Pareto distribution for MLE

qwertykeyboards

Hi, just wonderin if anyone can show me how to derive the mle of the pareto distribution. upcoming stats exam and this kinda question could feature. thanks a bunch.

silverside

basically

from wikipedia Pareto had two paramaters, m and k

density function /pdf: k m^k x^(-1-k)

log pdf: log k + k log m - (1+k) log x

log of joint pdf for n variables xi... xn: n log k + nk log m - (1+k) Sigma(1=1..n) log x_i

define y = geometric mean of x_i

log joint pdf = n log k + nk log m - n(1+k) log y

Differentiate this w.r.t. k or m and find local minimum (maximum?) to find MLE for m or k



does that help?

reason for taking log transformation is that log is a monotonic transformation -> local minimum of log == local minimum of function itself, and it makes the maths easier for the joint pdf (i think its called the likelihood function)

qwertykeyboards

that helps loads, think i have it now cheers!

silverside

you're welcome - I should be charging for grinds

qwertykeyboards

while i have you here, any chance you could explain to me how in the solution below step 2 is derived from step 1??

Poisson dist: f(X;m) = (e^-m m^X) / X!
L = Pi from i=1 up to n (f(Xi; m)
step 1 => L = Pi from i=1 up to n [(e^-m m^Xi) / Xi!]

step 2 => Log L = -nm + Sigma Xi log m - Sigma log Xi!

solution ends up showing mhat = i/n Sigma Xi

would appreciate if u cud show me how to get from step 1 to step 2 as i think a couple of steps might be missing.

thanks! :pac::eek:

silverside

I think all you need is the fact that

log of a product == sum of logs

out of interest, what course are you doing? I did a BSc / MSc in maths / stats / finance and must get around to doing an ad for tuition.

qwertykeyboards

ah yes, that explains it, puzzled by what was happening to the e in the equation but its ln so it'l become 1 (loge e^-Y = -Y(1) = -Y. but why does the n get multiplied by it? is that simply because of the limits on pi of i=1 up to n?

also, why are sigmas put in front of some and not others? is it just the parts with Xi that get the sigmas then yeah?

both Financial and Actuarial Mathematics and Quantitative Finance do these stats modules in dcu.