Maximum likelihood estimator

**wolverine21** · November 23rd 2008, 10:29 PM

5. A random sample of size n is taken from a distribution with pdf:

f(x;θ) = θx^(θ-1 ) 0 < x < 1; θ > 0.

Find the maximum likelihood estimator of θ.

I tried teaching myself out of the book but it has no examples. Can someone run me through a step by step way of solving this? I have nothing to reference and my notes have 2 lines on it.

**mr fantastic** · November 23rd 2008, 11:26 PM

Originally Posted by wolverine21

5. A random sample of size n is taken from a distribution with pdf:

f(x;θ) = θx^(θ-1 ) 0 < x < 1; θ > 0.

Find the maximum likelihood estimator of θ.

I tried teaching myself out of the book but it has no examples. Can someone run me through a step by step way of solving this? I have nothing to reference and my notes have 2 lines on it.

The likelihood function $L(x_1, x_2, \, .... \, x_n)$ is defined to be the joint pdf of the random variables $X_1, \, X_2, \, .... \, X_n$ .

Therefore $L(x_1, x_2, \, .... \, x_n) = \left( \theta x_1^{\theta - 1} \right) \cdot \left( \theta x_2^{\theta - 1} \right) \cdot .... \left( \theta x_n^{\theta - 1} \right) = \theta^n \, ( x_1 \cdot x_2 \cdot .... x_n)^{\theta - 1}$ .

The maximum likelihood estimate of $\theta$ is the value of $\theta$ that maximises $L(x_1, x_2, \, .... \, x_n)$ .

Since $\ln L$ is a monotonically increasing function of L, both L and $\ln L$ will be a maximum for the same value of $\theta$ . It's obviously easier in this instance to find the value of $\theta$ that maximises $\ln L$ :

$\ln L = n \ln \theta + (\theta - 1) \ln (x_1 \cdot x_2 \cdot .... x_n)$

$\Rightarrow \frac{d \ln L}{d\theta} = \frac{n}{\theta} + \ln (x_1 \cdot x_2 \cdot .... x_n)$ .

Now solve $\frac{dL}{d\theta} = 0$ for $\theta$ .

**aadbaluyot** · January 11th 2009, 02:13 AM

Please see my computation.. I don't know if it is correct. I have a question, Is there a possibility that it we can a negative in MLE?...thanks!

**mr fantastic** · January 11th 2009, 03:08 AM

Originally Posted by aadbaluyot

Please see my computation.. I don't know if it is correct. I have a question, Is there a possibility that it we can a negative in MLE?...thanks!

You are missing a log in the second line. See my post above.

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