Using R for Introductory Statistics : John Verzani

Goodness of fit 255theoretical densities and cumulativedistribution functions are drawnadded to the existing plot using lines(). The following commands produced Figure 9.2:> y = rnorm(20)> plot(density(y), main="Densities”) # densities> curve(dnorm(x), add=TRUE, lty=2)> plot(ecdf(y), main="C.d.f.s”) # c.d.f.s> curve(pnorm(x), add=TRUE, lty=2)If the data is from the population with c.d.f. F, then we would expect that F n is close to Fis some way. But what does “close” mean? In this context, we have two differentfunctions of x. Define the distance between them as the largest difference they have:D=maximum in x of |F n (x)-F(x)|.The surprising thing is that with only the assumption that F is continuous, D has a knownsampling distribution called the Kolmogorov-Smirnov distribution. This is illustrated inFigure 9.3, where the sampling distribution of the statistic for n=25 is simulated forseveral families of random data. In each case, we see the same distribution. This factallows us to construct a significance test using the test statistic D. In addition, a similartest can be done to compare two independent samples.The Kolmogorov-Smirnov goodness-of-fit testAssume X 1 , X 2 , …, X n is an i.i.d. sample from a continuous distribution with c.d.f. F(x).Let F n (x) be the empirical c.d.f. A significance test ofH 0 : F(x)=F 0 (x), H A :F(x)≠F 0 (x)Figure 9.3 Density estimates for samplingdistribution of the Kolmogorov-Smirnov statistic