Zero-augmented normal-kernel Vasicek-type distribution

Defines a zero-augmented normal-kernel Vasicek-type distribution for responses in $[0,1)$. The parameter $\nu$ is the probability of a structural zero. Conditional on a positive response, the distribution is NVASIM with mean $\mu$ and shape parameter $\sigma$.

Usage

d0NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

p0NVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

q0NVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

r0NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

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

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

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

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

ZANVASIM(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 $x=1$.
mu: Mean of the positive Vasicek component, in $(0,1)$.
sigma: Shape parameter of the positive Vasicek component, in $(0,1)$.
nu: Probability of a structural zero, 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

ZANVASIM() returns a gamlss.family object. The functions d0NVASIM(), p0NVASIM(), q0NVASIM(), and r0NVASIM() return density or probability mass values, cumulative probabilities, quantiles, and random observations, respectively. dZANVASIM(), pZANVASIM(), qZANVASIM(), and rZANVASIM() are equivalent names following the GAMLSS family-name convention.

Details

Let $Y_+\sim\mathrm{NVASIM}(\mu,\sigma)$ and let $0<\nu<1$. The zero-augmented distribution is defined by $$P(Y=0)=\nu$$ and $$f_Y(y)=(1-\nu)f_{Y_+}(y\mid\mu,\sigma),\quad 0<y<1.$$ Its cumulative distribution function is $$F_Y(y)=\nu+(1-\nu)F_{Y_+}(y\mid\mu,\sigma),\quad 0<y<1.$$ Consequently, $$E(Y)=(1-\nu)\mu$$ and $$\mathrm{Var}(Y)=(1-\nu)\mathrm{Var}(Y_+)+ \nu(1-\nu)\mu^2.$$

Thus, $\mu$ is the mean conditional on $Y>0$; it is not the marginal mean when $\nu>0$. The marginal mean is $(1-\nu)\mu$.

References

Mazucheli, J., Alves, B., Korkmaz, M. C., and Leiva, V. (2022). Vasicek quantile and mean regression models for bounded data: New formulation, mathematical derivations, and numerical applications. Mathematics, 10, 1389. doi:10.3390/math10091389

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 <- r0NVASIM(1000, mu = 0.60, sigma = 0.30, nu = 0.20)
mean(y == 0)
#> [1] 0.198
mean(y)
#> [1] 0.478592
(1 - 0.20) * 0.60
#> [1] 0.48

library(gamlss)
fit <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  family = ZANVASIM(),
  control = gamlss.control(trace = FALSE)
)
fitted(fit, what = "mu")[1]
#> [1] 0.5972124
fitted(fit, what = "sigma")[1]
#> [1] 0.2947561
fitted(fit, what = "nu")[1]
#> [1] 0.198