Asymptotic Methods in Statistical Inference - Statistics Centre

102
• For any symmetric, square integrable density
with variance 2 , ¯ ≥ 864
Proof: Put () =12
2 h R
2 () i 2
and () =
( ), with unit variance ( 2 =1). Then
() =12
∙Z
2 ()¸2
= ()
so we can take =1. Wearethento
minimize
Z
Z
2 () =1
2 () subject to
Z
() =1
with symmetric and non-negative.
It is sufficient that = (; ) minimize
Z
2 () +2
Z
2 () − 2
Z
()
unconditionally, for constants (‘Lagrange multipliers’)
, and satisfy the side conditions. (Why?
It is very instructive to write out the details of the
argument.)