Inference for the Standardized Median

Abstract

We carry out inference for the median, divided by a fixed interquantile range (IQR). It is a standardized effect or 'effect size' defined by 3 quantiles. Applied statisticians sometimes prefer effect sizes to raw effects, such as the median, because it is scalefree. Inference regarding the median itself has already been thoroughly investigated by a number of authors, including (McKean and Schrader (1984) and Sheather and McKean 1987). More generally, the problem of Studentizing a quantile estimator has been studied by (Bloch and Gastwirth 1968;Hall andSheather 1988 andSiddiqui 1960), amongst others.We are given a location-scale family F α,β (x) = F ((x − α)/β) where α, β > 0 are unknown. Assume F = F 0,1 has a continuous derivative f which is symmetric about 0 and is positive over a possibly infinite symmetric interval containing 0. Denote the quantile function of F by G = F −1 , its value at any 0 < r < 1 by x r = G(r), and its derivative at r by g(r) = {f (x r )} −1 . The rth quantile of F α,β is related to x r of F by α + βx r , and g α,β (r) = β g(r).For 0 < r < 0.5, a value to be chosen later, let the rth IQR of F α,β be β IQR r , where IQR r = x 1−r − x r . Also define the rth standardized median of F α,β by:th order statistic of a sample of size n from F . Let M = X ([n/2]) be the sample median, which is consistent for x 0.5 and for a fixed 0 < r < 1/2 define the rth sample IQR by, which is a consistent estimator of x 1−r − x r . We want to estimate the rth standardized median defined in (22.1) byδ r = M/R r . In the next Sect. 22.2 we derive a variance stabilizing transformation (VST) ofδ r . This leads to confidence intervals for δ r , whose effectiveness of coverage is evaluated R. G. Staudte ( )

Inference for quantile measures of kurtosis, peakedness, and tail weight

“…The methodology for finding a variance stabilizing transformation (VST) of a ratio of statistics, each of which is a finite linear combination of order statistics, has already been established for other ratios of linear combinations of quantiles in Staudte (2013aStaudte ( ,b, 2014, so here we only restate the required results. One first shows that Therefore nVar[κ p,r ] = q(E[κ p,r ]) where q(t) = a 0 + a 1 t + a 2 t 2 is a quadratic with constants:…”

Distribution‐free confidence intervals for the standardized median

“…In what follows, we restrict our attention to the case r = 0.25, not only because for ease of presentation but also because it is shown in Staudte () that this value is a good compromise between the competing criteria of efficiency and robustness to outliers. This means that we can simplify the notation further by writing IQR for IQR 0.25 , R instead of R 0.25 , and δ for δ 0.25 .…”

“…Inference for the standardized median is available for symmetric location‐scale families in Staudte (). Here, we drop both the assumption of symmetry and knowledge of the underlying density that generates the family.…”

“…Here, we drop both the assumption of symmetry and knowledge of the underlying density that generates the family. Instead, we extend the variance stabilization results of Staudte () to the asymmetric case and use it to obtain confidence intervals for the standardized median for the case of known F . Then, we estimate the constants required by this variance stabilizer for the case of unknown F and find sample sizes required by this two‐stage procedure to obtain distribution‐free confidence intervals with accurate coverage.…”