Question Problem 4 (2 points) Suppose that X1,..., Xn is a random sample from a probability density function in the one- parameter exponential family. That is ſa(6)b(x )e-[c@)d(a)], u < x < v f(3;6) = otherwise, where a and care functions that do not depend on 2, b and d are functions that do not depend on 0, and u and v are constants that do not depend on 0. Show that h=1d(Xi) is sufficient for 0. 1 Problem 5 (2 points) Let Y1, ..., Yn be a random sample from a uniform distribution on the interval (0,50). Derive the method of moments estimator for 0.
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Transcribed Image Text: Problem 4 (2 points) Suppose that X1,..., Xn is a random sample from a probability density function in the one- parameter exponential family. That is ſa(6)b(x )e-[c@)d(a)], u < x < v f(3;6) = otherwise, where a and care functions that do not depend on 2, b and d are functions that do not depend on 0, and u and v are constants that do not depend on 0. Show that h=1d(Xi) is sufficient for 0. 1 Problem 5 (2 points) Let Y1, ..., Yn be a random sample from a uniform distribution on the interval (0,50). Derive the method of moments estimator for 0.
More Transcribed Image Text: Problem 4 (2 points) Suppose that X1,..., Xn is a random sample from a probability density function in the one- parameter exponential family. That is ſa(6)b(x )e-[c@)d(a)], u < x < v f(3;6) = otherwise, where a and care functions that do not depend on 2, b and d are functions that do not depend on 0, and u and v are constants that do not depend on 0. Show that h=1d(Xi) is sufficient for 0. 1 Problem 5 (2 points) Let Y1, ..., Yn be a random sample from a uniform distribution on the interval (0,50). Derive the method of moments estimator for 0.
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The pdf is given by,{:[f(x;theta)=a(theta)b(x)e^(-[c(theta)d(x)],)","u <= x <= theta],[=0quad0.omega". "]:}The joint pdf of x_(1),dots,x_(n) is,f(x_;theta)=(a(theta))^(n)(prod_(i=1)^(n)b(x_(i)))e^(-[c(theta)sum_(i=1)^(n)d(x_(i))])Note that,{:[f(x^('),theta)=(a(theta))^(n)(prod_(i=1)^(n)b(x_(i)))e^(-sum_(i=1)^(n)[(theta)d(x_(i))])],[=(*(a(theta))^(n)e^(-c(theta)sum_(i=1)^(n)d(x_(i)))*(prod_(i=1)^(n)b(x_(i))):}],[=g(T;theta)*h(x)]:}where, T=sum_(i=1)^(n)d(x_(i)) and h(x_) hat(is)^(S)=prod_(i=1)^(n)b(x_(i)) which does not depend on the parameter ... See the full answer