Dynamical System Case Study 1 (symmetric 3d system)

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 case study, we explore the phase portrait of this system for $$\alpha = 18,\; m = 3$$.

Beeping Busy Beavers and twin prime conjecture

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

Hodges-Lehmann-Sen shift and shift confidence interval estimators

In the previous two posts (1, 2), I discussed the Hodges-Lehmann median estimator. The suggested idea of getting median estimations based on a cartesian product could be adopted to estimate the shift between two samples. In this post, we discuss how to build Hodges-Lehmann-Sen shift estimator and how to get confidence intervals for the obtained estimations. Also, we perform a simulation study that checks the actual coverage percentage of these intervals.

Statistical efficiency of the Hodges-Lehmann median estimator, Part 2

In the previous post, we evaluated the relative statistical efficiency of the Hodges-Lehmann median estimator against the sample median under the normal distribution. In this post, we extended this experiment to a set of various light-tailed and heavy-tailed distributions.

Statistical efficiency of the Hodges-Lehmann median estimator, Part 1

In this post, we evaluate the relative statistical efficiency of the Hodges-Lehmann median estimator against the sample median under the normal distribution. We also compare it with the efficiency of the Harrell-Davis quantile estimator.

Expected value of the maximum of two standard half-normal distributions

Let $$X_1, X_2$$ be i.i.d. random variables that follow the standard normal distribution $$\mathcal{N}(0,1^2)$$. In the previous post, I have found the expected value of $$\min(|X_1|, |X_2|)$$. Now it’s time to find the value of $$Z = \max(|X_1|, |X_2|)$$.

Expected value of the minimum of two standard half-normal distributions

Let $$X_1, X_2$$ be i.i.d. random variables that follow the standard normal distribution $$\mathcal{N}(0,1^2)$$. One day I wondered, what is the expected value of $$Z = \min(|X_1|, |X_2|)$$? It turned out to be a fun exercise. Let’s solve it together!

Unbiased median absolute deviation for n=2

I already covered the topic of the unbiased median deviation based on the traditional sample median, the Harrell-Davis quantile estimator, and the trimmed Harrell-Davis quantile estimator. In all the posts, the values of bias-correction factors were evaluated using the Monte-Carlo simulation. In this post, we calculate the exact value of the bias-correction factor for two-element samples.

Weighted trimmed Harrell-Davis quantile estimator

In this post, I combine ideas from two of my previous posts:

Thus, we are going to build a weighted version of the trimmed Harrell-Davis quantile estimator based on the highest density interval of the given width.

Let’s say we want to compare two samples $$x = \{ x_1, x_2, \ldots, x_n \}$$ and $$y = \{ y_1, y_2, \ldots, y_m \}$$ using the one-sided Mann–Whitney U test. Sometimes, we don’t have an opportunity to gather enough data and we have to work with small samples. Imagine that the size of both samples is six: $$n=m=6$$. We want to set the statistical level $$\alpha$$ to $$0.001$$ (because we really don’t want to get false-positive results). Is it a valid requirement? In fact, the minimum p-value we can observe with $$n=m=6$$ is $$\approx 0.001082$$. Thus, with $$\alpha = 0.001$$, it’s impossible to get a positive result. Meanwhile, everything is correct from the technical point of view: since we can’t get any positive results, the false positive rate is exactly zero which is less than $$0.001$$. However, it’s definitely not something that we want: with this setup the test becomes useless because it always provides negative results regardless of the input data.