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yamao
Posts:
1
Registered:
12/3/04


anybody can solve these problems?
Posted:
May 2, 2001 2:37 AM


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 Â¦Ã)x, Â¦Ã Â¦Ã
R1}. Say as much as you can about the maximum likelihood estimator of Â¦Ã and its asymptotic properties. For a given confidence level, find a large sample confidence interval for Â¦Ã that has a nonrandom length.
2: Do the same thing for the scale parameter family {g(x/ Â¦Ã)/ Â¦Ã x Â¦Ã
R1, Â¦Ã >0}
3: Given a random sample of size n from the normal distribution N (Â¦Ã , Â¦Ã2 ), where Â¦Ã is unknown and Â¦Ã2 is known. Consider the hypothesis H0: Â¦Ã =0 and H1: Â¦Ã Â¡Ã 0. Let Â¦Ã Â¦Ã
(0, 1) be the prior probability of H0 and 1Â¦Ã the prior probability of H1.Given that H1 is true, let the prior distribution of Â¦Ã be N(0, Â¦Ã2). Let Â¦Ã0 and Â¦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 Â¦Ã2 > Â¡Ã and as Â¦Ã2 > 0?
4: let M be a fixed borel measure on Rd and let t: Rd > Rp be measurable. Show that the set Â¦Â¨ ={Â¦Ã Â¦Ã
Rp  Â¡Ãexp(t(x)Â¦Ã)M(dx) < Â¡Ã} is convex if nonempty(use the geometric Â¨Carithmetric mean inequality: xay(1a) Â¡Ã ax+(1a)y for x>0, y>0, 0<a<1) Assume that the set Â¦Â¨ is nonempty and open. Define c on Â¦Â¨ by c(Â¦Ã)1=Â¡Ãexp(t(x)Â¦Ã)M(dx). Then f(xÂ¦Ã)=c(Â¦Ã)exp(t(x)Â¦Ã) is a parametric family of density functions called the exponential family generated by M and t. Â¦Â¨ 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).



