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Topic: anybody can solve these problems?
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yamao

Posts: 1
Registered: 12/3/04
anybody can solve these problems?
Posted: May 2, 2001 2:37 AM
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I have several problems here, could you show your idea?

question:
1: let g be a given density function on R, and consider the location
parameter family generated by translating g by s:
{g(x- ¦È)|x, ¦È ¦Å R1}. Say as much as you can about the maximum
likelihood estimator of ¦È and its asymptotic properties. For a given
confidence level, find a large sample confidence interval for ¦È that
has a nonrandom length.

2: Do the same thing for the scale parameter family
{g(x/ ¦È)/ ¦È |x ¦Å R1, ¦È >0}

3: Given a random sample of size n from the normal distribution
N (¦Ì , ¦Ò2 ), where ¦Ì is unknown and ¦Ò2 is known. Consider the
hypothesis H0: ¦Ì =0 and H1: ¦Ì ¡Ù 0. Let ¦Î ¦Å (0, 1) be the prior
probability of H0 and 1-¦Î the prior probability of H1.Given that H1
is true, let the prior distribution of ¦Ì be N(0, ¦Ó2). Let ¦Ã0 and
¦Ã1, respectively, be the costs of an incorrect decision when H0 and
H1 are true. Develop the bayes solution to the problem of deciding
between H0 and H1. what are the limiting decision rules as ¦Ó2 -> ¡Þ
and as ¦Ó2 -> 0?

4: let M be a fixed borel measure on Rd and let t: Rd -> Rp be
measurable. Show that the set ¦¨ ={¦È ¦Å Rp | ¡Òexp(t(x)¦È)M(dx) < ¡Þ}
is convex if nonempty(use the geometric ¨Carithmetric mean inequality:
xay(1-a) ¡Ü ax+(1-a)y for x>0, y>0, 0<a<1) Assume that the set ¦¨ is
nonempty and open. Define c on ¦¨ by
c(¦È)-1=¡Òexp(t(x)¦È)M(dx). Then
f(x|¦È)=c(¦È)exp(t(x)¦È) is a parametric family of density functions
called the exponential family generated by M and t. ¦¨ is called its
natural parameter space. Apply the theory of maximum likelihood
estimation to this situation as fully as you can (differentiation
under the integral sign is permissible).





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