Posts on Andrey Akinshin

In this post, we derive the exact equation for the quantile absolute deviation around the median of the Normal distribution.

This post proposes the standard quantile absolute deviation (SQAD) — a dispersion estimator with 54% Gaussian efficiency and 32% breakdown point, sitting between MAD and the Shamos estimator in the efficiency-robustness trade-off.

This post derives the asymptotic Gaussian efficiency of the quantile absolute deviation as a function of its quantile level.

This post evaluates the finite-sample Gaussian efficiency of the Rousseeuw–Croux $S_n$ and $Q_n$ estimators.

This post documents when MAD breaks down: heavy-tailed distributions, multimodal data, and large outlier contamination.

This post announces the preprint on finite-sample bias-correction factors for MAD based on the Harrell-Davis quantile estimator.

Change point detection is a popular task in various disciplines.

We consider the following dynamical system: $$ \begin{cases} \dot{x}_1 = L(a_1, k_1, x_3) - k_1 x_1,\\ \dot{x}_2 = L(a_2, k_2, x_1) - k_2 x_2,\\ \dot{x}_3 = L(a_3, k_3, x_2) - k_3 x_3, \end{cases} $$ where $L$ is a piecewise linear function: $$ L(a, k, x)…

This post introduces the “degenerate point” — the fraction of tied values at which a dispersion estimator collapses to zero — as a metric for comparing robustness to ties.

This post introduces the untied quantile absolute deviation — a modification of QAD that never collapses to zero on samples with tied values.

In one of the previous posts, I described the idea of the middle non-zero quantile absolute deviation.

Let’s consider a discrete distribution $X$ defined by its probability mass function $p_X(x)$.

In the previous post, we discussed the Gastwirth’s location estimator.

This post studies Gastwirth’s location estimator — a weighted sum of three quantiles designed for robustness.

Let’s consider the following dynamical system: $$ \begin{cases} \dot{x}_1 = f(x_3) - x_1,\\ \dot{x}_2 = f(x_1) - x_2,\\ \dot{x}_3 = f(x_2) - x_3, \end{cases} $$ where $f(x) = \alpha / (1+x^m)$ is a Hill function.

In this post, I use Beeping Busy Beavers to show that twin prime conjecture could be proven or disproven.

This post introduces the Hodges–Lehmann–Sen shift estimator for the location difference between two samples, together with confidence intervals built via the Wilcoxon signed-rank test inversion.

This post extends the efficiency study of the Hodges–Lehmann estimator to various light-tailed and heavy-tailed distributions.

This post evaluates the relative efficiency of the Hodges–Lehmann estimator against the sample median under the normal distribution.

Let $X_1, X_2$ be i.i.d. random variables that follow the standard normal distribution $\mathcal{N}(0,1^2)$.

Let $X_1, X_2$ be i.i.d. random variables that follow the standard normal distribution $\mathcal{N}(0,1^2)$.

This post derives the exact analytical bias-correction factor for MAD with $n = 2$.

In this post, I combine ideas from two of my previous posts: Trimmed Harrell-Davis quantile estimator: quantile estimator that provides an optimal trade-off between statistical efficiency and robustness Weighted quantile estimators: a general scheme that…

The Mann–Whitney U test is one of the most popular nonparametric null hypothesis significance tests.

In the previous post, I discussed different fence-based outlier detectors.

In previous posts, I discussed properties of Tukey’s fences and asymmetric decile-based outlier detector (Part 1, Part 2).

Since the beginning of previous year, I have been working on building a quantile estimator that provides an optimal trade-off between statistical efficiency and robustness.

In the previous post, I suggested an asymmetric decile-based outlier detector as an alternative to Tukey’s fences.

In the previous post, I covered some problems with the outlier detector based on Tukey fences.

Tukey’s fences is one of the most popular simple outlier detectors for one-dimensional number arrays.

This post replaces MAD with the middle non-zero QAD in the gamma effect size formula to handle discrete distributions with ties.

This post introduces the middle non-zero quantile absolute deviation (MNZQAD) as a scale measure robust to ties in discrete distributions.

This post derives MAD bias-correction factors using the trimmed Harrell-Davis quantile estimator, refining finite-sample behaviour further.

This post compares MAD and the Shamos estimator across various distributions and sample sizes.

In the previous posts, I discussed [[p2-quantile-estimator-intro]] (a sequential estimator which takes $O(1)$ memory and estimates a single predefined quantile), the moving P² quantile estimator ([[mp2-quantile-estimator]]) (a moving modification of P² wh…

I already covered the P² quantile estimator and its possible implementation improvements in several blog posts.

I have already written a few blog posts about the P² quantile estimator (which is a sequential estimator that uses $O(1)$ memory): [[p2-quantile-estimator]] [[p2-quantile-estimator-rounding-issue]] [[p2-quantile-estimator-initialization]] In this post, we…

Update: the estimator accuracy could be improved using a bunch of patches.

There are multiple ways to compute the “average” value of an array of numbers.

Let’s say we have a distribution $X$ that is given by its $s$-quantile values: $$ q_{X_1} = Q_X(p_1),\; q_{X_2} = Q_X(p_2),\; \ldots,\; q_{X_{s-1}} = Q_X(p_{s-1}) $$ where $Q_X$ is the quantile function of $X$, $p_j = j / s$.

Let’s say we have two distributions $X$ and $Y$.

Let’s say we have two distributions $X$ and $Y$, and we want to express the “absolute difference” between them.

Update: the final paper was published in Communications in Statistics - Simulation and Computation (DOI: 10.1080/03610918.2022.2050396).

Let’s consider the classic sample median.

I already discussed misleadingness of such metrics like standard deviation and skewness.

Skewness is a commonly used measure of the asymmetry of the probability distributions.

The Greenwald-Khanna quantile estimator is a classic sequential quantile estimator which has the following features: It allows estimating quantiles with respect to the given precision $\epsilon$.

Update: the estimator accuracy could be improved using a bunch of patches.

Traditional quantile estimators that are based on one or two order statistics are a common way to estimate distribution quantiles based on the given samples.

In the previous post, I discussed the problem of non-smooth quantile-respectful density estimation (QRDE) which is generated by the trimmed Harrell-Davis quantile estimator based on the highest density interval of the given width.