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Find the maximum likelihood estimator based on a sample of size \(n\) from the two-sided exponential distribution with PDF $$ f(x ; \theta)=\frac{1}{2} e^{-|x-\theta|}, \quad-\infty

Short Answer

Expert verified
The maximum likelihood estimator for \(\theta\) in the given two-sided exponential distribution is the sample median. Furthermore, this estimator is unbiased.
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Step 1: Formulate the Likelihood Function

The likelihood function is the joint probability of observing the given sample data. For the two-sided exponential distribution, the PDF is given as \(f(x ; \theta)=\frac{1}{2} e^{-|x-\theta|}\). Therefore, the likelihood of observing a sample \(x_1, x_2, ..., x_n\) is given by multiplying the individual probabilities, thus: \(L(\theta; x) = \prod_{i=1}^{n}f(x_i;\theta) = \prod_{i=1}^{n} \frac{1}{2}e^{-|x_i - \theta|}\).

Step 2: Calculating the Log-likelihood function

Maximizing product of exponential functions is non-trivial. We can simplify this problem by taking log of the likelihood function. This is known as the log-likelihood function which transforms the products into summations. So, the log-likelihood function becomes: \(l(\theta) = \log L(\theta; x) = \sum_{i=1}^{n} \log \left(\frac{1}{2}e^{-|x_i - \theta|}\right) = -n\log 2 - \sum_{i=1}^{n}|x_i - \theta|\).

Step 3: Maximizing the Log-likelihood

We calculate the derivative of the log-likelihood with respect to \(\theta\), our target parameter and equate it to zero to find the value of \(\theta\) that maximizes the function. However, due to the absolute value, the function isn't differentiable. Thus, we find that the sum of absolute differences is minimized when \(\theta\) is a median. Therefore, the maximum likelihood estimate of \(\theta\) is its sample median, i.e. \(\hat{\theta_{MLE}} = median(x_1, x_2, ..., x_n)\).

Step 4: Checking the Bias

To verify if the estimator \(\hat{\theta_{MLE}}\) is unbiased, we need to check if the Expected value of the estimator equals the parameter. For a single observation, the median of a two-sided exponential distribution is equal to the parameter \(\theta\). Therefore, assuming the data are i.i.d., the expectation of the sample median will also equal \(\theta\), hence, the estimator \(\hat{\theta_{MLE}}\) is unbiased.

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