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


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:

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

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 exponential and Cauchy 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 $x_0 \in [-100; 100]$.

Exponential distribution

Cauchy distribution

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. 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


References (2)