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.

References

• [Lehmann1963]
  Hodges, J. L., and E. L. Lehmann. “Estimates of Location Based on Rank Tests.” The Annals of Mathematical Statistics 34, no. 2 (June 1963): 598–611.
  DOI: 10.1214/aoms/1177704172
• [Bhattacharyya1977]
  Bhattacharyya, Helen T. “Nonparametric Estimation of Ratio of Scale Parameters.” Journal of the American Statistical Association 72, no. 358 (June 1977): 459–63.
  DOI: 10.1080/01621459.1977.10481021
• [Padgett1982]
  Padgett, W. J., and L. J. Wei. “Estimation of the Ratio of Scale Parameters in the Two Sample Problem with Arbitrary Right Censorship.” Biometrika 69, no. 1 (1982): 252–56.
  DOI: 10.1093/biomet/69.1.252
• [Price1996]
  Price, Robert Martin Jr. “Estimating the Ratio of Medians: Theory and Applications.” University of Wyoming, 1996.