# Moments and other quantities of a (possibly) truncated multivariate normal distribution Moments up to the specified orders of a possibly truncated `d`-dimensional normal distribution; the distribution must be non-degenerate. Each component variable can be truncated on one side (to the left or to the right) or on two sides or not truncated. After the initial stage, cumulants up to the fourth order and other quantities are computed, provided all moments of the required order had been computed in the first stage. ```r mom.mtruncnorm(powers=4, mean, varcov, lower, upper, cum = TRUE, ...) ``` ## Arguments - `powers`: a vector of non-negative integer values representing the required order of moments for each component variable, or a single such value, in which case this value is repeated for all component variables. - `mean`: a `d`-length vector representing the mean value of the pre-truncation normal random variable. If `d=length(mean)`, see Details for restrictions on `d`. - `varcov`: a symmetric positive definite matrix with dimensions `(d,d)` representing the variance matrix of the pre-truncation normal random variable. - `lower`: a `d`-vector representing the lower truncation values of the component variables; `-Inf` values are allowed. If missing, it is set equal to `rep(-Inf, d)`. - `upper`: a `d`-vector representing the upper truncation values of the component variables; `Inf` values are allowed. If missing, it is set equal to `rep(Inf, d)`. - `cum`: a logical value; if codeTRUE (default value), cumulants are other quantities are computed up to the minimum between the fourth order and the maximum possible order, given the available moments. - ``...``: additional arguments passed to `sadmvn`; see Details for a description. ## Details The maximal value of d is `20`. If this threshold is exceeded, `NA`s are returned. The constraint originates from the underlying function `sadmvn`. This function makes use of two workhorses, `recintab` and `mom2cum`, providing a user-friendly interface to these more basic tools. The first function computes an array of raw moments of the truncated normal; the second function translates them into cumulants and other quantities such as the Mardia's measures of skewness and kurtosis, unless `cum=FALSE`. See the documentation of these two underlying functions for additional information about the arguments and the returned quantities. The argument `...` is passed, via `recintab`, to `sadmvn` for regulation of its working. Not all `d` component variables need to be truncated. In fact, the function works also with no truncated components (just omit `lower` and `upper`), although for this case there exist known formulae to do the job. ## Returns A list with the following components: - **mom**: an array with raw moments as produced by `recintab`, followed by normalization; see its documentation for a description. - **cum1**: the vector of first-order cumulants, AKA the expected value or the mean value, which will be there provided `cum=TRUE` and all elements of `powers` are not less than 1. - **order2, ...**: additional lists with higher order terms up to order 4; these lists only exist when the available moments are of sufficiently high order. See `mom2cum` for a more detailed description. ## Author(s) Adelchi Azzalini ## See Also `recintab`, `mom2cum`, `sadmvn` ## Examples ```r mu <- c(1, -0.5, 0) Sigma <- toeplitz(1/(1:3)) lower <- c(-Inf, -3, -4) upper <- c(1.5, Inf, 2) m <- mom.mtruncnorm(1, mu, Sigma, lower, upper) print(m$cum1) # m <- mom.mtruncnorm(3, mu, Sigma, lower, upper) print(m$order3$gamma1.marginal) print(m$order3$gamma1.Mardia) # #-- # Example 2 of Leppard & Tallis (1989, Appl.Stat. vol.38, p.547) truncp <- c(0, 1, 2) U <- c(0, 0, 0) V <- 0.5*(diag(3) + matrix(1, 3, 3)) m <- mom.mtruncnorm(2, U, V, truncp) print(m$cum1, digits=9) print(m$order2$cum2, digits=9) ```

mom.mtruncnorm function