ABSTRACT
See Figure B.I. Since the tangents are perpendicular to a, their equations
A
must be
or
Lemma B.3.1 (Lehmann 1986, p. 90) Let 0 denote the parameter space and let 6 be a random vector whose distribution depends on 0 € Q. // (</>0(0) : 9 G 0} is a family of tests such that
Pe{<t>e(6] = 0} > 1 - a for each 9 £ 0, then
is a level 100(1 — a)% confidence set for 0. Note that in order to obtain a confidence set via this connection, a family
of tests, one for each hypothesized parameter value, is required. When the family of distributions is a location family of distributions, one can start with one or more tests for a particular hypothesized parameter value and employ equi variance to generate the family of tests. In the presence of an unknown (nuisance) scale parameter, usually this hypothesized parameter value is chosen so that a statistic whose distribution depends on neither the location parameters nor the scale parameter (i.e., a pivotal quantity) is available. Lemma B.3.2 Suppose the distribution of 0 — 6 does not depend on 9 — (0i, . . . , 6p). Consider a partition ©i, . . . , Qm of the parameter space, that is,
and @i n 0; = 0 for all i ^ j.
