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

**Update: this blog post is a part of research that aimed to build a statistically efficient and robust quantile estimator.
A preprint with final results is available on arXiv:
arXiv:2111.11776 [stat.ME].**

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

### 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.