Posts / Hodges-Lehmann ratio estimator vs. Bhattacharyya's scale ratio estimator


Previously, I discussed an idea of a ratio estimator based on the Hodges-Lehmann estimator. This idea looks so simple and natural that I was sure that it must have already been proposed and studied. However, when I started to search for it, it turned out that it was not as easy as I expected. Moreover, some papers attribute this idea to Bhattacharyya, which is not accurate. In this post, we discuss the difference between these two approaches.

The Hodges-Lehmann ratio estimator

For two samples $\mathbf{x} = ( x_1, x_2, \ldots, x_n )$ and $\mathbf{y} = ( y_1, y_2, \ldots, y_m )$, the classic Hodges-Lehmann location shift estimator $\operatorname{HL}$ is defined as follows:

$$ \operatorname{HL}(\mathbf{x}, \mathbf{y}) = \underset{1 \leq i \leq n,\,\, 1 \leq j \leq m}{\operatorname{median}} \left(x_i - y_j \right). $$

It seems natural to extend this idea and estimate the ratio between two samples as the median of the ratios of their elements. We call this approach the Hodges-Lehmann ratio estimator $\operatorname{HLR}$:

$$ \operatorname{HLR}(\mathbf{x}, \mathbf{y}) = \underset{1 \leq i \leq n,\,\, 1 \leq j \leq m}{\operatorname{median}} \left(x_i / y_j \right). $$

This ratio estimator can also be obtained from the classic location shift estimator using the log transformation:

$$ \operatorname{HLR}(\mathbf{x}, \mathbf{y}) = \exp \bigl( \operatorname{HL}(\log \mathbf{x}, \log \mathbf{y}) \bigr). $$

This approach is applicable only for distribution with positive support. Therefore, we assume that $x_i, y_j > 0$. If the second distribution is a scaled version of the first one ($k\cdot Y = X$), the ratio estimator estimates the scale ratio factor $k$.

Bhattacharyya’s scale ratio estimator

In bhattacharyya1977, a similar idea is considered. However, the paper investigates not the ratio of random variables but the ratio of their scale parameters. This approach is shift-invariant and, therefore, ignores the actual absolute values of the random variables. In Section 2, the author specifies an assumption of zero medians for both distributions:

$$ \operatorname{median}(X) = \operatorname{median}(Y) = 0. $$

Next, they introduce a definition of a relevant pair $(x_i, y_j)$. In each relevant pair, both $x_i$ and $y_j$ should be both positive or both negative. Finally, they introduce estimator $\hat{\Delta}$ defined as the median of ratios among only relevant pairs:

$$ \hat{\Delta}(\mathbf{x}, \mathbf{y}) = \underset{\substack{1 \leq i \leq n,\,\, 1 \leq j \leq m, \\ x_i \cdot y_j > 0}}{\operatorname{median}} \left( \frac{x_i}{y_j} \right). $$

Conclusion

It is easy to see that $\operatorname{HLR} \neq \hat{\Delta}$ in the general case. While they can be consistent in the case of pure scale transform ($k\cdot Y = X$), they estimate different parameters. The Hodges-Lehmann ratio estimator estimates the ratio between variable values from two distributions, while Bhattacharyya’s scale ratio estimator estimates the ratio between the scale parameters of these distributions.

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