- #1

madameclaws

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So I am struggling with this homework problem because I got burned out of another problem earlier today, and I just cannot get beyond what I have.

The problem is:

Let X be a continuous random variable with the pdf: f(x)=e^(-(x-θ)) , x > θ ,

and suppose we have a sample of size n , { X1, X2 , … , Xn }.

Is T = Min ( X1, X2 , … , Xn ) a consistent estimator for θ ?

From my class, I know that my cdf for this is F_T(t)= 1-e^(-n(t-θ)) and my pdf is f_t(t)=ne^(-n(t-θ)) where t>θ.

Now, to show that an estimator is consistent, I need to show that my E(T) is unbiased and my Var(T) as n->infinity goes to 0.

What I am currently stuck on is finding my E(T), as silly as that sounds.

I know my integral needs to be:

∫(from 0 to theta) t*ne^(-n(t-θ)) dt

So dusting off my integration by parts, I get my integral to be:

n∫(from 0 to theta) t*e^(-n(t-θ)) dt

[-t*e^(-n(t-θ))|(from 0 to theta)-(1/n)e^(-n(t-θ))|(from 0 to theta)]

[-θ*e^(-n(θ-θ))+0-e^(-n(θ-θ))+e^(nθ)]

[-θ-1/n+(1/n)e^(nθ)]

Which I am pretty much stuck on how to get an unbiased estimator out of that.

Thus I can pretty much assume I did something wrong somewhere and I need help.

Could someone please take a look at this and let me know where I am going wrong with this?

Thanks!

The problem is:

Let X be a continuous random variable with the pdf: f(x)=e^(-(x-θ)) , x > θ ,

and suppose we have a sample of size n , { X1, X2 , … , Xn }.

Is T = Min ( X1, X2 , … , Xn ) a consistent estimator for θ ?

## Homework Equations

From my class, I know that my cdf for this is F_T(t)= 1-e^(-n(t-θ)) and my pdf is f_t(t)=ne^(-n(t-θ)) where t>θ.

## The Attempt at a Solution

Now, to show that an estimator is consistent, I need to show that my E(T) is unbiased and my Var(T) as n->infinity goes to 0.

What I am currently stuck on is finding my E(T), as silly as that sounds.

I know my integral needs to be:

∫(from 0 to theta) t*ne^(-n(t-θ)) dt

So dusting off my integration by parts, I get my integral to be:

n∫(from 0 to theta) t*e^(-n(t-θ)) dt

[-t*e^(-n(t-θ))|(from 0 to theta)-(1/n)e^(-n(t-θ))|(from 0 to theta)]

[-θ*e^(-n(θ-θ))+0-e^(-n(θ-θ))+e^(nθ)]

[-θ-1/n+(1/n)e^(nθ)]

Which I am pretty much stuck on how to get an unbiased estimator out of that.

Thus I can pretty much assume I did something wrong somewhere and I need help.

Could someone please take a look at this and let me know where I am going wrong with this?

Thanks!

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