Maximum likelihood estimates can usually be found as turning points of the likelihood function (or equivalently the log-likelihood function) — i.e. by solving \(\ell'(\theta) = 0\). However this method does not work in a few examples.

Rectangular distribution

The following six values,

0.12   0.32   0.36   0.51   0.63   0.69

are a random sample from a rectangular distribution,

\[ X \;\; \sim \; \; \RectDistn(0, \;\beta) \]

This distribution has likelihood is

\[ L(\beta) \;\;=\;\; \prod_{i=1}^6 {f(x_i \;|\; \beta)} \;\;=\;\; \begin{cases} \left(\dfrac 1 {\beta}\right)^6 &\text{for } \beta \ge \max(x_1, \dots, x_6) \\[0.4em] 0 &\text{otherwise} \end{cases} \]

This is illustrated below for a few values of \(\beta\). The red lines give the values of \(f(x \;|\; \beta)\) at the data points; their product gives the likelihood.

When \(\beta\) is less than the maximum data value, 0.690, the pdf at this value is zero, so the likelihood is zero. As \(\beta\) increases above 0.690, the pdfs for all data values decrease, and so does the likelihood. The likelihood function is shown below.

The maximum likelihood estimate is at a discontinuity in the likelihood function not at a turning point, so the MLE cannot be found solving \(\ell'(\beta) = 0\).

Bias and standard error

The 2nd derivative of the log-likelihood function is undefined at the MLE and cannot be used to obtain an approximate standard error. However formulae for its mean and standard deviation can be found from first principles — we will derive them later.

\[ E\left[\hat{\beta}\right] \;=\; \frac n {n+1} \beta \spaced{and} \se\left(\hat{\beta}\right) \;=\; \sqrt {\frac n {(n+1)^2(n+2)}}\times \beta \]

The estimator is therefore biased but is consistent since its bias and standard error both tend to zero as \(n \to \infty\).