Ganch this might be right up your alley...

[Quebb Diesel]

Suppose x1, x2, x3 are i.i.d. Θeˆ(-Θx), x>0

find the UMVUE of eˆ-Θ

[Ganchrow]

I think you need to do your own homework assignments.

[Quebb Diesel]

its actually not a homework assignment...the prof said hed give some extra credit if you could figure it out the UMVUE w/ 3 observations and even more for n observations...im stumped...no pointers?!

[Ganchrow]

the prof said hed give some extra credit if you could figure it out the UMVUE w/ 3 observations!

You'd first need to spell out exactly what extra credit your professor is prepared to give me.

[Quebb Diesel]

im sure he could give you some pointers in black jack...

[Quebb Diesel]

ehh...wrote the function wrong anyway...

f(x|Θ)=Θeˆ(-Θx)/x!

[raiders72002]

Quebb- You have to put this stuff to good use. I flunked the first actuarial test by one point and then realized that wasn't my calling. Sports is where it's at.

[Quebb Diesel]

ha ha...i dont really understand the UMVUE that much...but i assume it is perfect for sports as long as you could create a strong algorithm...but if it was that easy plenty of people would be rich

from what i understand the UMVUE is the best estimator to use being that its unbiased and has a minimal variance...Ganch...which do you prefer...the UMVUE or MLE?

[pico]

try maximum likelihood method...log the function to seperate the terms.

[Quebb Diesel]

the prof wants the UMVUE of eˆ-Θ...

[Quebb Diesel]

i tried to evaluate the MLE of Θ and got 3/((x1+x2+x3)+1/╥xi!) so the MLE of e^-Θ would be eˆ-3/((x1+x2+x3)+1/╥xi!)...it doesnt look right to me though...i could be wrong

[Ganchrow]

im sure he could give you some pointers in black jack...

I almost take that as an insult.

ehh...wrote the function wrong anyway...

f(x|Θ)=Θeˆ(-Θx)/x!

Whether it's for extra credit or not this is clearly a homework assignment unrelated to sports betting. More importantly it's your homework assignment and not mine. I finished grad school almost 15 years ago and am not looking to go back.

With all due respect if you're really hoping to get an MS degree in stats, then this is a question you should be able to answer on your own without help from (of all places) a sports betting forum. The best nudge I can give you is to tell you to look at the pdf and try to figure out if it reminds you of anything. (Like maybe an exponential distribution ... ?)

[Ganchrow]

i tried to evaluate the MLE of Θ and got 3/((x1+x2+x3)+1/╥xi!) so the MLE of e^-Θ would be eˆ-3/((x1+x2+x3)+1/╥xi!)...it doesnt look right to me though...i could be wrong

You're close but you must have made a minor error somewhere.

If the probability density function of discrete random variable X given parameter Θ is given by:

f(x|Θ)=Θe^-Θx/x!

If we let λ= e^-Θ then by taking the log of both sides we also have Θ = -log(λ). This allows us to rewrite f(·):

g(λ) = -log(λ) * λ^x / x!

This gives us the log-likelihood function function (summation over n observations of x):

L(λ) = Σ log[-log(λ) * λ^x_i / x_i!]
L(λ) = Σ { log[-log(λ)] + x_i * log[λ] - log[x_i!] }

Taking the derivative of L(·) wrt λ and setting to zero gives us:

^dL/_dλ = Σ {¹/_(λ*log(λ)) + ^x_i / _λ}

so the maximum likelihood estimator of λ= e^-Θ would be given by:

λ_MLE = e^-n/_Σx
λ_MLE = e^-1/_x

where x is the sample mean.

I hope this also gives you a clue as to your original question ...

Anyway, as I mentioned before this a sports betting and not a math forum so I really do hope that going forward we can shy away from these types of questions ...

[Quebb Diesel]

You're close but you must have made a minor error somewhere.

If the probability density function of discrete random variable X given parameter Θ is given by:

f(x|Θ)=Θe^-Θx/x!

