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Logistic-kernel Vasicek-type distribution with quantile parameterization

LVASIQ.Rd

The function LVASIQ() defines the logistic-kernel Vasicek-type distribution as a gamlss.family object for conditional quantile regression. The functions dLVASIQ, pLVASIQ, qLVASIQ, and rLVASIQ give the density, distribution function, quantile function, and random generation. The parameter \(\mu\) is the conditional \(\tau\)-th quantile (\(0<\mu<1\)), \(\sigma\) is a shape parameter (\(0<\sigma<1\)), and \(\tau\in(0,1)\) is fixed by the user. The fixed level is supplied through the quantile argument.

Usage

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

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

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

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

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

Arguments

x

Vector of quantiles in \((0,1)\).

mu

Vector of \(\tau\)-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 LVASIQ().

log

Logical; if TRUE, the log-density is returned.

q

Vector of values in \([0,1]\) at which the cumulative distribution function is evaluated.

lower.tail

Logical; if TRUE, probabilities are \(P(X\le x)\).

log.p

Logical; if TRUE, probabilities p are given as log(p) or cumulative probabilities are returned on the log scale, as appropriate.

p

Vector of probabilities in \([0,1]\) on the probability scale.

n

Number of observations.

Link function for the \(\mu\) parameter.

Link function for the \(\sigma\) parameter.

Value

dLVASIQ gives the density, pLVASIQ the distribution function, qLVASIQ the quantile function, and rLVASIQ generates random deviates. LVASIQ() returns a gamlss.family object.

Details

Let \(\mathrm{logit}(u)= \log\left(\frac{u}{1-u}\right)\) and \(\Lambda(z)=\frac{1}{1+e^{-z}}\), with \(\lambda(z)= \Lambda(z)\left[1-\Lambda(z)\right]\). Define $$z = \sqrt{\frac{1-\sigma}{\sigma}} \left[\mathrm{logit}(x)-\mathrm{logit}(\mu)\right] +\mathrm{logit}(\tau).$$

Cumulative distribution function $$F(x\mid\mu,\sigma,\tau)=\Lambda(z).$$

Probability density function $$f(x\mid\mu,\sigma,\tau)= \sqrt{\frac{1-\sigma}{\sigma}} \frac{\lambda(z)}{x(1-x)}.$$

Quantile function $$Q(p\mid\mu,\sigma,\tau)= \Lambda\!\left\{ \mathrm{logit}(\mu) +\sqrt{\frac{\sigma}{1-\sigma}} \left[\mathrm{logit}(p)-\mathrm{logit}(\tau)\right] \right\}.$$

By construction \(Q(\tau)=\mu\), i.e. \(\mu\) is the \(\tau\)-th quantile. Note that, unlike the normal-kernel Vasicek distribution, the logistic kernel does not yield a closed-form mean; in particular \(E(X)\neq\mu\) in general.

The GAMLSS family uses analytical derivatives. For one observation, let $$a=\sqrt{\frac{1-\sigma}{\sigma}},\qquad d=\mathrm{logit}(y)-\mathrm{logit}(\mu),\qquad P=\Lambda\left\{ad+\mathrm{logit}(\tau)\right\},$$ and define $$V=P(1-P),\qquad b=\frac{1}{2\sigma(1-\sigma)},\qquad g=\frac{1}{\mu(1-\mu)}.$$ If \(\ell\) denotes the individual log-likelihood contribution, the first derivatives are $$\frac{\partial\ell}{\partial\mu} =-ag(1-2P)$$ and $$\frac{\partial\ell}{\partial\sigma} =-b\left\{1+ad(1-2P)\right\}.$$ The second and cross derivatives are $$\frac{\partial^2\ell}{\partial\mu^2} =g^2\left\{ a(1-2\mu)(1-2P)-2a^2V \right\},$$ $$\frac{\partial^2\ell}{\partial\mu\,\partial\sigma} =abg\left\{ (1-2P)-2adV \right\},$$ and $$\frac{\partial^2\ell}{\partial\sigma^2} =b^2\left\{ 2(1-2\sigma) +ad(3-4\sigma)(1-2P) -2a^2d^2V \right\}.$$ These expressions are evaluated directly by LVASIQ(); numerical differentiation is not used. The mean and variance components of the family object use numerical quadrature because the corresponding moments do not have elementary closed forms.

Note

The level supplied through quantile 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

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.

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 160–162.

Witzany, J. (2013). A note on the Vasicek's model with the logistic distribution. Ekonomický časopis (Journal of Economics), 61(10), 1053–1066.

Author

Josmar Mazucheli jmazucheli@gmail.com

Examples

set.seed(123)
x <- rLVASIQ(n = 1000, mu = 0.50, sigma = 0.25, quantile = 0.5)
S <- seq(min(x), max(x), length.out = 1000)

hist(x, prob = TRUE, main = "Logistic-kernel Vasicek-type")
lines(S, dLVASIQ(x = S, mu = 0.50, sigma = 0.25, quantile = 0.5), col = 2)


plot(ecdf(x))
lines(S, pLVASIQ(q = S, mu = 0.50, sigma = 0.25, quantile = 0.5), col = 2)


data <- data.frame(
  y = rLVASIQ(n = 100, mu = 0.50, sigma = 0.25, quantile = 0.50)
)
fit <- gamlss::gamlss(
    y ~ 1,
    data = data,
    family = LVASIQ(
      quantile = 0.50, mu.link = "logit", sigma.link = "logit"
    )
)
#> GAMLSS-RS iteration 1: Global Deviance = -55.5222 
#> GAMLSS-RS iteration 2: Global Deviance = -55.5263 
#> GAMLSS-RS iteration 3: Global Deviance = -55.5263 
fitted(fit, what = "mu")[1:5]
#> [1] 0.4838788 0.4838788 0.4838788 0.4838788 0.4838788