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Fundamentals of Mathematical Statistics

S.C. Gupta, V.K. Kapoor

Chapter 4: Chapter 4

Problem 41

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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

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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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TABLE OF CONTENTS :

TABLE OF CONTENTS

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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Chapter 4: Chapter 4

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

Step by step solution

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Step 1: Formulate the Likelihood Function

Step 2: Calculating the Log-likelihood function

Step 3: Maximizing the Log-likelihood

Step 4: Checking the Bias

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Chapter 9

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