Introduction to Probability and Statistics for Engineers and Scientists

*11.6The Kolmogorov–Smirnov Goodness of Fit Test for Continuous Data 505

To compute the value ofDfor a given data setYj=yj,j=1,...,n, lety(1),y(2),...,y(n)
denote the values of theyjin increasing order. That is,

y(j)=jth smallest ofy 1 ,...,yn

For example, ifn= 3 andy 1 =3,y 2 =5,y 3 =1, theny(1)=1,y(2)=3,y(3)=5. Since
Fe(x) can be written

Fe(x)=























0ifx<y(1)

1

n

ify(1)≤x<y(2)

..

.

j

n

ify(j)≤x<y(j+1)

..

.

1ify(n)≤x

we see thatFe(x) is constant within the intervals (y(j−1),y(j)) and then jumps by 1/nat
the pointsy(1),...,y(n). SinceF(x) is an increasing function ofxthat is bounded by 1, it
follows that the maximum value ofFe(x)−F(x) is nonnegative and occurs at one of the
pointsy(j),j=1,...,n(see Figure 11.3).
That is,

Maximum

x

{Fe(x)−F(x)}=Maximum

j=1,...,n

{

j

n

−F(y(j))

}

(11.6.1)

1

x

F(x)

Fe(x)

y(1) y(2) y(3)y(4) y(5)

FIGURE 11.3 n= 5.