Let \(X\) denote an identically and independently distributed continuous
variate with realizations \(x_i ~~ (1 \le i \le k)\).
Further, let the increasingly ordered realizations
denote \(x_{(1)} \le x_{(2)} \le \ldots \le x_{(n)}\). Then
the following model for a single maximum outlier can be proposed:
$$
x_{(i)} = \left\{
\begin{array}{lcl}
\mu + \epsilon_{(i)}, & \qquad & i = 1, \ldots, n - 1 \\
\mu + \Delta + \epsilon_{(n)} & & \\
\end{array} \right.$$
with \(\epsilon \approx N(0,\sigma)\). The null hypothesis,
H\(_0: \Delta = 0\) is tested against the alternative,
H\(_{\mathrm{A}}: \Delta > 0\).
For testing a single minimum outlier, the model can be proposed
as
$$
x_{(i)} = \left\{
\begin{array}{lcl}
\mu + \Delta + \epsilon_{(1)} & & \\
\mu + \epsilon_{(i)}, & \qquad & i = 2, \ldots, n \\
\end{array} \right.$$
The null hypothesis is tested against the alternative,
H\(_{\mathrm{A}}: \Delta < 0\).
The p-value is computed with the function pgrubbs.