Sensitivity curve of the Harrell-Davis quantile estimator, Part 3



In the previous posts (1, 2), I have explored the sensitivity curves of the Harrell-Davis quantile estimator on the normal distribution, the exponential distribution, and the Cauchy distribution. In this post, I build these sensitivity curves for some additional distributions.

The classic Harrell-Davis quantile estimator (see [Harrell1982]) is defined as follows:

\[Q_{\operatorname{HD}}(p) = \sum_{i=1}^{n} W_{\operatorname{HD},i} \cdot x_{(i)},\quad W_{\operatorname{HD},i} = I_{i/n}(\alpha, \beta) - I_{(i-1)/n}(\alpha, \beta), \]

where \(I_t(\alpha, \beta)\) is the regularized incomplete beta function, \(\alpha = (n+1)p\), \(\;\beta = (n+1)(1-p)\). In this post we consider the Harrell-Davis median estimator \(Q_{\operatorname{HD}}(0.5)\).

The standardized sensitivity curve (SC) of an estimator \(\hat{\theta}\) is given by

\[\operatorname{SC}_n(x_0) = \frac{ \hat{\theta}_{n+1}(x_1, x_2, \ldots, x_n, x_0) - \hat{\theta}_n(x_1, x_2, \ldots, x_n) }{1 / (n + 1)} \]

Thus, the SC shows the standardized change of the estimator value for situation, when we add a new element \(x_0\) to an existing sample \(\mathbf{x} = \{ x_1, x_2, \ldots, x_n \}\). In the context of this post, we perform simulations using the different distributions given by their quantile functions that we denote as \(F^{-1}\). Following the approach from [Maronna2019, Section 3.1], we define the sample \(\mathbf{x}\) as

\[\mathbf{x} = \Bigg\{ F^{-1}\Big(\frac{1}{n+1}\Big), F^{-1}\Big(\frac{2}{n+1}\Big), \ldots, F^{-1}\Big(\frac{n}{n+1}\Big) \Bigg\} \]

Now let’s explore the SC values for different sample sizes for \(x_0 \in [-100; 100]\).

Distributions

We consider the following distributions:

DistributionSupportSkewnessTailness
Uniform(a=0, b=1)\([0;1]\)SymmetricLight-tailed
Triangular(a=0, b=2, c=1)\([0;2]\)SymmetricLight-tailed
Triangular(a=0, b=2, c=0.2)\([0;2]\)Right-skewedLight-tailed
Beta(a=2, b=4)\([0;1]\)Right-skewedLight-tailed
Beta(a=2, b=10)\([0;1]\)Right-skewedLight-tailed
Normal(m=0, sd=1)\((-\infty;+\infty)\)SymmetricLight-tailed
Weibull(scale=1, shape=2)\([0;+\infty)\)Right-skewedLight-tailed
Student(df=3)\((-\infty;+\infty)\)SymmetricLight-tailed
Gumbel(loc=0, scale=1)\((-\infty;+\infty)\)Right-skewedLight-tailed
Exp(rate=1)\([0;+\infty)\)Right-skewedLight-tailed
Cauchy(x0=0, gamma=1)\((-\infty;+\infty)\)SymmetricHeavy-tailed
Pareto(loc=1, shape=0.5)\([1;+\infty)\)Right-skewedHeavy-tailed
Pareto(loc=1, shape=2)\([1;+\infty)\)Right-skewedHeavy-tailed
LogNormal(mlog=0, sdlog=1)\((0;+\infty)\)Right-skewedHeavy-tailed
LogNormal(mlog=0, sdlog=2)\((0;+\infty)\)Right-skewedHeavy-tailed
LogNormal(mlog=0, sdlog=3)\((0;+\infty)\)Right-skewedHeavy-tailed
Weibull(shape=0.3)\([0;+\infty)\)Right-skewedHeavy-tailed
Weibull(shape=0.5)\([0;+\infty)\)Right-skewedHeavy-tailed
Frechet(shape=1)\((0;+\infty)\)Right-skewedHeavy-tailed
Frechet(shape=3)\((0;+\infty)\)Right-skewedHeavy-tailed

Sensitivity curves

Here are the results of numerical simulations:




Conclusion

As we can see, for \(n \geq 15\) the actual impact of \(x_0\) is negligible, which makes the Harrell-Davis median estimator a practically reasonable choice regardless of the distribution. However, we should be careful with heavy distributions like the Pareto or LogNormal distributions.

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.