Cardioid Distribution Family Function

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

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)

Arguments

lmu, lrho

Parameter link functions applied to the $\mu$ and $\rho$ 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.

Details

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

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

Value

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

References

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

Author

T. W. Yee

Note

Fisher scoring using simulation is used.

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.

Examples

if (FALSE) { # \dontrun{
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)
} # }