*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.

# Quantile estimators based on k order statistics, Part 4: Adopting trimmed Harrell-Davis quantile estimator

In the previous posts, I discussed various aspects of quantile estimators based on k order statistics. I already tried a few weight functions that aggregate the sample values to the quantile estimators (see posts about an extension of the Hyndman-Fan Type 7 equation and about adjusted regularized incomplete beta function). In this post, I continue my experiments and try to adopt the trimmed modifications of the Harrell-Davis quantile estimator to this approach.

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 one of the previous posts. 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:

\[F(u) = \left\{ \begin{array}{lcrcllr} 0 & \textrm{for} & & & u & < & L_k, \\ G(u) & \textrm{for} & L_k & \leq & u & \leq & R_k, \\ 1 & \textrm{for} & R_k & < & u, & & \end{array} \right. \]

\[L_k = (h - 1) / (n - 1) \cdot (n - (k - 1)) / n, \quad R_k = L_k + (k-1)/n, \]

\[h = (n - 1)p + 1. \]

Now we just have to define the \(G: [0;1] \to [0;1]\) function that defines \(F\) values inside the window. We already discussed a few possible options for \(G\):

\[G_{HF7}(u) = (u - L_k)/(R_k-L_k). \]

\[G_{\textrm{Beta}}(u) = I_{(u - L_k)/(R_k-L_k)}(kp, k(1-p)). \]

Now it’s time to try the trimmed modifications of the Harrell-Davis quantile estimator (THD). In order to adjust THD, we should rescale the original regularized incomplete beta function:

\[G_{\textrm{THD}}(u) = (I_u - I_{L_k}) / (I_{R_k} - I_{L_k}), \quad I_x = I_x(p(n+1), (1-p)(n+1)) \]

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.

The considered estimator based on k order statistics is denoted as “KOS-THDk”. The estimator from the previous post based on the adjusted beta function is denoted as “KOS-Bk”.

Here are some of the statistical efficiency plots:

### Conclusion

The above plots are not so impressive: the suggested estimator has poor statistical efficiency. In the next post, we will try to make a few adjustments in order to solve this problem.