i++)
    {
        if(dp[i]>dplus)
        {
            dplus=dp[i];
        }
    }

    for(int i=1;i<n;i++)
    {
        if(dm[i]>dmin)
        {
             dmin=dm[i];
        }
    }
    System.out.println("D+="+dplus+"\nD-="+dmin);

    double D=Math.max(dplus,dmin);

    System.out.println("D="+D);

    System.out.println("Enter the critical value:");
    double da=s.nextDouble();

    if(D>da)
    {
        System.out.println("Hypothesis H0 is Rejected");
    }
    else
    {
    System.out.println("Hypothesis H0 is Accepted");
    }
}
}

Solved – use Kolmogorov-Smirnov to compare two empirical distributions

That is OK, and quite reasonable. It is referred to as the two-sample Kolmogorov-Smirnov test. Measuring the difference between two distribution functions by the supnorm is always sensible, but to do a formal test you want to know the distribution under the hypothesis that the two samples are independent and each i.i.d. from the same underlying distribution. To rely on the usual asymptotic theory you will need continuity of the underlying common distribution (not of the empirical distributions). See the Wikipedia page linked to above for more details.

In R, you can use the ks.test, which computes exact $p$-values for small sample sizes.

Best Answer

Related Solutions

Solved – Explanation of the Kolmogorov-Smirnov Test with applications in Java

Solved – use Kolmogorov-Smirnov to compare two empirical distributions

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