Increase the dispersion of a distribution

Increases the dispersion (spread) of a distribution by transforming its quantile function in a standardized Q-Q space.

Usage

widen(x, scale, knots = NULL, name = NULL)

Arguments

x: a distribution as a dist_fns S3 object
scale: acts as a multiplier for the log-odds difference from the median, effectively acting like an odds-ratio parameter. A scale > 1 increases dispersion (stretches quantiles away from the median in logit space), while 0 < scale < 1 decreases dispersion (compresses quantiles towards the median in logit space). The median value is preserved in the original parameter space.
knots: the number of knots in the transformed distribution, if it is not already an empirical CDF distribution.
name: a name for the widened distribution

Value

an empirical dist_fn with the same median and increased SD. This transformation will change the mean of skewed distributions.

Details

The transformation aims to increase the standard deviation by a factor scale while preserving the median. It operates on the internal Q-Q space representation (qx, qy) of an empirical distribution generated by empirical_cdf.

Applies a logit-space scaling transformation centred on the median quantile. This transformation modifies the quantile axis (qx) in the Q-Q space of an empirical distribution to change its dispersion while preserving the median value.

Let qx be the original quantile coordinate in [0, 1], and qmedian be the quantile corresponding to the median (e.g., qx_from_qy(0.5)). The transformation is defined as:

$$ qx_2 = \text{expit}\left( (\text{logit}(qx) - \text{logit}(qmedian)) \times \text{scale} + \text{logit}(qmedian) \right) $$

where $\text{logit}(x) = \log(x / (1 - x))$ and $\text{expit}(x) = 1 / (1 + \exp(-x))$ are the standard logit and inverse-logit functions, respectively.

Examples

d1 = as.dist_fns("norm",4,2)
w1 = widen(d1, scale=1.5)