Springer - Read

388 Appendix B Statistical Complements
B.3 Confidence Intervals
Estimation of a parameter or parameter vector by least squares or maximum likelihood
leads to a particular value, often referred to as a point estimate. It is clear that this
will rarely be exactly equal to the true value, and so it is important to convey some
idea of the probable accuracy of the estimator. This can be done using the notion of
confidence interval, which specifies a random set covering the true parameter value
with some specified (high) probability.
Example B.3.1 If X (X1,...,Xn) ′ is a vector of independent N µ, σ 2 random variables, we saw
in Section B.2 that the random variable Xn 1
n
n
i1 Xi is the maximum likelihood
estimator of µ. This is a point estimator of µ. To construct a confidence interval for
µ from Xn, we observe that the random variable
Xn − µ
S/ √ n
has Student’s t-distribution with n − 1 degrees of freedom, where S is the sample
standard deviation, i.e., S2 1 n 2
n−1 i1
Xi − Xn . Hence,
P −t1−α/2 < Xn − µ
S/ √ n