Page 396 - Probability and Statistical Inference

7. Point Estimation 373

sufficient for θ. We wish to estimate T(θ) = θ unbiasedly. Since U has its pdf
given by nu θ I(0 < u < θ) we have E [X ] = n(n + 1) θ. Hence, E [(n +
n-1 -n
-1
θ
θ
n:n
-1
1)n X ] = θ so that W = (n + 1)n X is an unbiased estimator of θ. Thus,
-1
n:n
n:n
by the Lehmann-Scheffé Theorems, (n + 1)n X is the UMVUE for θ. Here,
-1
n:n
the domain space χ for the X variable depends on ? itself. Hence, an approach
through the Cramér-Rao inequality is not feasible. !
Example 7.5.10 Suppose that X , ..., X are iid N(µ, σ ) where µ and s are
2
n
1
both unknown, −∞ < µ < ∞, 0 < σ < ∞ and χ = ℜ. Let us write θ = (µ, θ ) ∈
2
Θ = ℜ × ℜ and we wish to estimate T (θ) = µ unbiasedly. Recall from the
+
2
Example 6.6.7 that U = ( , S ) is a complete sufficient statistic for θθ θθ θ. Obvi-
ously, is an unbiased estimator of T (θ) whereas is a function of U only.
Thus, by the Lehmann-Scheffé Theorems, is the UMVUE for µ. Similarly,
2
2
one can show that S is the UMVUE for σ . The situation here involves two
unknown parameters µ and σ. Hence, an approach through the Cramér-Rao
inequality as stated will not be appropriate. !
Example 7.5.11 (Example 7.5.10 Continued) Suppose that X , ..., X are iid
1
n
χ
N(µ, σ ) where µ and σ are both unknown, −∞ < µ < ∞, 0 < σ < ∞ and = ℜ.
2
2
+
Let us write θθ θθ θ = (µ, θ ) ∈ Θ = ℜ × ℜ and we wish to estimate T(?) = µ + s
unbiasedly. Now, U = ( , S ) is a complete
2
sufficient statistic for θθ θθ θ. From (2.3.26) for the moments of a gamma variable,
it follows that E [S] = a s where Thus
θ n
is an unbiased estimator of T(θ) and W depends only on U.
Hence, by the Lehmann-Scheffé Theorems, is the UMVUE for µ +
σ. Similarly one can derive the UMVUE for the parametric function µσ where
?
? is any real number. For k < 0, however, one should be particularly cautious
about the minimum required n. We leave this out as Exercise 7.5.7. !
Example 7.5.12 (Example 4.4.12 Continued) Suppose that X , ..., X are
1
n
-1
iid with the common negative exponential pdf f(x; θθ θθ θ) = σ exp{−(x − µ)/σ}I(x
> µ) where µ and σ are both unknown, θθ θθ θ = (µ, σ) ∈ Θ = ℜ × ℜ , n = 2. Here,
+
we wish to estimate T(θ) = µ + σ, that is the mean of the population, unbi-
asedly. Recall from Exercises 6.3.2 and 6.6.18 that
is a complete sufficient statistic for θθ θθ θ. Now,
is an unbiased estimator of T(θ) and note that we can rewrite = Xn:1
+ which shows that depends only on U. Thus, by the
Lehmann-Scheffé Theorems, is the UMVUE for µ+σ. Exploiting (2.3.26),
k
one can derive the UMVUE for the parametric function µσ where K is any
real number. For ? < 0, however, one should be particularly cautious about the
minimum required n. We leave this out as Exercise 7.5.8. !