Given n
realizations \(x_i=(x_{i1}, \ldots,x_{id})\),
\(i \in \left\lbrace 1, \ldots,n \right\rbrace \)
of a random vector X
, the pseudo-observations are defined via
\(u_{ij}=r_{ij}/(n+1)\) for
\(i \in \left\lbrace 1, \ldots,n \right\rbrace\)
and
\(j \in \left\lbrace 1, \ldots,d \right\rbrace \), where
\(r_{ij}\) denotes the rank of \(x_{ij}\) among all \(x_{kj}\),
\(k \in \left\lbrace 1, \ldots,n \right\rbrace \).
The pseudo-observations can thus also be computed by component-wise applying
the empirical distribution functions to the data and scaling the result by
\(n/(n+1)\). This asymptotically negligible scaling factor is used to force
the variates to fall inside the open unit hypercube, for example, to avoid
problems with density evaluation at the boundaries.
When lower_tail = FALSE
, then pseudo_obs()
simply returns
1 - pseudo_obs()
.