Weighted Quantile Estimation for a Weighted Mixture Distribution


Update: this blog post is a part of research that aimed to build weighed versions of various quantile estimators. A preprint with final results is available on arXiv: arXiv:2304.07265 [stat.ME]. Some information in this blog post can be obsolete: please, use the preprint as the primary reference.

Let $\mathbf{x} = \{ x_1, x_2, \ldots, x_n \}$ be a sample of size $n$. We assign non-negative weight coefficients $w_i$ with a positive sum for all sample elements:

$$ \mathbf{w} = \{ w_1, w_2, \ldots, w_n \}, \quad w_i \geq 0, \quad \sum_{i=1}^{n} w_i > 0. $$

For simplification, we also consider normalized (standardized) weights $\overline{\mathbf{w}}$:

$$ \overline{\mathbf{w}} = \{ \overline{w}_1, \overline{w}_2, \ldots, \overline{w}_n \}, \quad \overline{w}_i = \frac{w_i}{\sum_{i=1}^{n} w_i}. $$

In the non-weighted case, we can consider a quantile estimator $\operatorname{Q}(\mathbf{x}, p)$ that estimates the $p^\textrm{th}$ quantile of the underlying distribution. We want to build a weighted quantile estimator $\operatorname{Q}(\mathbf{x}, \mathbf{w}, p)$ so that we can estimate the quantiles of a weighed sample.

In this post, we consider a specific problem of estimating quantiles of a weighted mixture distribution.

For example, we can consider three distributions given by their cumulative distribution functions (CDFs) $F_X$, $F_Y$, and $F_Z$ with weight coefficients $w_X$, $w_Y$, and $w_Z$. Their weighted mixture is given by $F=\overline{w}_X F_X + \overline{w}_Y F_Y + \overline{w}_Z F_Z$. Let us say that we have samples $\mathbf{x}$, $\mathbf{y}$, and $\mathbf{z}$ from $F_X$, $F_Y$, and $F_Z$; and we want to estimate the quantile function $F^{-1}$ of the mixture distribution $F$. If each sample contains a sufficient number of elements, we can consider a straightforward approach:

  1. Obtain estimations $\hat{F}^{-1}_X$, $\hat{F}^{-1}_Y$, $\hat{F}^{-1}_Z$ of the distribution quantile functions based on the given samples;
  2. Invert quantile functions and obtain estimations $\hat{F}_X$, $\hat{F}_Y$, $\hat{F}_Z$ of the CDFs for each distribution;
  3. Combine these CDFs and build an estimation $\hat{F}=\overline{w}_X\hat{F}_X+\overline{w}_Y\hat{F}_Y+\overline{w}_Z\hat{F}_Z$ of the mixture CDF;
  4. Invert $\hat{F}$ and get the estimation $\hat{F}^{-1}$ of the mixture distribution quantile function.

The approach performs well only when the sample sizes are large enough so that we can efficiently estimate sample quantiles.