spline_cdf: Approximate density function, CDF, or quantile function on the interior of provided quantiles by representing the distribution as a sum of a discrete part at any duplicated `qs` and a continuous part for which the CDF is estimated using a monotonic Hermite spline. See details for more.

Description

Approximate density function, CDF, or quantile function on the interior of provided quantiles by representing the distribution as a sum of a discrete part at any duplicated qs and a continuous part for which the CDF is estimated using a monotonic Hermite spline. See details for more.

Usage

spline_cdf(ps, qs, tail_dist, fn_type = c("d", "p", "q"), n_grid = 20)

Value

a function to evaluate the PDF, CDF, or quantile function.

Arguments

ps: vector of probability levels
qs: vector of quantile values corresponding to ps
tail_dist: name of parametric distribution for the tails
fn_type: the type of function that is requested: "d" for a PDF, "p" for a CDF, or "q" for a quantile function.
n_grid: grid size to use when augmenting the input qs to obtain a finer grid of points along which we form a piecewise linear approximation to the spline. n_grid evenly-spaced points are inserted between each pair of consecutive values in qs. The default value is 20. This can be set to NULL, in which case the piecewise linear approximation is not used. This is not recommended if the fn_type is "q".

Details

The CDF of the continuous part of the distribution is estimated using a monotonic degree 3 Hermite spline that interpolates the quantiles after subtracting the discrete distribution and renormalizing. In theory, an estimate of the quantile function could be obtained by directly inverting this spline. However, in practice, we have observed that this can suffer from numerical problems. Therefore, the default behavior of this function is to evaluate the "stage 1" CDF estimate corresponding to discrete point masses plus monotonic spline at a fine grid of points, and use the "stage 2" CDF estimate that linearly interpolates these points with steps at any duplicated q values. The quantile function estimate is obtained by inverting this "stage 2" CDF estimate. When the distribution is continuous, we can obtain an estimate of the PDF by differentiating the CDF estimate, resulting in a discontinuous "histogram density". The size of the grid can be specified with the n_grid argument. In settings where it is desirable to obtain a continuous density function, the "stage 1" CDF estimate can be used by setting n_grid = NULL.