Weighted trimmed Harrell-Davis quantile estimator

In this post, I combine ideas from two of my previous posts:

Thus, we are going to build a weighted version of the trimmed Harrell-Davis quantile estimator based on the highest density interval of the given width.

Simple trimmed Harrell-Davis quantile estimator

The concept of this estimator is fully covered in my recent paper [Akinshin2022]. Here I just briefly recall the basic idea.

Let \(x\) be a sample with \(n\) elements: \(x = \{ x_1, x_2, \ldots, x_n \}\). We assume that all sample elements are sorted (\(x_1 \leq x_2 \leq \ldots \leq x_n\)) so that we could treat the \(i^\textrm{th}\) element \(x_i\) as the \(i^\textrm{th}\) order statistic \(x_{(i)}\). Based on the given sample, we want to build an estimation of the \(p^\textrm{th}\) quantile \(Q(p)\).

The classic Harrell-Davis quantile estimator (see [Harrell1982]) suggests the following approach:

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

where \(I_x(\alpha, \beta)\) is the regularized incomplete beta function, \(\alpha = (n+1)p\), \(\;\beta = (n+1)(1-p)\).

When we switch to the trimmed modification of this estimator, we perform summation only within the highest density interval \([L;R]\) of \(\operatorname{Beta}(\alpha, \beta)\) of size \(D\) (as a rule of thumb, we can use \(D = 1 / \sqrt{n}\)):

\[Q_{\operatorname{THD}} = \sum_{i=1}^{n} W_{\operatorname{THD},i} \cdot x_i, \quad W_{\operatorname{THD},i} = F_{\operatorname{THD}}(i / n) - F_{\operatorname{THD}}((i - 1) / n), \]

\[F_{\operatorname{THD}}(x) = \begin{cases} 0 & \textrm{for }\, x < L,\\ \big( I_x(\alpha, \beta) - I_L(\alpha, \beta) \big) / \big( I_R(\alpha, \beta) \big) - I_L(\alpha, \beta) \big) \big) & \textrm{for }\, L \leq x \leq R,\\ 1 & \textrm{for }\, R < x. \end{cases} \]

Thus, we use only sample elements with the highest weight coefficients (\(W_{\operatorname{THD},i}\)) and ignore sample elements with small weight coefficients. It allows us to get a high statistical efficiency (which is close to the efficiency of the classic Harrell-Davis quantile estimator) and a good robustness level (in most cases, outliers have zero impact on the final result).

Weighted trimmed Harrell-Davis quantile estimator

Let’s assign weights \(w = \{ w_1, w_2, \ldots, w_n \}\) to all sample elements. Now we would like to patch the above equations so that they take these weights into account.

First of all, we should calculate the effective sample size for the weighted sample using the Kish’s approach:

\[n^* = \frac{\Big( \sum_{i=1}^n w_i \Big)^2}{\sum_{i=1}^n w_i^2 }. \]

The \(\alpha\) and \(\beta\) coefficients should be also properly updated:

\[\alpha^* = (n^*+1)p,\; \beta^* = (n^*+1)(1-p). \]

The highest density interval \([L;R]\) of \(\operatorname{Beta}(\alpha, beta)\) should be also updated to the highest density interval \([L^*;R^*]\) of \(\operatorname{Beta}(\alpha^*, \beta^*)\).

In the original Harrell-Davis quantile estimator and its trimmed modification, we used \(l_i = (i-1)/n\) and \(r_i = i/n\) as borders for a segment of Beta distribution which is used to determine \(W_{\operatorname{HD},i}\) / \(W_{\operatorname{THD},i}\). In the weighted case, we define these values using the given weights:

\[\left\{ \begin{array}{rcc} l^*_i & = & \dfrac{s_{i-1}(w)}{s_n(w)},\\ r^*_i & = & \dfrac{s_i(w)}{s_n(w)}, \end{array} \right. \]

where \(s_i(w) = \sum_{j=1}^{i} w_j\) (assuming \(s_0(w) = 0\)).

Next, we define a new truncated Beta distribution for the weighted sample:

\[F^*_{\operatorname{THD}}(x) = \begin{cases} 0 & \textrm{for }\, x < L^*,\\ \big( I_x(\alpha^*, \beta^*) - I_L(\alpha^*, \beta^*) \big) / \big( I_R(\alpha^*, \beta^*) \big) - I_L(\alpha^*, \beta^*) \big) \big) & \textrm{for }\, L^* \leq x \leq R^*,\\ 1 & \textrm{for }\, R^* < x. \end{cases} \]

Finally, we are ready to write down the final equation for the weighted trimmed Harrell-Davis quantile estimator:

\[Q_{\operatorname{THD}}^* = \sum_{i=1}^{n} W^*_{\operatorname{THD},i} \cdot x_i, \quad W^*_{\operatorname{THD},i} = F^*_{\operatorname{THD}}(r_i^*) - F^*_{\operatorname{THD}}(l_i^*). \]

This approach could be easily adopted for quantile exponential smoothing and dispersion exponential smoothing without additional efforts.