One-augmented normal-kernel Vasicek-type distribution

Defines a one-augmented normal-kernel Vasicek-type distribution for responses in $(0,1]$. The parameter $\nu$ is the probability at one. Conditional on an observation in $(0,1)$, the distribution is NVASIM with mean $\mu$ and shape parameter $\sigma$.

Usage

d1NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

p1NVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

q1NVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

r1NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

dOANVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

pOANVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qOANVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

rOANVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

OANVASIM(mu.link = "logit", sigma.link = "logit", nu.link = "logit")

Arguments

x: Vector of values in $[0,1]$ at which the density or probability mass is evaluated. The distribution has support $(0,1]$, and the returned value is zero at $x=0$.
mu: Mean of the continuous Vasicek component, in $(0,1)$.
sigma: Shape parameter of the continuous Vasicek component, in $(0,1)$.
nu: Probability at one, in $(0,1)$.
log: Logical; if TRUE, log probabilities or log densities are returned.
q: Vector of values in $[0,1]$ at which the cumulative distribution function is evaluated.
lower.tail: Logical; if TRUE, probabilities are $P(Y\leq y)$; otherwise, they are $P(Y>y)$.
log.p: Logical; if TRUE, probabilities are supplied or returned on the log scale.
p: Vector of probabilities.
n: Number of observations. If length(n) > 1, its length is taken to be the number required.
mu.link: Link function for $\mu$.
sigma.link: Link function for $\sigma$.
nu.link: Link function for $\nu$.

Value

OANVASIM() returns a gamlss.family object. The functions d1NVASIM(), p1NVASIM(), q1NVASIM(), and r1NVASIM() return probability mass or density values, cumulative probabilities, quantiles, and random observations, respectively. dOANVASIM(), pOANVASIM(), qOANVASIM(), and rOANVASIM() are equivalent names following the GAMLSS family-name convention.

Details

Let $Y_c\sim\mathrm{NVASIM}(\mu,\sigma)$ and let $0<\nu<1$. The BEOI-type one-augmented distribution is defined by $$P(Y=1)=\nu$$ and $$f_Y(y)=(1-\nu)f_{Y_c}(y\mid\mu,\sigma),\quad 0<y<1.$$ Consequently, $$E(Y)=\nu+(1-\nu)\mu$$ and $$\mathrm{Var}(Y)=(1-\nu)\mathrm{Var}(Y_c)+ \nu(1-\nu)(1-\mu)^2.$$ Thus, $\mu=E(Y\mid 0<Y<1)$ is the mean of the continuous component, whereas $\nu+(1-\nu)\mu$ is the marginal mean.

References

Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions. Statistical Papers, 51, 111–126.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape. Applied Statistics, 54(3), 507–554.

Examples

set.seed(123)
y <- r1NVASIM(1000, mu = 0.60, sigma = 0.30, nu = 0.20)
mean(y == 1)
#> [1] 0.198
mean(y)
#> [1] 0.6782138
0.20 + (1 - 0.20) * 0.60
#> [1] 0.68

if (FALSE) { # \dontrun{
library(gamlss)
fit <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  family = OANVASIM(),
  control = gamlss.control(trace = FALSE)
)
} # }