Page 401 - Probability and Statistical Inference

378 7. Point Estimation

2
θ while obviously so is T itself. Now, , S are independently distributed and
hence we claim the following:

Thus, the Rao-Blackwellized versions of T and T’ are respectively
and Clearly W and W’ are both unbiased estimators for the
unknown parameter θ. But observe that V (W) = n θ while V (W ′ ) = (a −
-1 2
-2
θ θ n
1)θ . Next, note that Γ(x + 1) = xΓ(x) for x > 0, and evaluate
2
V (W′ ). Look at the Table 7.6.1.
θ
Table 7.6.1. Comparing the Two Variances V (W) and V (W’)
θ
θ
n θ V (W) Exact θ (V (W′ ) Approx. θ V (W′ )
-2
-2
-2
θ
θ
θ
2 .5000 (π/2) − 1 .5708
3 .3333 (4/π) − 1 .2732
4 .2500 (3π/8) − 1 .1781
5 .2000 {32/(9π)} − 1 .1318
Upon inspecting the entries in the Table 7.6.1, it becomes apparent that W
is better than W′ in the case n = 2. But when n = 3, 4, 5, we find that W′ is
better than W.
For large n, however, V (W′ ) can be approximated. Using (1.6.24) to
θ
approximate the ratios of gamma functions, we obtain

Now, we apply (7.6.1) with x = ½n, a = -½ and b = 0 to write

In other words, from (7.6.2) it follows that for large n, we have

Using MAPLE, the computed values of θ V (W′ ) came out to be 2.6653 ×
-2
θ
10 , 1.7387 × 10 , 1.2902 × 10 , 1.0256 × 12 , 5.0632 × 10 and 2.5157 ×
-2
3
-2
-2
-2
10 respectively when n = 20, 30, 40, 50, 100 and 200. The approximation
-3
given by (7.6.3) seems to work well for n ≥ 40.