Question Suppose that Y1, Y2, ... ,Yn constitute a random sample from the density function п ? f(yle= {*)220 e-(y=0), y>0, elsewhere, = 10, where 0 is an unknown, positive constant. a) Find an estimator ê, for by the method of moments. (b) Find an estimator ôz for @ by the method of maximum likelihood. (c) Adjust ô7 and 62 so that they are unbiased. Find the efficiency of the adjusted ôn relative to the adjusted ô2.
CT2F9V The Asker · Probability and Statistics
Transcribed Image Text: Suppose that Y1, Y2, ... ,Yn constitute a random sample from the density function п ? f(yle= {*)220 e-(y=0), y>0, elsewhere, = 10, where 0 is an unknown, positive constant. a) Find an estimator ê, for by the method of moments. (b) Find an estimator ôz for @ by the method of maximum likelihood. (c) Adjust ô7 and 62 so that they are unbiased. Find the efficiency of the adjusted ôn relative to the adjusted ô2.
More Transcribed Image Text: Suppose that Y1, Y2, ... ,Yn constitute a random sample from the density function п ? f(yle= {*)220 e-(y=0), y>0, elsewhere, = 10, where 0 is an unknown, positive constant. a) Find an estimator ê, for by the method of moments. (b) Find an estimator ôz for @ by the method of maximum likelihood. (c) Adjust ô7 and 62 so that they are unbiased. Find the efficiency of the adjusted ôn relative to the adjusted ô2.
Community Answer
TSQPPI
{:[=int_(0)^(theta)ze^(-z)dz],[=sqrt2=1.],[E(y)=1+theta". "],[mu_(1)=1+theta]:}Replace u_(1) by bar(y) to get moments estionatorbar(y)-1= hat(theta)hat(theta)_(1) is unbiasid estimator of theta.{:[E( bar(y)-1)=E( bar(y))-1],[=1+theta-1=theta]:}{:[V(y-theta)=E(y-theta)^(2)-E^(2)(y-theta)],[=>v(y)=E(y-theta)^(2)-E^(2)(y-theta)],[E(y-theta)^(2)=int_(theta)^(oo)(y-theta)^(2)e^(-(y-theta))dy],[=int_(0)^(oo)z^(22)e^(-z)dz∣{:[y-theta=z],[dy=dz]:}],[=sqrt3=2],[V(y)=2-1=1],[v( hat(O)_(1))=v( bar(y)-1)=v( bar(y))=(v(Y_(i)))/(n)],[=(1)/(n)". "],[=e^(-sum_(i=1)^(n)(y_(i)-theta))I(theta < y_((1)) < cdots < y_((n)))]:}we need to minimise sum_(i=1)(y_(i)-theta)So the ... See the full answer