Stat Theory: Need to Prove Consistent Estimator

[madameclaws]

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 ?

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!

 

[Ray Vickson]

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 ?

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!

Easiest way:
$$\int_{\theta}^{\infty} t f(t-\theta) \; dt = \int_{\theta}^{\infty} (t - \theta + \theta) f(t-\theta) \; dt\\ = \theta \int_{\theta}^{\infty} f(t-\theta)\; dt + \int_{\theta}^{\infty} (t-\theta)f(t-\theta) \; dt\\ = \theta + \int_0^{\infty} s f(s) \; ds.$$

 

[madameclaws]

Hi Ray,
I am not understanding how you went from ∫tf(t-θ) to what you have presented below.
Could you provide further details in the steps you have listed below?
Also, should have I made my integral from theta to infinity instead of 0 to theta?

Thanks!

 

[Ray Vickson]

Hi Ray,
I am not understanding how you went from ∫tf(t-θ) to what you have presented below.
Could you provide further details in the steps you have listed below?
Also, should have I made my integral from theta to infinity instead of 0 to theta?

Thanks!

Sorry, I cannot do more. I gave the steps in detail, one-by-one. And no: the final integral goes from 0 to ∞ because we have changed variables from t to s = t-θ. That was the whole point: we reduce the problem to a standard form that is already familiar (or should be).

 

[madameclaws]

Hi Ray,

What I am not understanding is how you have t-theta+theta and then in the next step just eliminated that down to theta outside of the integral.
I just need to understand your thinking behind it because it doesn't make sense to me.

Thanks!

 

[statdad]

"Now, to show that an estimator is consistent, I need to show that my E(T) is unbiased and my Var(T) as n->infi"

A consistent estimator is merely one that converges in probability - it doesn't have to be unbiased. Thus you need to show that |T - theta| converges to zero in probability (T is your estimator).