# MLE -- Exponential likelihood

• Apr 22nd 2010, 10:14 PM
n3mo
MLE -- Exponential likelihood
I am trying to determine what the log-likelihood would be for the following 3 parameter exponential:

\$\displaystyle f(t) = a(1-e^{-bt}) + c\$

Can anyone help me out with this one? I'm trying to learn how to do MLE in R, and I'm 99% of the way there... but this log-likelihood is eluding me...

Thanks :)
• Apr 23rd 2010, 10:15 PM
matheagle
what the range of t, i.e., the support of your density
• Apr 24th 2010, 12:34 AM
GnomeSain
Quote:

Originally Posted by n3mo
I am trying to determine what the log-likelihood would be for the following 3 parameter exponential:

\$\displaystyle f(t) = a(1-e^{-bt}) + c\$

Can anyone help me out with this one? I'm trying to learn how to do MLE in R, and I'm 99% of the way there... but this log-likelihood is eluding me...

Thanks :)

I don't know how to program in R, but if you understand the fundamentals, it becomes a matter of syntax.

Suppose you have n independent data points, \$\displaystyle x_1,x_2,....,x_n\$. The likelihood function is the the product \$\displaystyle \prod_{k=1}^nf(x_n)\$ which is the joint probability function of these observations as well as a function of a, b and c, the unknown parameters. You estimate these parameters by maximizing this probability through various optimization techniques (ie Lagrangian). Now, you can also log the likelihood for ease of compuation. Since log is an increasing function, maximizing the log of the likelihood is the same as maximizing the likelihood.