Computational Statistics

CASE STUDY: DISTRIBUTION DIAGNOSTICS 11

Example 1.2: Distribution Function

Instead of the distribution P we consider its distribution function F = F

P

, where

F (x) = P (X ≤ x).

For an empirical distribution P

n

of n observations X

1

, . . . , X

n

, the corresponding em-

pirical distribution function is

F

n

(x) = #{i : X

i

≤ x}/n.

Example 1.3: Histogram

We select disjoint test sets A

j

, j = 1, . . . , J, covering the range of X. For example, for

the uniform distribution on (0, 1) we can choose the intervals

A

j

=



j − 1

J

,

j

J

i

as test sets.

Instead of the distribution P we consider the vector



P (A

j

)



j=1,...,J

. Its graphical rep-

resentation is called a histogram. The empirical ...