Quantile estimators based on k order statistics, Part 1: Motivation



Update: this blog post is a part of research that aimed to build a statistically efficient and robust quantile estimator. A preprint with final results is available on arXiv: arXiv:2111.11776 [stat.ME].

It’s not easy to choose a good quantile estimator. In my previous posts, I considered several groups of quantile estimators:

Unfortunately, all of these estimators have significant drawbacks (e.g., poor statistical efficiency or poor robustness). In this post, I want to discuss all of the advantages and disadvantages of each approach and suggest another family of quantile estimators that are based on k order statistics.

All posts from this series:

Properties of quantile estimators

First of all, we should define the properties of quantile estimators that are important in practice.

  • Statistical efficiency
    This metric is one of the most important because the main task of a quantile estimator is to provide quantile estimators that are closed to the true quantile values. Note that the classic definition of statistical efficiency doesn’t always provide a reliable measure of an estimator. Since it’s based on the mean square error (MSE), it’s not robust, and it could be easily corrupted in the case of heavy-tailed distributions. As an alternative, we could consider the whole distribution of absolute estimation errors. This approach is not so convenient as the classic one because we have to operate with a function instead of a single number. Nevertheless, this function provides much more stable values that are mostly not so sensitive to outliers.
  • Robustness
    Heavy-tailness is a frequent property of real-life distributions. With such a property, we could expect to have extreme outliers that could corrupt the estimations. With a low breakdown point, even a few outliers may distort estimations for all required quantiles and significantly reduce the statistical efficiency. That’s why it’s so important to have robust estimators.
  • Customizability of the trade-off between efficiency and robustness
    Unfortunately, it’s impossible to get the maximum statistical efficiency in the light-tailed case (which means that we should use more order statistics) and get a low breakdown point that protects us against outliers in the heavy-tailed case (which means that we should use fewer order statistics) at the same time. We always have to deal with a trade-off between statistical efficiency and robustness. It’s important to have the ability to customize this trade-off. If we could control the breakdown point, we would adjust the estimator settings using the a priori knowledge about distribution properties.
  • Computational efficiency
    In practice, it’s important to estimate quantiles fast. A slow quantile estimator could easily become a bottleneck of software that analyzes your data. If we need a single order statistics, we could get it using O(n) algorithmic complexity (e.g., using the Fast Deterministic Selection by Andrei Alexandrescu). If we need more order statistics, we may need O(n*log(n)) complexity to sort the whole sample. Don’t forget that complexity is not the only measure of performance; the computational constant is also important. For example, the Harrell-Davis quantile estimator involves getting values of the Beta function, which is not a fast operation.

Now let’s discuss different groups of quantile estimators according to the above properties.

Quantile estimators based on 1 or 2 order statistics

In this group, we have “traditional” quantile estimators which peak a single element from a sample (Hyndman-Fan Type1-3) or calculate a linear interpolation of two subsequent order statistics (Hyndman-Fan Type4-9).

  • Statistical efficiency
    In simple cases, quantile estimators from the Hyndman-Fan classification provide “good enough” efficiency. However, it’s not always optimal since these estimators use at most two sample elements and ignore the rest. The efficiency could be noticeably improved if we take the other sample elements into account.
  • Robustness
    The asymptotical breakdown point of the \(p^\textrm{th}\) quantile is min(p, 1-p). Thus, these estimators are robust.
  • Customizability of the trade-off between efficiency and robustness
    It’s impossible to change the trade-off.
  • Computational efficiency
    These estimators are fast because a single order statistic could be obtained using O(n) algorithmic complexity using the Fast Deterministic Selection by Andrei Alexandrescu.

Quantile estimators based on all order statistics

In this group, we have the Harrell-Davis quantile estimator, the Sfakianakis-Verginis quantile estimator, and the Navruz-Özdemir quantile estimator. All of them estimate a quantile as a weighted sum of all order statistics. This means that it’s not a good idea to use them with samples from heavy-tailed distribution: a single extreme outlier could corrupt estimated values of all quantiles.

  • Statistical efficiency / Robustness
    These estimators typically have a decent statistical efficiency with samples from light-tailed distributions. Unfortunately, the breakdown point is zero for all of the estimators because the estimation involves all sample elements with non-zero weights. Thus, these estimators are not robust, which means extremely low statistical efficiency for samples from heavy-tailed distributions.
  • Customizability of the trade-off between efficiency and robustness
    It’s impossible to change the trade-off.
  • Computational efficiency
    These estimators are pretty slow. In addition to the O(n*log(n)) sorting complexity, we have to perform O(n) summation (that involves heavy calculations of the Beta or Binomial function).

Quantile estimators based on a variable number of order statistics

In this group, we have the trimmed and winsorized modifications of the Harrell-Davis quantile estimator (THD and WHD). The idea behind them is simple: we should drop all the order statistics with small weight coefficients. In the case of light-tailed distributions, these terms don’t have a noticeable impact on the result. In the case of heavy-tailed distributions, these terms could corrupt the result regardless of how small the corresponding weight coefficient is.

  • Statistical efficiency / Robustness / Customizability
    The trade-off between statistical efficiency and robustness could be customized using the trimming percentage. In special cases, the above estimators could become the classic Harrell-Davis quantile estimator (the trimming percentage is zero) or a traditional quantile estimator (the trimming percentage is one). Unfortunately, it’s easy to achieve the required robustness because the breakdown point depends not only on the trimming percentage but also on the required quantile position.
  • Computational efficiency
    These quantile estimators are much faster than the classic Harrell-Davis quantile estimator because they require only several values of the Beta function (instead of O(n) values).

While these estimators is a good step forward in terms of customizability, they have an important drawback: the quantile function is not continuous. This introduces some troubles with:

Thus, it would be nice to have another customizable quantile estimator without the above drawbacks.

Quantile estimators based on k order statistics

One of the main problems of the winsorized and trimmed modifications of the Harrell-Davis quantile estimator is the lack of ability to directly customize the breakdown point. This issue could be resolved using an approach that always uses k order statistics around the required quantile. Let’s briefly discuss the properties of such an estimator.

  • Statistical efficiency / Robustness / Customizability
    Using the known k value, we could easily get the asymptotical breakdown point: \(\max(\min(p, 1-p) - k/2n), 0)\). The higher k we use, the higher efficiency (in the light-tailed case) we have (because we use more data from the sample). The smaller k we use, the higher robustness we have (because several corrupted measurements typically will not be able to spoil the estimation). By setting k=n, we can get the Harrell-Davis quantile estimator. By setting k=1 or k=2, we can get the traditional quantile estimators from the Hyndman-Fan classification.
  • Computational efficiency
    The performance of the suggested approach is similar to the previous section: using the sorted data, we have to perform O(k) operations.

Thus, quantile estimators based on k order statistics could be a bridge between the traditional quantile estimators and the Harrell-Davis quantile estimators. However, with a proper weight function, we could avoid the disadvantages of the THD/WHD and obtain a continuous and smooth quantile function (more about it in the next post).

Conclusion

In the scope of this post, we discussed only an idea of a quantile estimator based on k order statistics. It looks like it could resolve issues related to other popular quantile estimators listed above. In the next post, we will continue exploring the possibilities of using this approach. We will discuss the weight function that allows transforming k order statistics to a single estimation number. As usual, we will also perform Monte-Carlo simulations that evaluate the actual statistical efficiency of the suggested estimators.