Page 398 - Probability and Statistical Inference

7. Point Estimation 375

is, the CRLB from (7.5.18) for the variance of unbiased estimators of T(θ)
would be θ (n + n α) . Obviously, is sufficient for
-1
2
1 2
θ and E {U} = (n + n α)θ so that (n + n α) U is an unbiased estimator of θ.
-1
θ
1
2
1
2
2
2
-1
-1
-2
2
But, note that V {(n + n α) U} = (n + n α) (n θ + n αθ ) = θ (n + n α) ,
θ
1
2
1
2
1
2
2
1
the same as the CRLB. Hence, we conclude that (n + n α) -1
1 2
is the UMVUE of θ. !
Example 7.5.14 Suppose that X , ..., X are iid N(µ, 1) with the un-
1n1
11
χ
known parameter µ ∈ (−∞, ∞) and = (−∞, ∞). Also, suppose that X , ...,
21
χ
2
X are iid N(2µ, σ ) involving the same unknown parameter µ and = (−∞,
2n2
∞), but σ(> 0) is assumed known. Let us assume that the X ’s are indepen-
1
dent of the X ’s. By direct calculations we have I (µ) = 1 and I (µ) = 4σ .
-2
X2
X1
2
That is, the CRLB from (7.5.18) for the variance of unbiased estimators of
T(µ) = µ is given by (n + 4n σ ) . Obviously,
-2 -1
1 2
is sufficient for µ and E {U} = (n + 4n σ )µ so that (n + 4n σ ) U is an
-2
-2 -1
1
2
1
2
µ
unbiased estimator of µ. But, note that V {(n + 4n σ ) U} = (n + 4n σ )
-2 -
-2 -1
µ
2
1
1
2
-2
2 (n + 4n σ ) = (n + 4n σ ) , the same as the CRLB. Hence, we conclude
-2 -1
1 2 1 2
that (n + 4n σ ) is the UMVUE of µ.!
-2 -1
1 2
7.5.4 Evaluation of Conditional Expectations
Thus far we aimed at finding the UMVUE of a real valued parametric function
T(θ). With some practice, in many problems one will remember the well-
known UMVUE for some of the parametric functions. Frequently, such
UMVUE’s will depend on complete sufficient statistics. Combining these in-
formation we may easily find the expressions of special types of conditional
expectations.
Suppose that T and U are statistics and we wish to find the expression for
E [T | U = u]. The general approach is involved. One will first derive the
θ
conditional distribution of T given U and then directly evaluate the integral
∫ tf(t | U = u; θ)dt). In some situations we can avoid this cumbersome
T
process. The following result may be viewed as a restatement of the Lehmann-
Scheffé Theorems but it seems that it has not been included elsewhere in its
present form.
Theorem 7.5.5 Suppose that X , ..., X are iid real valued random vari-
n
1
χ ⊆
ables with the common pmf or pdf given by f(x; θ) with x ∈ ℜ, θ ∈ Θ ⊆
ℜ . Consider two statistics T and U where T is real valued but U may be
k
vector valued such that E [T] = T(θ), a real valued parametric function.
θ
Suppose that U is complete sufficient for θ and a known real valued