Quantile estimators based on k order statistics, Part 8: Winsorized Harrell-Davis quantile estimator

In the previous post, we have discussed the trimmed modification of the Harrell-Davis quantile estimator based on the highest density interval of size \(\sqrt{n}/n\). This quantile estimator showed a decent level of statistical efficiency. However, the research wouldn’t be complete without comparison with the winsorized modification. Let’s fix it!

All posts from this series:

• Quantile estimators based on k order statistics, Part 1: Motivation (August 3, 2021)
• Quantile estimators based on k order statistics, Part 2: Extending Hyndman-Fan equations (August 10, 2021)
• Quantile estimators based on k order statistics, Part 3: Playing with the Beta function (August 17, 2021)
• Quantile estimators based on k order statistics, Part 4: Adopting trimmed Harrell-Davis quantile estimator (August 24, 2021)
• Quantile estimators based on k order statistics, Part 5: Improving trimmed Harrell-Davis quantile estimator (August 31, 2021)
• Quantile estimators based on k order statistics, Part 6: Continuous trimmed Harrell-Davis quantile estimator (September 7, 2021)
• Quantile estimators based on k order statistics, Part 7: Optimal threshold for the trimmed Harrell-Davis quantile estimator (September 14, 2021)
• Quantile estimators based on k order statistics, Part 8: Winsorized Harrell-Davis quantile estimator (September 21, 2021)

The approach

The general idea is the same that was used in one of the previous posts. We express the estimation of the \(p^\textrm{th}\) quantile as a weighted sum of all order statistics:

\[\begin{gather*} q_p = \sum_{i=1}^{n} W_{i} \cdot x_i,\\ W_{i} = F(r_i) - F(l_i),\\ l_i = (i - 1) / n, \quad r_i = i / n, \end{gather*} \]

where \(F\) is a CDF function of a specific distribution. In the case of the Harrell-Davis quantile estimator, we use the Beta distribution. Thus, \(F\) could be expressed via regularized incomplete beta function \(I_x(\alpha, \beta)\):

\[F_{\operatorname{HD}}(u) = I_u(\alpha, \beta), \quad \alpha = (n+1)p, \quad \beta = (n+1)(1 - p). \]

In the case of the winsorized Harrell-Davis quantile estimator, we use a part of the Beta distribution inside the \([L,\, R]\) window. In addition, we use the tails of the Beta distribution, but we replace “tailed elements” with elements that correspond to \(L\) and \(R\) positions. Thus, \(F\) could be expressed as rescaled regularized incomplete beta function inside the given window:

\[F_{\operatorname{WHD}}(u) = \left\{ \begin{array}{lcrcllr} 0 & \textrm{for} & & & u & < & L, \\ F_{\operatorname{HD}}(u) & \textrm{for} & L & \leq & u & \leq & R, \\ 1 & \textrm{for} & R & < & u. & & \end{array} \right. \]

In the previous post, we discussed the idea of choosing \(L\) and \(R\) as the highest density interval of the given width \(R-L\):

\[R-L = \frac{\sqrt{n}}{n}. \]

Numerical simulations

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:

• HD: Classic Harrell-Davis quantile estimator
• THD-SQRT: The described in the previous post trimmed modification of the Harrell-Davis quantile estimator based on highest density interval of size \(\sqrt{n}/n\).
• WHD-SQRT: The described above winsorized modification of the Harrell-Davis quantile estimator based on highest density interval of size \(\sqrt{n}/n\).

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. 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 and winsorized Harrell-Davis quantile estimators 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 (\(\textrm{MSE}\)):

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

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.

We are also going to use the following distributions:

• Uniform(0,1): Continuous uniform distribution; \(a = 0,\, b = 1\)
• Tri(0,1,2): Triangular distribution; \(a = 0,\, c = 1,\, b = 2\)
• Tri(0,0.2,2): Triangular distribution; \(a = 0,\, c = 0.2,\, b = 2\)
• Beta(2,4): Beta distribution; \(\alpha = 2,\, \beta = 4\)
• Beta(2,10): Beta distribution; \(\alpha = 2,\, \beta = 10\)
• Normal(0,1^2): Standard normal distribution; \(\mu = 0,\, \sigma = 1\)
• Weibull(1,2): Weibull distribution; \(\lambda = 1\;\textrm{(scale)},\, k = 2\;\textrm{(shape)}\)
• Student(3): Student distribution; \(\nu = 3\;\textrm{(degrees of freedom)}\)
• Gumbel(0,1): Gumbel distribution; \(\mu = 0\;\textrm{(location)},\, \beta = 1\;\textrm{(scale)}\)
• Exp(1): Exponential distribution; \(\lambda = 1\;\textrm{(rate)}\)
• Cauchy(0,1): Standard Cauchy distribution; \(x_0 = 0\;\textrm{(location)},\,\gamma = 1\;\textrm{(scale)}\)
• Pareto(1,0.5): Pareto distribution; \(x_m = 1\;\textrm{(scale)},\, \alpha = 0.5\;\textrm{(shape)}\)
• Pareto(1,2): Pareto distribution; \(x_m = 1\;\textrm{(scale)},\, \alpha = 2\;\textrm{(shape)}\)
• LogNormal(0,1^2): Log-normal distribution; \(\mu = 0, \sigma = 1\)
• LogNormal(0,2^2): Log-normal distribution; \(\mu = 0, \sigma = 2\)
• LogNormal(0,3^2): Log-normal distribution; \(\mu = 0, \sigma = 3\)
• Weibull(1,0.5): Weibull distribution; \(\lambda = 1\;\textrm{(scale)},\, k = 0.5\;\textrm{(shape)}\)
• Weibull(1,0.3): Weibull distribution; \(\lambda = 1\;\textrm{(scale)},\, k = 0.3\;\textrm{(shape)}\)
• Frechet(0,1,1): Frechet distribution; \(m=0\;\textrm{(location)},\, s = 1\;\textrm{(scale)},\, \alpha = 1\;\textrm{(shape)}\)
• Frechet(0,1,3): Frechet distribution; \(m=0\;\textrm{(location)},\, s = 1\;\textrm{(scale)},\, \alpha = 3\;\textrm{(shape)}\)

Simulation Results

Conclusion

One of the biggest drawbacks of the winsorized modification of the Harrell-Davis quantile estimator is the stair-like pattern which we can observe in the above images. Such a phenomenon could be easily explained. If we enumerate all the quantile values from 0 to 1, the \([L;R]\) window moves from left to right. At some specific moments, \(L\) and \(R\) cross the border between elements. Once they do that, the estimator switches the index of the winsorized elements that get “extra” weight. Thus, the winsorized modification of the Harrell-Davis quantile estimator is not smooth, and it has low statistical efficiency around these “switching points.”