Nonparametric Methods in Statistics with SAS Applications

146 UNIVARIATE PROBABILITY DENSITY ESTIMATION

and triangular. They are

Normal kernel: K(x) =

1

√

2

π

exp



−

1

2

x

2



, −∞ < x < ∞.

6

-

0

−1

1

x

K(x)

1/

√

2

π

Quadratic (or Epanechnikov) kernel: K(x) =



3

4



1 −x

2



, if |x| ≤ 1,

0, otherwise

.

6

-

0

−1

1

x

K(x)

3/4

Triangular kernel: K(x) =



1 −|x| if |x| ≤ 1,

0, otherwise

.

6

-

0

−1

1

x

K(x)

1

• The option C= lists up to five values of the standardized bandwidth c that is related

to the bandwidth

λ

by the formula

λ

= c Q n

−1/5

where Q is the interquartile range

of the data. The optimal standardized bandwidth that SAS computes by default is the

value of c that minimizes the approximate mean integrated squared error (AMISE)

defined by

AMISE =

(cQ)

4

4n

4/5



