Zero-and-one-augmented normal-kernel Vasicek-type distribution

Defines a normal-kernel Vasicek-type distribution augmented by point masses at zero and one. Conditional on an observation in $(0,1)$, the continuous component is NVASIM with mean $\mu$ and shape parameter $\sigma$.

Usage

d01NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1, log = FALSE)

p01NVASIM(
  q,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

q01NVASIM(
  p,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

r01NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1)

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

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

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

rZOANVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1)

ZOANVASIM(
  mu.link = "logit",
  sigma.link = "logit",
  nu.link = "logit",
  tau.link = "logit"
)

Arguments

x: Vector of values in $[0,1]$ at which the density or probability mass is evaluated. The distribution has support $[0,1]$.
mu: Mean of the continuous Vasicek component, in $(0,1)$.
sigma: Shape parameter of the continuous Vasicek component, in $(0,1)$.
nu: Probability at zero, $\nu=P(Y=0)$, in $(0,1)$.
tau: Conditional probability at one among nonzero observations, $\tau=P(Y=1\mid Y>0)$, in $(0,1)$. This parameter is unrelated to the fixed quantile level used by NVASIQ(), LVASIQ(), and HVASIQ().
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 $\nu$.
tau.link: Link function for $\tau$.

Value

ZOANVASIM() returns a four-parameter gamlss.family object. The functions d01NVASIM(), p01NVASIM(), q01NVASIM(), and r01NVASIM() return probability mass or density values, cumulative probabilities, quantiles, and random observations, respectively. dZOANVASIM(), pZOANVASIM(), qZOANVASIM(), and rZOANVASIM() are equivalent names following the GAMLSS family-name convention.

Details

Let $Y_c\sim\mathrm{NVASIM}(\mu,\sigma)$. Write $p_0$, $p_1$, and $p_c$ for the probabilities of zero, one, and the continuous component. The sequential BEOI-type parameterization is $$\nu=P(Y=0),\qquad \tau=P(Y=1\mid Y>0).$$ Hence, $$p_0=\nu,\qquad p_1=(1-\nu)\tau,\qquad p_c=(1-\nu)(1-\tau).$$ The distribution is $$P(Y=0)=p_0,\qquad P(Y=1)=p_1$$ and $$f_Y(y)=p_c f_{Y_c}(y\mid\mu,\sigma),\quad 0<y<1.$$ Its marginal mean and variance are $$E(Y)=(1-\nu)\left[\tau+(1-\tau)\mu\right]$$ and $$\mathrm{Var}(Y)= (1-\nu)\left[(1-\tau)\left\{\mathrm{Var}(Y_c)+\mu^2\right\} +\tau\right] -\left\{(1-\nu)\left[\tau+(1-\tau)\mu\right]\right\}^2.$$ Logit links for $\nu$ and $\tau$ guarantee valid probabilities. In the closure of the parameter space, $\nu=0$ gives the BEOI-type one-augmented model and $\tau=0$ gives the zero-augmented model. The implemented distribution functions and GAMLSS family use the open parameter space $0<\nu<1$ and $0<\tau<1$; these nested models are therefore limiting cases rather than admissible interior parameter values.

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 <- r01NVASIM(
  1000, mu = 0.60, sigma = 0.30, nu = 0.20, tau = 0.25
)
c(zero = mean(y == 0), one = mean(y == 1))
#>  zero   one 
#> 0.198 0.198 
mean(y)
#> [1] 0.5587152
(1 - 0.20) * (0.25 + (1 - 0.25) * 0.60)
#> [1] 0.56

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

## Real-data example with observations at both boundaries
data("aep", package = "vasicekreg")
aep$inappropriate <- with(aep, noinap / los)
fit_aep <- gamlss(
  inappropriate ~ sex + ward + year + age + loglos,
  sigma.formula = ~ loglos,
  nu.formula = ~ loglos,
  tau.formula = ~ loglos,
  family = ZOANVASIM(),
  data = aep,
  control = gamlss.control(n.cyc = 200, trace = FALSE)
)
} # }