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I am trying to find the maximum likelihood estimators and the method of moments of the following:

g(x;w;s) = pdf =  1/we^((-x-s)/(w)), for x > s

For finding the MEE of s and w, I know I have to solve the following two functions:

(1) xbar = mu 
(2) 1/n summation Xi^2 = mu^2 + var

I found the E[x] = mu = integral from s to infinity x*1/we^((-x-s)/w)dx = 

e^(-2s/w)(w+s)

I then found E[X^2] = var = integral from s to infinity x^2*1/we^((-x-s)/w)dx = 

e^(-2s/w)(2w^2+2ws+s^2)

From here I now have my two equations:

(1) xbar = e^(-2s/w)(w+s)
(2) 1/n summation Xi^2 = mu^2 + e^(-2s/w)(2w^2+2ws+s^2)

Now I know I need to solve the two systems of equations, but I am having a difficult time solving them. I wanted to solve the first equation for either w or s, and they substitute into the second equation, but I cannot figure it out. I am wondering if I integrated on the correct bounds? It made sense to me, but it might be wrong. So since I couldn't get any further using the MME, I attempted the method of maximum likelihood and I got the following:

P(X1=x1, X2=x2,...,Xn=xn)
= P(X1=x1)P(X2=x2)...P(Xn=xn)
= g(x1;w;s)g(x2;w;s)...g(xn;w,s)
= 1/we^((-x1-s)/w) * 1/we^((-x2-s)/w) *... * 1/we^((-xn-s)/w)
= 1/w^ne^(-1/w summation xi+s)

From here I am again stuck and I'm unsure if I even did this correctly. I don't know if what I did makes sense to anyone, but any help would be greatly appreciated! :)

Thank everyone! Lizzie

p.s. I'm sorry it's hard to read my math problem in this format I typed it, I'm unfamiliar with this website.

mapleleaf
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  • I'm voting to close this question as off-topic because it is about statistics / [math.se] instead of programming or software development. – Pang Aug 30 '16 at 04:24

1 Answers1

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Recheck your integrals. E[x] = mu != x*1/we^((-x-s)/w)dx but rather is equal to x*1/we^((-(x-s)/w)dx = s + w. I guess all your other errors follow because of the mistake in carrying forward the sign: -(x-s) = -x + s.

By the way, this density function corresponds to the shifted exponential distribution. Because if you take X to be exponentially distributed with mean w then X-s has density given by g.

Deepak Nag
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