quapdq4: Quantile Function of the Polynomial Density-Quantile4 Distribution

Description

This function computes the quantiles of the Polynomial Density-Quantile4 distribution (PDQ4) given parameters ($\xi$, $\alpha$, and $\kappa$) computed by parpdq4. The quantile function for $0 < \kappa < 1$ is $$x(F) = \xi + \alpha \biggl[\log\biggl(\frac{F}{1-F}\biggr) - 2\kappa\;\mathrm{atanh}(\kappa[2F-1])\biggr] \mbox{\ and}$$ for $-\infty < \kappa < 0$ is $$x(F) = \xi + \alpha \biggl[\log\biggl(\frac{F}{1-F}\biggr) + 2\kappa\;\mathrm{atan}(\kappa[2F-1])\biggr] \mbox{,}$$ where $x(F)$ is the quantile for nonexceedance probability $F$, $\xi$ is a location parameter, $\alpha$ is a scale parameter, and $\kappa$ is a shape parameter. The range of the distribution is $-\infty < x < \infty$.

Usage

quapdq4(f, para, paracheck=TRUE)

Value

Quantile value for nonexceedance probability $F$.

Arguments

f: Nonexceedance probability ($0 \le F \le 1$).
para: The parameters from parpdq4 or vec2par.
paracheck: A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the quantile function in the context of TL-moments with nonzero trimming.

Author

W.H. Asquith

Details

The PDQ4 was proposed by Hosking (2007) with the core justification of maximizing entropy and that “maximizing entropy subject to a set of constraints can be regarded as deriving a distribution that is consistent with the information specified in the constraints while making minimal assumptions about the form of the distribution other than those embodied in the constraints.” The PDQ4 is that family constrained to the $\lambda_1$, $\lambda_2$, and $\tau_4$ values of the L-moments. (See also the Polynomial Density-Quantile3 function for constraint on $\lambda_1$, $\lambda_2$, and $\tau_3$ values of the L-moments, quapdq3.)

The PDQ4 is a symmetrical distribution ($\tau_3 = 0$ everywhere) that has maximum entropy conditional on having specified values for the L-moments of $\lambda_1$, $\lambda_2$, and $\lambda_4 = \tau_4\lambda_2$ with $\lambda_3 = \tau_3 = 0$. The tails of the PDQ4 are exponentially decreasing and the distribution could be useful in distributional analysis with data showing similar tail characteristics. The attainable L-kurtosis range is $-1/4 < \tau_4 < 1$ with the sign change from negative to positive of $\kappa$ occurring at $\tau_4 = 1/6$. Finally, PDQ4 generalizes the logistic distribution, which is the special case $\kappa \rightarrow 0$, and contains distributions both lighter-tailed ($\kappa < 0$) and heavier-tailed ($\kappa > 0$) than the logistic.

References

Hosking, J.R.M., 2007, Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics: Journal of Statistical Planning and Inference, v. 137, no. 9, pp. 2,870--2891, tools:::Rd_expr_doi("10.1016/j.jspi.2006.10.010").