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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 = 0.25, q = 0.75, drop.na = FALSE)

Arguments

x

A numeric vector.

p

A numeric value between 0 and 0.5 (p = 0.25 by default).

q

A numeric value between 0.5 and 1 (q = 0.75 by default).

drop.na

Logical value indicating whether to remove missing values (NA) or not.

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 calculated when applying quartile skewness to either the left half or the right half of the probability mass, divided at the median of the univariate distribution. The left quantile weight (LQW) is the proportion of the data below the median, divided by the expected proportion (0.5) if the data were normally distributed. The right quantile weight (RQW) is the proportion of the data above the median, divided by 0.5.

Interpretation of Quantile Weights:

  • Values closer to 0 indicate lighter tails compared to the normal distribution.

  • Values closer to 1 signify heavier tails compared to the normal distribution.

  • Values significantly greater than 1 suggest the presence of outliers or extreme values in the respective tail.

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

vec <- c(-100, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100)
# non-robust approach
moments::kurtosis(vec)
#> [1] 6.474793

# robust approach
quantile_weight(vec)
#> # A tibble: 1 × 2
#>     LQW   RQW
#>   <dbl> <dbl>
#> 1 0.961 0.937