cardioid: Cardioid Distribution Family Function

Description

Estimates the two parameters of the cardioid distribution by maximum likelihood estimation.

Usage

cardioid(lmu = extlogitlink(min = 0, max = 2*pi),
         lrho = extlogitlink(min = -0.5, max = 0.5),
         imu = NULL, irho = 0.3, nsimEIM = 100, zero = NULL)

Value

An object of class "vglmff" (see

vglmff-class). The object is used by modelling functions such as vglm, rrvglm

and vgam.

Arguments

lmu, lrho: Parameter link functions applied to the $μ$ and $ρ$ parameters, respectively. See Links for more choices.
imu, irho: Initial values. A NULL means an initial value is chosen internally. See CommonVGAMffArguments for more information.
nsimEIM, zero: See CommonVGAMffArguments for more information.

Author

T. W. Yee

Warning

Numerically, this distribution can be difficult to fit because of a log-likelihood having multiple maximums. The user is therefore encouraged to try different starting values, i.e., make use of imu and irho.

Details

The two-parameter cardioid distribution has a density that can be written as $f (y; μ, ρ) = \frac{1}{2 π} (1 + 2 ρ \cos (y - μ))$ where $0 < y < 2 π$ , $0 < μ < 2 π$ , and $- 0.5 < ρ < 0.5$ is the concentration parameter. The default link functions enforce the range constraints of the parameters.

For positive $ρ$ the distribution is unimodal and symmetric about $μ$ . The mean of $Y$ (which make up the fitted values) is $π + (ρ / π) ((2 π - μ) \sin (2 π - μ) + \cos (2 π - μ) - μ \sin (μ) - \cos (μ))$ .

References

Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics, Singapore: World Scientific.

Examples

Run this code

if (FALSE) {
cdata <- data.frame(y = rcard(n = 1000, mu = 4, rho = 0.45))
fit <- vglm(y ~ 1, cardioid, data = cdata, trace = TRUE)
coef(fit, matrix=TRUE)
Coef(fit)
c(with(cdata, mean(y)), head(fitted(fit), 1))
summary(fit)
}

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