Consider a random sample X1,..., Xn from the population distribution with density f(x; 8) = 0x°-1, r € (0, 1), and parameter 0 e (0, 00). (1) Verify that the Assumptions (1) and (2) of the Cramér-Rao Theorem (Theorem 2.22) are met.

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Consider a random sample X1,..., Xn from the population distribution with density
f(x; 8) = 0x°-1, r € (0, 1),
and parameter 0 e (0, 00).
(1) Verify that the Assumptions (1) and (2) of the Cramér-Rao Theorem (Theorem 2.22)
are met.
Transcribed Image Text:Consider a random sample X1,..., Xn from the population distribution with density f(x; 8) = 0x°-1, r € (0, 1), and parameter 0 e (0, 00). (1) Verify that the Assumptions (1) and (2) of the Cramér-Rao Theorem (Theorem 2.22) are met.
Theorem 2.22
( Cramér - Rao Inequality)
Y. is
a random sauple
Suppore that
from Po, o e
X, -
Let T(X,.., Xn)
be au un bi ased eskmator of 8 (6), e.
Eo T
Let X~ Pe. Assume that the coudi'titious
(D, (2), (3)
below hold:
P. has a
pdf I pmf
(1) For all O E
and
f (x; @)
of
exists for all o E
e O and all x E N.
such that
P(XE No)
for all & G@.
2) VoE O
Đ logf (x,@)
and
3 logt
(x , )))
I(® € (0,00)
for all e E +
Here, I(6) is called
F'SHER
INFORMATION
Transcribed Image Text:Theorem 2.22 ( Cramér - Rao Inequality) Y. is a random sauple Suppore that from Po, o e X, - Let T(X,.., Xn) be au un bi ased eskmator of 8 (6), e. Eo T Let X~ Pe. Assume that the coudi'titious (D, (2), (3) below hold: P. has a pdf I pmf (1) For all O E and f (x; @) of exists for all o E e O and all x E N. such that P(XE No) for all & G@. 2) VoE O Đ logf (x,@) and 3 logt (x , ))) I(® € (0,00) for all e E + Here, I(6) is called F'SHER INFORMATION
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