ABSTRACT
In earlier chapters, especially in Chapter 3 and Chapter 4, we have seen how to evaluate an estimator in terms of bias, variance, and risk. For instance, given iid observations X1,X2, . . . ,Xn from a N(μ, σ2) population where both μ and σ are unknown, the variance σ2 can be estimated by its MLE
σ̂2ML = 1 n
(Xi −X)2 = 1 n SX (say)
where X is the sample mean. Using the fact that SX/σ2 follows χ2n−1 distribution, it is now easy to find the bias and variance of σ̂2ML as
Bias of σ̂2ML = E [ σ̂2ML
] − σ2
= ( n− 1 n
) σ2 − σ2
= −σ 2
n (5.1.1)
Variance of σ̂2ML = E [(
σ̂2ML − E [ σ̂2ML
])2] =
2(n− 1) n2
σ4
Using SX / σ2 as a pivot, one can find a (1− α)-level confidence interval
for σ2 as [ SX
]
tail (α/2)-probability cutoff points of χ2n−1 distribution. The length of this confidence interval is
SX
( 1
χ2n−1, (1−α/2) − 1
)
which is a random variable. Since the length plays an important role in evaluating a confidence interval, one can find the average (or expected) length of the above confidence interval which turns out to be
(n− 1) (
− 1 χ2n−1, α/2
) σ2 (5.1.2)
and this can be compared with the expected length of any other (1− α)-level confidence interval for σ2.
