Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 9 / b

For this rate parametrization of the Laplace distribution, $\mu$ is its mean and median. By symmetry, $\mathbb{E}(X-\mu )=0$, while $|X-\mu |$ has exponential distribution with rate $\lambda$, soThe integral follows by two applications of integration by parts. The central limit theorem therefore givesWith known $\lambda$ and the supplied cutoff 2, an approximate 95% confidence interval isThis known-rate Laplace location confidence interval uses the sample mean; the later maximum-likelihood estimator of location need not be used to construct this valid asymptotic confidence interval.

By independence, the likelihood function is the product of the individual probability density functions:Writing $A(\mu )={\sum }_i|x_i-\mu |$, the log-likelihood function is $\ell =n\ln \lambda -n\ln 2-\lambda A(\mu )$. Here $\lambda$ is a rate, the reciprocal of the scale in the usual Laplace distribution notation.

The joint maximum-likelihood estimators are functions of the observations selecting a maximizing parameter pair:The estimates after observation maximize the same likelihood function with $X_i=x_i$. Maximization may be nonunique, as for an even sample size, or may fail to yield a finite admissible pair, as in the one-observation case below. The definition does not turn a boundary supremum into an attained maximum.

For each fixed $\lambda >0$, maximizing the likelihood function over $\mu$ is equivalent to minimizing $A(\mu )={\sum }_i|x_i-\mu |$. Order the distinct observations as $x_{(1)}<\cdots <x_{(n)}$ and write $n=2m+1$. Between successive order statistics, with exactly $k$ observations to the left,Thus $A$ decreases up to $x_{(m+1)}$ and increases after it. Its unique minimum is at the sample median:At a fixed $\mu$ with $A(\mu )>0$, the log-likelihood function has derivative $n/\lambda -A(\mu )$ and second derivative $-n/{\lambda }^2<0$, so its unique maximizing rate is $n/A(\mu )$. The maximized likelihood function is $(n/[2eA(\mu )]{)}^n$, which increases when $A(\mu )$ decreases. Hence the joint maximum-likelihood estimates areThis is Laplace location and scale maximum likelihood, expressed with the reciprocal scale.

A finite rate maximum-likelihood estimate requires a positive total absolute deviation. For distinct observations and odd $n\ge 3$, that condition holds. For $n=1$, however, $\mu =x_1$ gives $A(\mu )=0$ and $L=\lambda /2\to \infty$ as $\lambda \to \infty$. There is no finite joint maximum-likelihood estimate in the stated parameter space; the formula with zero denominator is not an admissible estimate.

Let $n=2m\ge 2$. Between the two middle order statistics, $k=m$ and the slope $A^{\prime }(\mu )=2m-n$ is zero. To the left $A$ decreases and to the right it increases. Consequently all joint maximum-likelihood estimates arewhere the constant minimum isThe strict positivity follows from distinct observations. Thus the location maximum-likelihood estimate is not unique, while the rate maximum-likelihood estimate is unique and the same for every location in the displayed interval. Choosing the arithmetic midpoint of the two middle values is the usual sample median convention and gives one member of this entire maximizing family.