Sensitivity curve of the Harrell-Davis quantile estimator, Part 2

Andrey Akinshin · 2022-10-04

In the previous post, I have explored the sensitivity curves of the Harrell-Davis quantile estimator on the normal distribution. In this post, I continue the same investigation on the exponential and Cauchy distributions.

The classic Harrell-Davis quantile estimator (see harrell1982) is defined as follows:

QHD(p)=i=1nWHD,ix(i),WHD,i=Ii/n(α,β)I(i1)/n(α,β),

where It(α,β) is the regularized incomplete beta function, α=(n+1)p, β=(n+1)(1p). In this post we consider the Harrell-Davis median estimator QHD(0.5).

The standardized sensitivity curve (SC) of an estimator θ^ is given by

SCn(x0)=θ^n+1(x1,x2,,xn,x0)θ^n(x1,x2,,xn)1/(n+1)

Thus, the SC shows the standardized change of the estimator value for situation, when we add a new element x0 to an existing sample x={x1,x2,,xn}. In the context of this post, we perform simulations using the exponential and Cauchy distributions given by their quantile functions that we denote as F1. Following the approach from [Maronna2019, Section 3.1], we define the sample x as

x={F1(1n+1),F1(2n+1),,F1(nn+1)}

Now let’s explore the SC values for different sample sizes for x0[100;100].

Exponential distribution

Cauchy distribution

Conclusion

As we can see, for n15 the actual impact of x0 is negligible, which makes the Harrell-Davis median estimator a practically reasonable choice. This conclusion is relevant not only for the normal distribution (as shown in the previous post), but also for the exponential distribution and the Cauchy distribution.

References

  • [Harrell1982]
    Harrell, F.E. and Davis, C.E., 1982. A new distribution-free quantile estimator. Biometrika, 69(3), pp.635-640.
    https://doi.org/10.2307/2335999
  • [Maronna2019]
    Maronna, Ricardo A., R. Douglas Martin, Victor J. Yohai, and Matías Salibián-Barrera. Robust statistics: theory and methods (with R). John Wiley & Sons, 2019.