Solved – Value at $D_\max$ from Kolmogorov-Smirnov test in R

distributionskolmogorov-smirnov testr

I am comparing two distributions using the Kolmogorov-Smirnov test [ks.test()] in R and would like to know what the numerical value is where $D_\text{max}$ occurs. It's not in the output of ks.test() as far as I can see, so I was wondering if anyone had any ideas about how I could find that out. Thanks!

Best Answer

Something like that? Dmax occurs at the value max.at.

set.seed(12345)

x <- rnorm(10000, 5, 5)
y <- rnorm(10000, 7, 6.5)

# remove any missings from the data

x <- x[!is.na(x)]
y <- y[!is.na(y)]

ecdf.x <- ecdf(x)
ecdf.y <- ecdf(y)

plot(ecdf.x, xlim=c(min(c(x,y)), max(c(x,y))), verticals=T, cex.lab=1.2, cex.axis=1.3,
     las=1, col="skyblue4", lwd=2, main="")

plot(ecdf.y, verticals=T, add=T, do.points=FALSE, cex.lab=1.2,
     cex.axis=1.3, col="red", lwd=2)

n.x <- length(x)
n.y <- length(y)

n <- n.x * n.y/(n.x + n.y)
w <- c(x, y)

z <- cumsum(ifelse(order(w) <= n.x, 1/n.x, -1/n.y))

max(abs(z)) # Dmax
[1] 0.1664

ks.test(x,y)$statistic # the same
     D 
0.1664

max.at <- sort(w)[which(abs(z) == max(abs(z)))]
[1] 9.082877

# Draw vertical line

abline(v=max.at, lty=2)

lines(abs(z)~sort(w), col="purple", lwd=2)

legend("topleft", legend=c("x", "y", "|Distance|"), col=c("skyblue4", "red", "purple"), lwd=c(2,2,2), bty="n")

Dmax visualization

Related Solutions

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 – Kolmogorov-Smirnov two-sample $p$-values

Under the null hypothesis, the asymptotic distribution of the two-sample Kolmogorov–Smirnov statistic is the Kolmogorov distribution, which has CDF

$$\operatorname{Pr}(K\leq x)=\frac{\sqrt{2\pi}}{x}\sum_{i=1}^\infty e^{-(2i-1)^2\pi^2/(8x^2)} \>.$$

The $p$-values can be calculated from this CDF - see Section 4 and Section 2 of the Wikipedia page on the Kolmogorov–Smirnov test.

You seem to be saying that a non-parametric test statistic shouldn't have a distribution - that's not the case - what makes this test non-parametric is that the distribution of the test statistic does not depend on what continuous probability distribution the original data come from. Note that the KS test has this property even for finite samples as shown by @cardinal in the comments.

Best Answer

Related Solutions

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

Solved – Kolmogorov-Smirnov two-sample $p$-values

Related Question