# Weighted Hodges-Lehmann location estimator and mixture distributions

The classic non-weighted Hodges-Lehmann location estimator of a sample $$\mathbf{x} = (x_1, x_2, \ldots, x_n)$$ is defined as follows:

$\operatorname{HL}(\mathbf{x}) = \underset{1 \leq i \leq j \leq n}{\operatorname{median}} \left(\frac{x_i + x_j}{2} \right),$

where $$\operatorname{median}$$ is the sample median. Previously, we have defined a weighted version of the Hodges-Lehmann location estimator as follows:

$\operatorname{WHL}(\mathbf{x}, \mathbf{w}) = \underset{1 \leq i \leq j \leq n}{\operatorname{wmedian}} \left(\frac{x_i + x_j}{2},\; w_i \cdot w_j \right),$

where $$\mathbf{w} = (w_1, w_2, \ldots, w_n)$$ is the vector of weights, $$\operatorname{wmedian}$$ is the weighted median. For simplicity, in the scope of the current post, Hyndman-Fan Type 7 quantile estimator is used as the base for the weighted median.

In this post, we consider a numerical simulation in which we compare sampling distribution of $$\operatorname{HL}$$ and $$\operatorname{WHL}$$ in a case of mixture distribution.

### Numerical simulation

We consider the following mixture of two normal distribution:

$\frac{1}{3} \mathcal{N}(0, 1) + \frac{2}{3} \mathcal{N}(10, 1).$

For the sample size of $$n=10$$, we build two following sampling distributions:

• $$\operatorname{HL}(\mathbf{x})$$, where $$\mathbf{x}$$ is randomly taken from $$\frac{1}{3} \mathcal{N}(0, 1) + \frac{2}{3} \mathcal{N}(10, 1)$$;
• $$\operatorname{WHL}(\mathbf{x}, \mathbf{w})$$, where
• $$x_1, x_2, x_3, x_4, x_5$$ are randomly taken from $$\mathcal{N}(0, 1)$$,
• $$x_6, x_7, x_8, x_9, x_{10}$$ are randomly taken from $$\mathcal{N}(10, 1)$$,
• $$\mathbf{w} = (1, 1, 1, 1, 1, 2, 2, 2, 2, 2)$$.

Here are the corresponding sampling distribution density plots:

As we can see, the rebalancing of observed sub-samples from existing modes led to higher statistical efficiency and the lack of bimodality.