Answer to Question #199030 in Statistics and Probability for Sam

Question #199030

A random sample of size n is drawn from a uniform population over (θ-1/3,θ+1/3),Obtain maximum likelihood estimator ofθ.


1
Expert's answer
2022-01-31T16:29:28-0500

likelihood function for a uniform distribution:

L(θ)=Πi=1nf(xi;a,b)=Πi=1n1(ba)n=Πi=1n1(ba)n=Πi=1n1(θ+1/3θ+1/3)n=Πi=1n1(2/3)nL(\theta)=\Pi^n_{i=1}f(x_i;a,b)=\Pi^n_{i=1}\frac{1}{(b-a)^n}=\Pi^n_{i=1}\frac{1}{(b-a)^n}=\Pi^n_{i=1}\frac{1}{(\theta +1/3-\theta +1/3)^n}=\Pi^n_{i=1}\frac{1}{(2/3)^n}


log-likelihood function:

l(θ)=nlog(2/3)l(\theta)=-nlog(2/3)


log-likelihood function does not depend on θ\theta , so MLE for θ\theta does not exist


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