Let X1, X2, dots, Xn (n ge 2) be a random sample from the following probability density function f(x) = frac12 e^-x - mu, -infty < x < infty, where mu in (-infty, infty) is an unknown parameter. Let barX = 1/n sumi=1^n Xi and hatmu denote the maximum likelihood estimator of mu, whenever it exists. Then which of the following statements is/are correct?

Q56Section ST (Statistics (Q11–Q65))56 of 65

MLE for a Laplace location

Let $X_1, X_2, \dots, X_n$ ( $n \ge 2$ ) be a random sample from the following probability density function $f(x) = \frac12 e^{-|x - \mu|}$ , $-\infty < x < \infty$ , where $\mu \in (-\infty, \infty)$ is an unknown parameter. Let $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ and $\hat{\mu}$ denote the maximum likelihood estimator of $\mu$ , whenever it exists. Then which of the following statements is/are correct?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Select every correct option, then submit.

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