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Quantile Tail Weight Measure

Source: R/quantile_weight.R
quantile_weight.Rd

This function calculates the left quantile weight (LQW) and the right quantile weight (RQW) for a given numeric vector. These weights serve as robust measures of tail heaviness, providing insights into the distribution's behavior in the left and right tails, respectively.

Usage

quantile_weight(x, p = 0.125, q = 0.875, drop.na = FALSE)

Arguments

x

A numeric vector.

p

A numeric value between 0 and 0.5 (p = 0.125 by default).

q

A numeric value between 0.5 and 1 (q = 0.875 by default).

drop.na

Logical value indicating whether to remove missing values (NA). If FALSE (default) and x contains missing values, NA is returned.

Value

A tibble with two numeric columns:

  • LQW: Left quantile weight.

  • RQW: Right quantile weight.

Details

The quantile weights, comprising the left quantile weight (LQW) and the right quantile weight (RQW), are robust measures of tail weight in probability distributions. They have a breakdown value of 12.5%, meaning that they are resistant to the influence of up to 12.5% of outliers or contaminated data.

The concept of quantile weights is derived from quartile skewness, introduced by D.V. Hinkley in 1975. Quartile skewness measures the skewness or asymmetry of a distribution by comparing the differences between the quartiles, which are robust measures of location and scale.

Specifically, the quantile weights are obtained by applying quartile skewness to either the left half or the right half of the probability mass, divided at the median of the univariate distribution (Brys et al. 2006): $$LQW(p) = -\frac{Q((1-p)/2) + Q(p/2) - 2Q(0.25)}{Q((1-p)/2) - Q(p/2)}$$ $$RQW(q) = \frac{Q((1+q)/2) + Q(1-q/2) - 2Q(0.75)}{Q((1+q)/2) - Q(1-q/2)}$$ where \(Q\) is the quantile function of the data. The defaults \(p = 0.125\) and \(q = 0.875\) give a breakdown value of 12.5%.

Interpretation of Quantile Weights:

  • At the normal distribution, LQW = RQW = 0.2 (approximately).

  • Larger values indicate heavier tails than the normal distribution, and smaller values lighter tails.

References

  • Brys, G., Hubert, M., and Struyf, A. (2006). Robust measures of tail weight. Computational Statistics & Data Analysis, 50(3):733-759

  • Hinkley, D.V., (1975). On power transformations to symmetry. Biometrika, 62(1):101–111.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
# iron contents (mg/kg): a few samples are far above the others
fe <- forageLIBS$Fe[!is.na(forageLIBS$Fe)]
moments::kurtosis(fe)   # classical
#> [1] 25.01662
quantile_weight(fe)     # robust, left and right tails
#> # A tibble: 1 × 2
#>      LQW   RQW
#>    <dbl> <dbl>
#> 1 0.0174 0.687