文档介绍：AMS570 Quiz 5
Name: _______________________________________________ID: __________________________________________
This is a close-book exam, due 1:40pm. Please do NOT leave after the quiz – lecture will start after the quiz.
Let X1,…,Xn be a random sample from an exponential distribution exp θ with pdf:
f x;θ=1θexp-xθ, where θ>0 and x>0,
Please derive
The maximum likelihood estimator (MLE) for θ.
Is the above MLE unbiased for θ?
The method of moment estimator (MOME) for θ.
Please derive the distribution of the first order statistic X1, where X1= min(X1,…,Xn), and further show whether Y=nX1 is an unbiased estimator of θ or not.
Solution:
The likelihood function is
L=i=1nf xi;θ=i=1n1θexp-xiθ=1θnexp-i=1nxiθ
The log likelihood function is
l=lnL=ln1θnexp-i=1nxiθ=-nlnθ-i=1nxiθ
Solving
∂l∂θ=-nθ+i=1nxiθ2=0
We obtain the MLE for θ:
θ=X
Since
EX=EX=0∞x1θexp-xθdx=θ
We know the MLE θ=X is an unbiased estimator for θ.
Now we derive the method of moment estimator (MOME) for θ. Since we have only one parameter, θ, to estimate, the equation of the first population moment and sample moment will suffice. That is, solving: EX=θ=X, we obtain the MOME: θ=X
Now we derive the general formula for the pdf of the first order statistic as follows:
PX1>x=PX1>x,…,Xn>x=i=1nPXi>x
Therefore we have