SUNdistr-op function

Operations on SUNdistr-class objects

Given an object of SUNdistr-class, or possibly two such things in some cases, the functions perform various operations, and produce a new object of the same class. package

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affineTransSUNdistr(object, a, A, name, compNames, HcompNames, drop = TRUE)
conditionalSUNdistr(object, comp, values, eventType = "=", name, drop = TRUE) 
convolutionSUNdistr(object1, object2, name, compNames, HcompNames) 
joinSUNdistr(object1, object2, name, compNames, HcompNames) 
marginalSUNdistr(object, comp, name, drop=TRUE)

Arguments

• object, object1, object2: objects of class SUNdistr
• a: a numeric vector; see Details
• A: a numeric matrix; see Details
• name: an optional character string with the name of the returned distribution
• compNames: an optional vector of character strings with the names of the component variables of the returned distribution
• HcompNames: an optional vector of character strings with the names of the hidden variables of the returned distribution
• drop: a logical value (default: TRUE) relevant only in the case m=1. When both m=1 and drop=TRUE, the returned object is of class either SECdistrUv or SECdistrMv, depending on the dimension of the returned object, and family "SN" or "ESN", as appropriate.
• comp: a vector of integers representing the selected components
• values: a numeric vector which identifies the conditioning event
• eventType: a single character value which indicates the type of the conditioning event, as described in the Details section; possible values are "=" (default) and ">"

Details

For an object which represents the distribution of a multivariate SUN random variable $Y$ of dimension d, say, a number of operations are possible, producing a new object of the same class. This object could have been created by makeSUNdistr

or it could be the outcome from some previous call to one of the functions described here.

The function affineTransSUNdistr computes the distribution of $a+A'Y$, provided A is a full-rank matrix with nrow(A)=d and length(a)=ncol(A). See equation (7.6) of Azzalini & Capitanio (2014).

The function marginalSUNdistr builds a SUN distribution from the components selected by the comp vector.

A conditional distribution can be computed using conditionalSUNdistr

for two type of events, selected by eventType. The "=" case corresponds to the event $Y₁=y₁$ where $Y₁$ is the subset of components identified by the comp argument, $y₁$ is vector specified by the values argument and the equality sign must hold for each component. See equation (7.6) of Azzalini & Capitanio (2014).

If conditionalSUNdistr is used with eventType=">", the conditiong refers to the event $Y₁>y₁$, where the inequality must be interpreted components-wise; see Arellano-Valle & Azzalini (2021) for the underlying mathematical result. If the conditional distribution is required for the reverse inequality condition, "<" say, this is equivalent to consideration of the event $-Y₁>-y₁$. The corresponding distribution can be obtained in two steps: first a new variable is constructed reversing the sign of the required components using affineTransSUNdistr; then conditionalSUNdistr is applied to this new variable with the ">" condition and values $-y₁$. More complex conditions, where the "<" and ">" signs are mixed for different component varables, can be handled similarly, by introducing a square matrix A for affineTransSUNdistr

having an appropriate combination of 1s' and -1's on its main diagonal, and 0's elsewhere, and matching changes of sign to the components of $y₁$.

Functions convolutionSUNdistr and joinSUNdistr operate under the assumptions that object1 and object2 refer to independent variables. Specifically, convolutionSUNdistr computes the convolution of the two objects (i.e. the distribution of the sum of two independent variables), which must have the same dimension d. Function joinSUNdistr combines two objects into a joint distribution.

If the arguments name, compNames and HcompNames

are missing, they are composed from the supplied arguments.

Returns

an object of SUNdistr-class

References

Arellano-Valle, R. B. and Azzalini, A. (2021). Some properties of the unified skew-normal distribution. Statistical Papers, tools:::Rd_expr_doi("https://doi.org/10.1007/s00362-021-01235-2")

and arXiv:2011.06316

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.

Author(s)

Adelchi Azzalini

Note

The present structure and user interface of this function, and of other ones related to the SUN distribution, must be considered experimental, and they might possibly change in the future.

See Also

SUNdistr-base, makeSUNdistr, SUNdistr-class

Examples

xi <- c(1, 0, -1)
Omega <- matrix(c(2,1,1, 1,3,1, 1,1,4), 3, 3)
Delta <- matrix(c(0.72,0.20, 0.51,0.42, 0.88, 0.94), 3, 2, byrow=TRUE)
Gamma <- matrix(c(1, 0.8, 0.8, 1), 2, 2)
dp3 <- list(xi=xi, Omega=Omega, Delta=Delta, tau=c(-0.5, 0), Gamma=Gamma)
sun3 <- makeSUNdistr(dp=dp3, name="SUN3", compNames=c("x", "w", "z"))
#
a <- c(1,-2)
A <- matrix(1:6, 3, 2)
sun2at <- affineTransSUNdistr(sun3, a, A, "SUN2at", compNames=c("at1", "at2"))
sun2m <- marginalSUNdistr(sun3, comp=c(1,3), name="SUN2m")
sun1c <- conditionalSUNdistr(sun3, comp=c(1,3), values=c(1.1, 0.8),
            eventType=">", name="SUN1c", drop=FALSE)
#
Omega <- matrix(c(5, 1, 1, 6), 2, 2)
Delta <- matrix(c(0.30, 0.50, 0.50, 0.85), 2, 2, byrow=TRUE)
Gamma <- matrix(c(1, 0.18, 0.18, 1), 2, 2)
tau <- c(0.4, -0.8)
dp2 <- list(x=c(1, 0), Omega=Omega, Delta=Delta, tau=tau, Gamma=Gamma)
sun2 <-  makeSUNdistr(dp=dp2, name="SUN2", compNames=c("u", "v"))
#
sun2conv <- convolutionSUNdistr(sun2, sun2m, name="SUN2sum")
sun5 <- joinSUNdistr(sun3, sun2)