# Optimal threshold of the trimmed Harrell-Davis quantile estimator

The traditional quantile estimators (which are based on 1 or 2 order statistics) have great robustness. However, the statistical efficiency of these estimators is not so great. The Harrell-Davis quantile estimator has much better efficiency (at least in the light-tailed case), but it’s not robust (because it calculates a weighted sum of all sample values). I already wrote a post about trimmed Harrell-Davis quantile estimator: this approach suggest dropping some of the low-weight sample values to improve robustness (keeping good statistical efficiency). I also perform a numerical simulations that compare efficiency of the original Harrell-Davis quantile estimator against its trimmed and winsorized modifications. It’s time to discuss how to choose the optimal trimming threshold and how it affects the estimator efficiency.

### Simulation design

The relative efficiency value depends on five parameters:

- Target quantile estimator
- Baseline quantile estimator
- Estimated quantile \(p\)
- Sample size \(n\)
- Distribution

As target quantile estimators, we use the trimmed Harrell-Davis quantile estimators with different trimming percentage values.

The conventional baseline quantile estimator in such simulations is the traditional quantile estimator that is defined as a linear combination of two subsequent order statistics. To be more specific, we are going to use the Type 7 quantile estimator from the Hyndman-Fan classification or HF7 ([Hyndman1996]). It can be expressed as follows (assuming one-based indexing):

\[Q_{HF7}(p) = x_{(\lfloor h \rfloor)}+(h-\lfloor h \rfloor)(x_{(\lfloor h \rfloor+1)})-x_{(\lfloor h \rfloor)},\quad h = (n-1)p+1. \]

Thus, we are going to estimate the relative efficiency of the trimmed Harrell-Davis quantile estimator with different percentage values against the traditional quantile estimator HF7. For the \(p^\textrm{th}\) quantile, the classic relative efficiency can be calculated as the ratio of the estimator mean squared errors (\(\textrm{MSE}\)):

\[\textrm{Efficiency}(p) = \dfrac{\textrm{MSE}(Q_{HF7}, p)}{\textrm{MSE}(Q_{HD}, p)} = \dfrac{\operatorname{E}[(Q_{HF7}(p) - \theta(p))^2]}{\operatorname{E}[(Q_{HD}(p) - \theta(p))^2]} \]

where \(\theta(p)\) is the true value of the \(p^\textrm{th}\) quantile. The \(\textrm{MSE}\) value depends on the sample size \(n\), so it should be calculated independently for each sample size value.

Finally, we should choose the distributions for sample generation. I decided to choose 5 light-tailed distributions and 5 heavy-tailed distributions

distribution | description |
---|---|

U(0,1) | Uniform distribution on [0;1] |

Beta(2,10) | Beta distribution with a=2, b=10 |

N(0,1^2) | Normal distribution with mu=0, sigma=1 |

Weibull(1,2) | Weibull distribution with scale=1, shape=2 |

Exp(1) | Exponential distribution |

Cauchy(0,1) | Cauchy distribution with location=0, scale=1 |

Pareto(1, 0.5) | Pareto distribution with xm=1, alpha=0.5 |

LogNormal(0,3^2) | Log-normal distribution with mu=0, sigma=3 |

Weibull(1,2) | Weibull distribution with scale=1, shape=0.5 |

Exp(1) + Outliers | 95% of exponential distribution with rate=1 and 5% of uniform distribution on [0;10000] |

Here are the probability density functions of these distributions:

For each distribution, we are going to do the following:

- Enumerate all the percentiles and calculate the true percentile value \(\theta(p)\) for each distribution
- Enumerate different sample sizes (from 3 to 40)
- Generate a bunch of random samples, estimate the percentile values using all estimators, calculate the relative efficiency of all target quantile estimators quantile estimator.

### Simulation results

Here are the animated results of the simulation:

Below you can find static images for different trimming percentages and sample size values.

### Conclusion

It seems there is no “optimal” threshold value. The statistical efficiency heavily depends on the underlying distributions. For light-heavy distributions, a small (or even zero) trimming percentage is preferable because it provides the highest efficiency. However, for heavy-tailed distributions, it makes sense to increase the trimming percentage value in order to improve robustness.

Based on my experience, if you expect to have some outlier values, it’s a good idea to set the trimming percentage value between 1% and 10%.

### References

**[Harrell1982]**

Harrell, F.E. and Davis, C.E., 1982. A new distribution-free quantile estimator.*Biometrika*, 69(3), pp.635-640.

https://doi.org/10.2307/2335999**[Hyndman1996]**

Hyndman, R. J. and Fan, Y. 1996. Sample quantiles in statistical packages,*American Statistician*50, 361–365.

https://doi.org/10.2307/2684934