Confidence Intervals for Median Absolute Deviations

Confidence intervals for the median absolute deviation (MAD) of a single sample (madCI), for the difference of two MADs (madDiffCI), and for the squared ratio of two MADs (madRatioCI). Two methods are available throughout: an asymptotic interval based on the generalized lambda distribution (GLD), and a parallel bootstrap interval.

Usage

madCI(
  x,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("classic", "boot"),
  gldMethod = "TM",
  na.rm = FALSE,
  ...
)

madDiffCI(
  x,
  y,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("classic", "boot"),
  gldMethod = "TM",
  na.rm = FALSE,
  ...
)

madRatioCI(
  x,
  y,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("classic", "boot"),
  gldMethod = "TM",
  na.rm = FALSE,
  ...
)

Arguments

x: A non-empty numeric vector (first or only sample).
conf.level: Confidence level of the interval. A single numeric value in $(0, 1)$. Default 0.95.
sides: A character string specifying the side of the interval: "two.sided" (default), "left", or "right". Partial matching is supported. "left" sets uci = Inf; "right" sets lci = -Inf.
method: A character string selecting the CI method: "classic" (asymptotic GLD-based, default) or "boot" (parallel bootstrap).
gldMethod: A character string passed to .asv.mad() selecting the GLD estimation method. One of "ML", "MPS", "TM" (default), "SM", "TL", "Lmom", "DLA", or "Mom". See gld::fit.fkml(). Used only when method = "classic".
na.rm: Logical. Should missing values be removed before computation? Default FALSE.
...: Further arguments passed to the bootstrap engine when method = "boot": R, type, parallel, ncpus. See Details.
y: A non-empty numeric vector (second sample). Required for madDiffCI and madRatioCI.

Value

A named numeric vector with three elements:

est: point estimate:
$\mathrm{mad}(x)$ for madCI
$\mathrm{mad}(x) - \mathrm{mad}(y)$ for madDiffCI
$(\mathrm{mad}(x)/\mathrm{mad}(y))^2$ for madRatioCI
lci: lower confidence bound.
uci: upper confidence bound.

Details

Classic method (method = "classic")

All three functions follow Arachchige & Prendergast (2019) and base the interval on the asymptotic variance of the MAD, approximated by fitting a GLD to the data via .asv.mad(). The GLD estimation method is selected with gldMethod.

For madDiffCI the asymptotic variances of the two samples are combined as $\widehat{\mathrm{ASV}}(x)/n_x + \widehat{\mathrm{ASV}}(y)/n_y$.

For madRatioCI the interval is constructed on the log scale via the delta method and back-transformed to guarantee positivity: $$ \exp\!\Bigl(\log\hat\theta \;\pm\; z_{\alpha/2} \,\sqrt{\widehat{\mathrm{Var}}(\hat\theta)}\,/\,\hat\theta\Bigr), \quad \hat\theta = \bigl(\mathrm{MAD}(x)/\mathrm{MAD}(y)\bigr)^2. $$

The classic method is fast and accurate for large samples but may undercover for small or heavy-tailed distributions.

Bootstrap method (method = "boot")

Data are resampled $R$ times using a parallel Rcpp worker and a percentile or BCa interval is returned. For the two-sample functions the two samples are resampled independently. Bootstrap arguments are passed through ... and extracted via .extractBootArgs():

R: Number of bootstrap replicates (default 999).
type: CI type: "perc" or "bca" (default).
parallel: Parallelisation: "no", "multicore", or "snow" (default "no").
ncpus: Number of CPUs for parallel bootstrap (default getOption("boot.ncpus", 1L)).

Note

Based on code by Arachchige Chandima N. P. G. and Prendergast Luke A., adapted to conform to package standards.

References

Arachchige, C. N. P. G., & Prendergast, L. A. (2019). Confidence intervals for median absolute deviations. arXiv:1910.00229 [math.ST].

Examples

set.seed(1)
x <- rlnorm(100)
y <- rlnorm(200, meanlog = 1.2)

# single sample
madCI(x)
#>       est       lci       uci 
#> 0.9264988 0.7075985 1.1453991 
madCI(x, sides = "left")
#>       est       lci       uci 
#> 0.9264988 0.7427919       Inf 
madCI(x, method = "boot", R = 499, type = "bca")
#>       est       lci       uci 
#> 0.9264988 0.6922272 1.2100449 

# two-sample difference
madDiffCI(x, y)
#>       est       lci       uci 
#> -1.819975 -2.372101 -1.267849 
madDiffCI(x, y, method = "boot", R = 499, type = "perc")
#>       est       lci       uci 
#> -1.819975 -2.328010 -1.104668 

# two-sample squared ratio
madRatioCI(x, y)
#>        est        lci        uci 
#> 0.11379909 0.06247871 0.20727434 
madRatioCI(x, y, method = "boot", R = 499, type = "bca")
#>        est        lci        uci 
#> 0.11379909 0.05335922 0.25521800