# 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.