Thread: log likelihood in Pareto distribution

1. log likelihood in Pareto distribution

I'm having some trouble finding the log-likelihood in these questions:

Question 1
Pareto distribution is:
f(yt) = θ / (yt ^ (1+θ))

What is the log-likelihood, log L (θ)

Question 2
Geometric distribution:
f (yt) = θ (1 − θ)^yt, yt = 0, 1, 2, · · ·
where θ > 0, is an unknown parameter. Let {y1, y2, ..., yT } be iid observations from the geometric distribution.

What is the log-likelihood?

I also need to take derivatives of both. Would appreciate some help, I'm stuck!

2. Hello,

Let $\displaystyle \mathbf{x}=(x_1,\dots,x_n)$ a n-sample following a Pareto distribution.

Then the likelihood function, which is the pdf of the n-tuple, will be :

$\displaystyle L(\theta,\mathbf{x})=\prod_{i=1}^n f(\theta,x_i)=\frac{\theta^n}{\left(\prod_{i=1}^n x_i\right)^{1+\theta}}$

So the log is $\displaystyle \log\theta^n-\log \left(\prod_{i=1}^n x_i\right)^{1+\theta}=n\log\theta-(1+\theta)\log \left(\prod_{i=1}^n x_i\right)=n\log\theta-(1+\theta)\sum_{i=1}^n \log(x_i)$

now differentiate with respect to $\displaystyle \theta$...

Same thing for the other one.

3. Thanks Moo! I wasn't so far off Q1 after all.

Having more trouble with Q2 though, can anyone give me a bit more advice? Thanks

4. Originally Posted by Denny Crane
Thanks Moo! I wasn't so far off Q1 after all.

Having more trouble with Q2 though, can anyone give me a bit more advice? Thanks
What have you tried? Where are you stuck?

5. Outsourcing your 306-317 Econometrics homework? Nice.

6. Originally Posted by jrt12
Outsourcing your 306-317 Econometrics homework? Nice.
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Thread closed for the time being.