A variation of quantile() that can be applied to weighted samples.

weighted_quantile(
x,
probs = seq(0, 1, 0.25),
weights = NULL,
n = NULL,
na.rm = FALSE,
type = 7
)

weighted_quantile_fun(x, weights = NULL, n = NULL, na.rm = FALSE, type = 7)

## Arguments

x

numeric vector: sample values

probs

numeric vector: probabilities in $$[0, 1]$$

weights

Weights for the sample. One of:

• numeric vector of same length as x: weights for corresponding values in x, which will be normalized to sum to 1.

• NULL: indicates no weights are provided, so unweighted sample quantiles (equivalent to quantile()) are returned.

n

Presumed effective sample size. If this is greater than 1 and continuous quantiles (type >= 4) are requested, flat regions may be added to the approximation to the inverse CDF in areas where the normalized weight exceeds 1/n (i.e., regions of high density). This can be used to ensure that if a sample of size n with duplicate x values is summarized into a weighted sample without duplicates, the result of weighted_quantile(..., n = n) on the weighted sample is equal to the result of quantile() on the original sample. One of:

• NULL: do not make a sample size adjustment.

• numeric: presumed effective sample size.

• function or name of function (as a string): A function applied to weights (prior to normalization) to determine the sample size. Some useful values may be:

• "length": i.e. use the number of elements in weights (equivalently in x) as the effective sample size.

• "sum": i.e. use the sum of the unnormalized weights as the sample size. Useful if the provided weights is unnormalized so that its sum represents the true sample size.

na.rm

logical: if TRUE, corresponding entries in x and weights are removed if either is NA.

type

integer between 1 and 9: determines the type of quantile estimator to be used. Types 1 to 3 are for discontinuous quantiles, types 4 to 9 are for continuous quantiles. See Details.

## Value

weighted_quantile() returns a numeric vector of length(probs) with the estimate of the corresponding quantile from probs.

weighted_quantile_fun() returns a function that takes a single argument, a vector of probabilities, which itself returns the corresponding quantile estimates. It may be useful when weighted_quantile() needs to be called repeatedly for the same sample, re-using some pre-computation.

## Details

Calculates weighted quantiles using a variation of the quantile types based on a generalization of quantile().

Type 1--3 (discontinuous) quantiles are directly a function of the inverse CDF as a step function, and so can be directly translated to the weighted case using the natural definition of the weighted ECDF as the cumulative sum of the normalized weights.

Type 4--9 (continuous) quantiles require some translation from the definitions in quantile(). quantile() defines continuous estimators in terms of $$x_k$$, which is the $$k$$th order statistic, and $$p_k$$, which is a function of $$k$$ and $$n$$ (the sample size). In the weighted case, we instead take $$x_k$$ as the $$k$$th smallest value of $$x$$ in the weighted sample (not necessarily an order statistic, because of the weights). Then we can re-write the formulas for $$p_k$$ in terms of $$F(x_k)$$ (the empirical CDF at $$x_k$$, i.e. the cumulative sum of normalized weights) and $$f(x_k)$$ (the normalized weight at $$x_k$$), by using the fact that, in the unweighted case, $$k = F(x_k) \cdot n$$ and $$1/n = f(x_k)$$:

Type 4

$$p_k = \frac{k}{n} = F(x_k)$$

Type 5

$$p_k = \frac{k - 0.5}{n} = F(x_k) - \frac{f(x_k)}{2}$$

Type 6

$$p_k = \frac{k}{n + 1} = \frac{F(x_k)}{1 + f(x_k)}$$

Type 7

$$p_k = \frac{k - 1}{n - 1} = \frac{F(x_k) - f(x_k)}{1 - f(x_k)}$$

Type 8

$$p_k = \frac{k - 1/3}{n + 1/3} = \frac{F(x_k) - f(x_k)/3}{1 + f(x_k)/3}$$

Type 9

$$p_k = \frac{k - 3/8}{n + 1/4} = \frac{F(x_k) - f(x_k) \cdot 3/8}{1 + f(x_k)/4}$$

Then the quantile function (inverse CDF) is the piece-wise linear function defined by the points $$(p_k, x_k)$$.

weighted_ecdf()