Some multivariate singular unified skew-normal distributions and their application

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X-(r,X- ...,X- r + k) = (X-(r), ..., X(r + K))(T) from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X-(r + j,X- ...,X- r + k) given X-(r,X- ...,X- r + j - 1), X-(r,X- ...,X- r + k) given (X-(r) > t)and X((r, ..., r + k))given (X(r + k) < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.