40 observed losses have been recorded in thousands of dollars and are =
grouped as follows:
Interval ($000) Number of losses
(1,4/3) 16
[4/3,2) 10
[2,4) 10
[4,infinity) 4
The null hypothesis (Ho) is that the random variable X underlying the =
observed losses, in thousands, has the density function
f(x) =3D 1/(x squared) where 1<x<infinity
Since exact value of the losses are not available, it is not possible to =
compute the exact value of the Kolmogorov-Smirnov statistic used to test =
Ho. However, it is possible to put bounds on the value of this statistic=
.. =
Based on the information above, determine the smallest possible value and=
=
the largest possible value of the Kolmogorov-Smirnov statistic used to te=
st
Ho.
A. Smallest possible value =3D 0.10, Largest possible value =
=3D
0.25
B. Smallest possible value =3D 0.10, Largest possible value =3D 0.40=
C. Smallest possible value =3D 0.15, Largest possible value =3D =
0.25
D. Smallest possible value =3D 0.15, Largest possible value =3D 0.40=
E. Smallest possible value =3D 0.25, Largest possible value =
=3D
0.40
My attempt at the solution:
I computed F(x) =3D 1 - 1/x
F(1) =3D 0
F(4/3) =3D 1/4
Upper bound of (1,4/3) =3D 0.25
Lower bound of (1,4/3) =3D 0
Fn(x) =3D 0.40 =
By following the book procedure, I get K-S =3D 0.25 by using upper bound
and K-S =3D 0.40=
by
using lower bound.
I get 0.25 as the lowest possible answer but the manual specifies 0.15 as=
=
the lowest possible answer. Which is correct and why ? =
<
The lowest possible value is .15. Whatever "manual" you are using is
correct. ( In the future it would help to be more specific about what you=
mean by manual. )
This is obtained by just comparing the empirical and theoretical
distributions at the boundary points
of 1, 4/3 ,2 , 4 and infinity. Since we don't know the empirical
distribution function at any other values, for example 3.9999, we cannot
make any other comparisons. (In the case of ungrouped data we know the =
empirical distribution function every value of x.) However, since the K-=
S
stat is the maximum over all x, we know it must be .15 or more.
Note this was one of the harder problems, since applying the K-S stat to
ungrouped data is not specifically covered in the textbook.
Howard Mahler