Hyperbolic-secant-kernel Vasicek-type quantile distribution

HVASIQ() defines the hyperbolic-secant-kernel Vasicek-type distribution as a gamlss.family object for conditional quantile regression. The functions dHVASIQ(), pHVASIQ(), qHVASIQ(), and rHVASIQ() provide the density, distribution function, quantile function, and random generation.

Usage

dHVASIQ(x, mu, sigma, quantile = 0.5, log = FALSE)

pHVASIQ(q, mu, sigma, quantile = 0.5, lower.tail = TRUE, log.p = FALSE)

qHVASIQ(p, mu, sigma, quantile = 0.5, lower.tail = TRUE, log.p = FALSE)

rHVASIQ(n, mu, sigma, quantile = 0.5)

HVASIQ(quantile = 0.5, mu.link = "logit", sigma.link = "logit")

Arguments

x: Vector of quantiles in $(0,1)$.
mu: Vector of conditional $\tau$-th quantiles, $0<\mu<1$.
sigma: Vector of shape parameter values, $0<\sigma<1$.
quantile: Fixed quantile level $\tau\in(0,1)$ represented by $\mu$, used in the distribution functions and in HVASIQ().
log: Logical; if TRUE, returns the log-density.
q: Vector of values in $[0,1]$ at which the cumulative distribution function is evaluated.
lower.tail: Logical; if TRUE, probabilities are $P(X\leq x)$.
log.p: Logical; if TRUE, probabilities are supplied or returned on the logarithmic scale.
p: Vector of probabilities in $[0,1]$ on the probability scale.
n: Number of observations.
mu.link: Link function for $\mu$.
sigma.link: Link function for $\sigma$.

Value

dHVASIQ() returns the density, pHVASIQ() the distribution function, qHVASIQ() the quantile function, and rHVASIQ() random deviates. HVASIQ() returns a gamlss.family object.

Details

Let $$H(w)=\frac{2}{\pi}\arctan\{\exp(w)\},\qquad h(w)=\frac{1}{\pi\cosh(w)},$$ and $$q(u)=H^{-1}(u)= \log\left\{\tan\left(\frac{\pi u}{2}\right)\right\}.$$ For fixed $\tau\in(0,1)$, define $$a=\sqrt{\frac{1-\sigma}{\sigma}},\qquad z=a\{q(x)-q(\mu)\}+q(\tau).$$ The cumulative distribution function and density are $$F(x\mid\mu,\sigma,\tau)=H(z)$$ and $$f(x\mid\mu,\sigma,\tau)=a\frac{h(z)}{h\{q(x)\}}.$$ The quantile function is $$Q(p\mid\mu,\sigma,\tau)= H\left[q(\mu)+\sqrt{\frac{\sigma}{1-\sigma}} \{q(p)-q(\tau)\}\right].$$ Consequently, $Q(\tau)=\mu$; $\mu$ is exactly the conditional $\tau$-th quantile and is not the conditional mean in general.

The GAMLSS family uses analytical derivatives. For one observation, let $$d=q(y)-q(\mu),\quad z=ad+q(\tau),\quad T=\tanh(z),\quad S=\operatorname{sech}^2(z),$$ $$b=\frac{1}{2\sigma(1-\sigma)},\qquad g=\frac{\pi}{\sin(\pi\mu)}.$$ If $\ell$ is the individual log-likelihood contribution, then $$\frac{\partial\ell}{\partial\mu}=agT,$$ $$\frac{\partial\ell}{\partial\sigma}=b(adT-1),$$ $$\frac{\partial^2\ell}{\partial\mu^2} =g^2\{-a\cos(\pi\mu)T-a^2S\},$$ $$\frac{\partial^2\ell}{\partial\mu\,\partial\sigma} =-abg(T+adS),$$ and $$\frac{\partial^2\ell}{\partial\sigma^2} =b^2\{2(1-2\sigma)-(3-4\sigma)adT-a^2d^2S\}.$$ The conditional mean and variance do not have elementary closed forms and are evaluated by numerical integration of the quantile function.

The fixed level is supplied through the quantile argument. It is stored in the family definition and embedded as a numeric literal in the family components used by GAMLSS; no global variable is required.

References

Dunn, P. K. and Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5(3), 236–244.

Fischer, M. J., Hui, A. and Hösle, S. (2017). wHS-type distributions with application to finance. Journal of Statistics and Management Systems, 20(1), 67–89. doi:10.1080/09720510.2016.1190575

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

Author

Josmar Mazucheli jmazucheli@gmail.com

Examples

set.seed(123)
y <- rHVASIQ(500, mu = 0.60, sigma = 0.30, quantile = 0.25)

fit <- gamlss::gamlss(
    y ~ 1,
    sigma.formula = ~ 1,
    family = HVASIQ(quantile = 0.25),
    control = gamlss::gamlss.control(trace = FALSE)
)
fitted(fit, what = "mu")[1]
#> [1] 0.5968614