Approximate the Cramer’s distance between a pair of distributions
F and G that are represented by a collection of unequally-spaced quantiles.

`calc_cramers_dist_unequal_space(q_F, tau_F, q_G, tau_G, approx_rule)`

## Arguments

- q_F
vector containing the quantiles of F

- tau_F
vector containing the probability levels corresponding to
the quantiles of F.

- q_G
vector containing the quantiles of G

- tau_G
vector containing the probability levels corresponding to
the quantiles of G.

- approx_rule
string specifying which formula to use
for approximation. Valid rules are "left_sided_riemann" and
"trapezoid_riemann". See Details for more information.

## Value

a single value of approximated pairwise Cramér distance
between q_F and q_G

## Details

This function accommodates cases when the lengths of
`q_F`

and `q_G`

are not equal and when `tau_F`

and `tau_G`

are not equal.
A Riemann sum is used to approximate a pairwise Cramér distance.
The approximation formula for "left_sided_riemann" is
$$ \text{CD}(F,G) \approx \left\{\sum^{2K-1}_{j=1}(\tau^F_j-\tau^G_j)^2(q_{i+1}-q_i)\right\}
$$
and the approximation formula for "trapezoid_riemann" is
$$ \text{CD}(F,G) \approx \left\{\frac{1}{(K+1)^2}\sum^{2K-1}_{i=1}\frac{(\tau^F_j-\tau^G_j)^2+(\tau^F_{j+1}-\tau^G_{j+1})^2}{2}(q_{i+1}-q_i)\right\}
$$
where \(q_i\) is an element in a vector of an ordered pooled quantiles
of `q_F`

and `q_G`

and \(\tau^F_j\) and \(\tau^G_j\) are defined as
the probability level of a quantile in `q_F`

when \(q_i\) comes from \(F\) and
the probability level of a quantile in `q_G`

when \(q_i\) comes from \(G\),
respectively.

## Examples

```
f_vector <- 1:9
tau_F_vector <- tau_G_vector <- seq(0.1, 0.9, 0.1)
g_vector <- seq(4, 20, 2)
calc_cramers_dist_unequal_space(f_vector, tau_F_vector, g_vector, tau_G_vector, "left_sided_riemann")
#> [1] 2.51
```