VGAM (version 1.1-8)

cens.gumbel: Censored Gumbel Distribution

Description

Maximum likelihood estimation of the 2-parameter Gumbel distribution when there are censored observations. A matrix response is not allowed.

Usage

cens.gumbel(llocation = "identitylink", lscale = "loglink", iscale = NULL,
            mean = TRUE, percentiles = NULL, zero = "scale")

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Arguments

llocation, lscale

Character. Parameter link functions for the location and (positive) \(scale\) parameters. See Links for more choices.

iscale

Numeric and positive. Initial value for \(scale\). Recycled to the appropriate length. In general, a larger value is better than a smaller value. The default is to choose the value internally.

mean

Logical. Return the mean? If TRUE then the mean is returned, otherwise percentiles given by the percentiles argument.

percentiles

Numeric with values between 0 and 100. If mean=FALSE then the fitted values are percentiles which must be specified by this argument.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The value (possibly values) must be from the set {1,2} corresponding respectively to \(location\) and \(scale\). If zero=NULL then all linear/additive predictors are modelled as a linear combination of the explanatory variables. The default is to fit the shape parameter as an intercept only. See CommonVGAMffArguments for more information.

Author

T. W. Yee

Warning

Numerical problems may occur if the amount of censoring is excessive.

Details

This VGAM family function is like gumbel but handles observations that are left-censored (so that the true value would be less than the observed value) else right-censored (so that the true value would be greater than the observed value). To indicate which type of censoring, input extra = list(leftcensored = vec1, rightcensored = vec2) where vec1 and vec2 are logical vectors the same length as the response. If the two components of this list are missing then the logical values are taken to be FALSE. The fitted object has these two components stored in the extra slot.

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

See Also

gumbel, gumbelff, rgumbel, guplot, gev, venice.

Examples

# Example 1
ystar <- venice[["r1"]]  # Use the first order statistic as the response
nn <- length(ystar)
L <- runif(nn, 100, 104)  # Lower censoring points
U <- runif(nn, 130, 135)  # Upper censoring points
y <- pmax(L, ystar)  # Left  censored
y <- pmin(U, y)      # Right censored
extra <- list(leftcensored = ystar < L, rightcensored = ystar > U)
fit <- vglm(y ~ scale(year), data = venice, trace = TRUE, extra = extra,
            fam = cens.gumbel(mean = FALSE, perc = c(5, 25, 50, 75, 95)))
coef(fit, matrix = TRUE)
head(fitted(fit))
fit@extra

# Example 2: simulated data
nn <- 1000
ystar <- rgumbel(nn, loc = 1, scale = exp(0.5))  # The uncensored data
L <- runif(nn, -1, 1)  # Lower censoring points
U <- runif(nn,  2, 5)  # Upper censoring points
y <- pmax(L, ystar)  # Left  censored
y <- pmin(U, y)      # Right censored
if (FALSE) par(mfrow = c(1, 2)); hist(ystar); hist(y);
extra <- list(leftcensored = ystar < L, rightcensored = ystar > U)
fit <- vglm(y ~ 1, trace = TRUE, extra = extra, fam = cens.gumbel)
coef(fit, matrix = TRUE)