If we let λ= e^-Θ then by taking the log of both sides we also have Θ = -log(λ). This allows us to rewrite f(·):

g(λ) = -log(λ) * λ^x / x!

This gives us the log-likelihood function function (summation over n observations of x):

L(λ) = Σ log[-log(λ) * λ^x_i / x_i!]
L(λ) = Σ { log[-log(λ)] + x_i * log[λ] - log[x_i!] }

Taking the derivative of L(·) wrt λ and setting to zero gives us:

^dL/_dλ = Σ {¹/_(λ*log(λ)) + ^x_i / _λ}

so the maximum likelihood estimator of λ= e^-Θ would be given by:

λ_MLE = e^-n/_Σx
λ_MLE = e^-1/_x

where x is the sample mean.

I hope this also gives you a clue as to your original question ...

Anyway, as I mentioned before this a sports betting and not a math forum so I really do hope that going forward we can shy away from these types of questions ...

substituting Θ makes the MLE 2000 times easier! i understand ML estimators but am still trying to understand the UMVUE concept...

PS: i did not substitute Θ, but made an error in the derivative of the likelihood function. d/dΘ of ln(╥xi!) = 0 not 1/╥xi! !!! calculus isnt a race!

sorry about that...im curious though...which estimator do you prefer?

im not too sure if you are in here to inform people about statistics or if you gamble any? id like to see some of your work and concepts regarding sports statistics though, its new to me actually seen real statistic work applied to sports! do you keep track of how successful your analysis is?

[Quebb Diesel]

I almost take that as an insult.




Whether it's for extra credit or not this is clearly a homework assignment unrelated to sports betting. More importantly it's your homework assignment and not mine. I finished grad school almost 15 years ago and am not looking to go back.

With all due respect if you're really hoping to get an MS degree in stats, then this is a question you should be able to answer on your own without help from (of all places) a sports betting forum. The best nudge I can give you is to tell you to look at the pdf and try to figure out if it reminds you of anything. (Like maybe an exponential distribution ... ?)

def part of the exponential family...i just need to break down this rao blackwell theorem down!

im sure you do just well in blackjack as well too...no insults over here! i pretty sure my prof concentrates in sequential patterns and nonparametric statistics and have heard stories about him killing it in vegas on the black jack tables though...

[BigBollocks]

Ganch how do you recall so much after being away for 15 years? A lot of this stuff I learned and then forgot six months after having a desk job.

[Quebb Diesel]

Ganch how do you recall so much after being away for 15 years? A lot of this stuff I learned and then forgot six months after having a desk job.

im assuming none of this will come in handy later on down the road now...am i going to be pointing and clicking on a computer the rest of my life?!

[Ganchrow]

im curious though...which estimator do you prefer?

It all on depends on how many beers I have in me.

im not too sure if you are in here to inform people about statistics or if you gamble any?

Well I certainly don't gamble for fun if that's what you mean ...

I do you keep track of how successful your analysis is?

I do indeed. I also consider that information proprietary.

[Ganchrow]

def part of the exponential family...i just need to break down this rao blackwell theorem down!

Cool. Keep on plucking away.

im sure you do just well in blackjack as well too...no insults over here! i pretty sure my prof concentrates in sequential patterns and nonparametric statistics and have heard stories about him killing it in vegas on the black jack tables though...

I've written some good blackjack software but that's really about it.

Without question Justin is the man at SBR for practical blackjack knowledge.

[Ganchrow]

Ganch how do you recall so much after being away for 15 years? A lot of this stuff I learned and then forgot six months after having a desk job.

After grad school I worked as a Wall Street quant for many years. That helped.

It's not about remembering everything ... it's about remembering just enough to figure out what you need to look up.

[raiders72002]

I've written some good blackjack software but that's really about it.

Without question Justin is the man at SBR for practical blackjack knowledge.

I'd be interested in what type of counting methods you guys use from single deck on up playing live.

[raiders72002]

There are a few guys on the forums that I know of that are banned from playing blackjack. They were working with a team.