# Quantile estimators based on k order statistics, Part 3: Playing with the Beta function

*Communications in Statistics - Simulation and Computation*(DOI: 10.1080/03610918.2022.2050396). A preprint is available on arXiv: arXiv:2111.11776 [stat.ME]. Some information in this blog post can be obsolete: please, use the official paper as the primary reference.

In the previous two posts, I discussed the idea of quantile estimators based on k order statistics. A already covered the motivation behind this idea and the statistical efficiency of such estimators using the extended Hyndman-Fan equations as a weight function. Now it’s time to experiment with the Beta function as a primary way to aggregate k order statistics into a single quantile estimation!

All posts from this series:

- Quantile estimators based on k order statistics, Part 1: Motivation
*(2021-08-03)* - Quantile estimators based on k order statistics, Part 2: Extending Hyndman-Fan equations
*(2021-08-10)* - Quantile estimators based on k order statistics, Part 3: Playing with the Beta function
*(2021-08-17)* - Quantile estimators based on k order statistics, Part 4: Adopting trimmed Harrell-Davis quantile estimator
*(2021-08-24)* - Quantile estimators based on k order statistics, Part 5: Improving trimmed Harrell-Davis quantile estimator
*(2021-08-31)* - Quantile estimators based on k order statistics, Part 6: Continuous trimmed Harrell-Davis quantile estimator
*(2021-09-07)* - Quantile estimators based on k order statistics, Part 7: Optimal threshold for the trimmed Harrell-Davis quantile estimator
*(2021-09-14)* - Quantile estimators based on k order statistics, Part 8: Winsorized Harrell-Davis quantile estimator
*(2021-09-21)*

## The approach

The general idea is the same that was used in the previous post. We express the estimation of the $p^\textrm{th}$ quantile as follows:

$$ \begin{gather*} q_p = \sum_{i=1}^{n} W_{i} \cdot x_i,\\ W_{i} = F(r_i) - F(l_i),\\ l_i = (i - 1) / n, \quad r_i = i / n, \end{gather*} $$where F is a CDF function of a specific distribution. The distribution has non-zero PDF only inside a window $[L_k, R_k]$ that covers at most k order statistics:

$$ F(u) = \left\{ \begin{array}{lcrcllr} 0 & \textrm{for} & & & u & < & L_k, \\ G\Big((u - L_k)/(R_k-L_k)\Big) & \textrm{for} & L_k & \leq & u & \leq & R_k, \\ 1 & \textrm{for} & R_k & < & u, & & \end{array} \right. $$$$ L_k = (h - 1) / (n - 1) \cdot (n - (k - 1)) / n, \quad R_k = L_k + (k-1)/n, $$$$ h = (n - 1)p + 1. $$Now we just have to define the $G: [0;1] \to [0;1]$ function that defines $F$ values inside the window. In the previous post, where we used the extension of Hyndman-Fan Type 7 equation, we used just the most simple linear function:

$$ G_{HF7}(u) = u. $$In this post, we are going to try the Beta distribution (which is used in the Harrell-Davis quantile estimator). The CDF of the Beta distribution is the regularized incomplete beta function) $I_x(\alpha, \beta)$. We will try this idea with $\alpha=kp, \beta = k(1-p)$:

$$ G(u) = I_u(kp, k(1-p)). $$With such values, the suggested estimator becomes the exact copy of the Harrell-Davis quantile estimator for $k=n+1$. Let’s perform some numerical simulations to check the statistical efficiency of this estimator.

## Numerical simulations

We are going to take the same simulation setup that was declared in this post. Briefly speaking, we evaluate the classic MSE-based relative statistical efficiency of different quantile estimators on samples from different light-tailed and heavy-tailed distributions using the classic Hyndman-Fan Type 7 quantile estimator as the baseline.

Here is the animated version of the simulations (the considered estimators based on k order statistics are denoted as “KOS-Bk”):

And here are static images of the result for different sample sizes:

## Conclusion

In this post, we discussed a quantile estimator that is based on k order statistics aggregated using the Beta function. It seems that this estimator is a good step in the right direction: it’s better than the traditional Hyndman-Fan Type 7 quantile estimator for the samples from light-tailed distributions (however, it’s worse than the Harrell-Davis quantile estimator). Also, it’s more robust than the Harrell-Davis quantile estimator in the case of heavy-tailed distributions. Moreover, we could specify the desired breakdown point by customizing the k value.

In the next post, we are going to try one more weight function.