4.8.1 Fixed-Size Confidence Bounds

Gleser (1965) has extended Chow and Robbins' (1965) results to the linear regression problem Let y ₁, y ₂,... be a sequence of independent observations with

(4.8.1)

where ? ? is an unknown 1 p vector, x ⁽ⁱ⁾ a known p 1 column vector, and ? _i a random error having an unknown distribution function F with mean 0 and finite, but unknown variance ? ². We wish to find a region W in p-dimensional Euclidean space such that P( ? ? W)=1 ? ? and such that the length of the interval cut off on the _i-axis by W has width=2 d, i=1, 2,..., p. As has already been noted for p=1, no fixed-sample size procedure meeting the requirements exists. Hence, we are led to consider sequential procedures.

When ? is Known

Since the least-squares (Gauss-Markov) estimate of ? has, component-wise, uniformly minimum variance among all linear unbiased estimates of ?, has good asymptotic properties (such as consistency) and performs reasonably well against non-linear unbiased estimates, the least squares estimate of ? would be a natural candidate to use in the construction of our confidence region. It is well-known that the least squares estimate of ? is given by

(4.8.2)

where , X _n=( x ⁽¹⁾, x ⁽²⁾,..., x ^{( n )}