Weighted quantile estimation for a weighted mixture distribution

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.

The source code of this post and all the relevant files are available on GitHub.