Derive the asymptotic distribution of maximum likelihood estimator

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Suppose that $X_1,...,X_n$ is a random sample from a distribution with pdf :



$$ f(x,theta)= fractheta^32x^2e^-theta x space for space 0,x,infty$$



I found that $l(theta)=-nlog(2)+3nlog(theta) +2sum X_i -theta sum X_i$



Then $fracd^2dtheta^2=frac-3ntheta^2$ therefore fisher information $I(theta)= frac3ntheta^2$



How can i derive the asymptotic distribution from here ?




convergence asymptotics estimation estimation-theory fisher-information




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  • What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
    – StubbornAtom
    Jul 30 at 15:03














up vote
0
down vote

favorite












Suppose that $X_1,...,X_n$ is a random sample from a distribution with pdf :



$$ f(x,theta)= fractheta^32x^2e^-theta x space for space 0,x,infty$$



I found that $l(theta)=-nlog(2)+3nlog(theta) +2sum X_i -theta sum X_i$



Then $fracd^2dtheta^2=frac-3ntheta^2$ therefore fisher information $I(theta)= frac3ntheta^2$



How can i derive the asymptotic distribution from here ?




convergence asymptotics estimation estimation-theory fisher-information




share|cite|improve this question



















  • What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
    – StubbornAtom
    Jul 30 at 15:03












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose that $X_1,...,X_n$ is a random sample from a distribution with pdf :



$$ f(x,theta)= fractheta^32x^2e^-theta x space for space 0,x,infty$$



I found that $l(theta)=-nlog(2)+3nlog(theta) +2sum X_i -theta sum X_i$



Then $fracd^2dtheta^2=frac-3ntheta^2$ therefore fisher information $I(theta)= frac3ntheta^2$



How can i derive the asymptotic distribution from here ?




convergence asymptotics estimation estimation-theory fisher-information




share|cite|improve this question











Suppose that $X_1,...,X_n$ is a random sample from a distribution with pdf :



$$ f(x,theta)= fractheta^32x^2e^-theta x space for space 0,x,infty$$



I found that $l(theta)=-nlog(2)+3nlog(theta) +2sum X_i -theta sum X_i$



Then $fracd^2dtheta^2=frac-3ntheta^2$ therefore fisher information $I(theta)= frac3ntheta^2$



How can i derive the asymptotic distribution from here ?





convergence asymptotics estimation estimation-theory fisher-information





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 11:14









user1607

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  • What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
    – StubbornAtom
    Jul 30 at 15:03
















  • What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
    – StubbornAtom
    Jul 30 at 15:03















What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
– StubbornAtom
Jul 30 at 15:03




What is your MLE? We have the result $$sqrtn(hat theta_MLE-theta)sim ANleft(0,frac1E_thetaleft[fracpartialpartialthetaln f_theta(X_1)right]^2right)$$ provided some regularity conditions are met.
– StubbornAtom
Jul 30 at 15:03















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