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Let $X_1,\ldots,X_n$ be a sample from the distribution whose probability density function is:

$$f(x)=\begin{cases}\frac13e^{-\frac13(x-\theta)}&, x\ge\theta \\ 0&,x<\theta\end{cases}$$

Find the maximum likelihood estimator of $\theta$.

So the PDF is equal to: $$f(x\mid \theta) = \frac{1}{3}e^{-\frac{1}{3}(x-\theta)}1_{\{x\geq \theta\}}$$

and thereby the likelihood will be:

$$\mathcal{L} = \prod_{i=1}^n f(x\mid \theta)= \prod_{i=1}^n \frac{1}{3}e^{-\frac{1}{3}(x-\theta)}1_{\{x\geq \theta\}}$$

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It's a common mistake.

Your pdf is

$$f(x| \theta) = \frac{1}{3}e^{-\frac{1}{3}(x-\theta)}1_{\{x\geq \theta\}}$$

Use this pdf in the likelihood (along with the indicator) and see for yourself that the MLE estimate of $\theta$ is $\min\limits_{i=1}^{n} X_i$.

Edit:

$$L(\theta|x_{1},\ldots,x_{n}) = \left(\frac{1}{3}\right)^n e^{-\frac{1}{3n}(\bar{x}-\theta)}\prod_{i=1}^{n}1_{\{x_i\geq \theta\}}$$

Now, $\theta$ should satisfy $x_i \geq \theta \ \forall i$ otherwise likelihood will be zero. Therefore, $\theta \leq \min\limits_{i=1}^{n}x_i$. As, $\theta$ decreases further the exponential part decreases due to the negative sign in its power and therefore, the likelihood decreases.

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    $\begingroup$ P.S. It is not always possible to compute the minima/maxima using derivative tests. Sometimes plotting graph or simple reasoning does the job. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Aug 16 '17 at 15:23
  • $\begingroup$ so based on this the likelihood I should be able to see that the MLE estimate is $\min\limits_{i=1}^{n} X_i$ ? $\endgroup$ – Daniel Robotics Aug 16 '17 at 15:41
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    $\begingroup$ Yes. Just plot the likelihood function as a function of $\theta$ and you will see it. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Aug 16 '17 at 15:47
  • $\begingroup$ Sorry if I seem annoying, but I don't know how to plot the likelihood function as a function of θ $\endgroup$ – Daniel Robotics Aug 16 '17 at 19:50
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    $\begingroup$ Check edit please. However, I strongly recommend to plot the likelihood for better vizualization. $\endgroup$ – Dhruv Kohli - expiTTp1z0 Aug 16 '17 at 20:05

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