statistics

Let $X$ be a random variable in $\mathbb{R}^d$ with law $\mu$. We denote by $\mathcal{P}(\mathbb{R}^d)$ the set of all Borel probability measures on $\mathbb{R}^d$. Assume that $\mu \in ...

We propose a family of weighted statistics based on the CUSUM process of the WLS residuals for the online detection of changepoints in a Random Coefficient Autoregressive model, using both the standard CUSUM and the Page-CUSUM process. We derive the asymptotics under the null of no changepoint for all possible weighing schemes, including the case of the standardized CUSUM, for which we derive a D…

This study proposes two novel time-varying model-averaging methods for time-varying parameter regression models. When the number of predictors is small, we propose a novel time-varying complete subset-averaging (TVCSA) procedure, where the optimal time-varying subset size is obtained by minimizing the local leave- h -out cross-validation criterion. The TVCSA method is asymptotically optimal for a…

We study the uniform convergence rates of nonparametric estimators for a probability density function and its derivatives when the density has a known pole. Such situations arise in some structural microeconometric models, for example, in auction, labor, and consumer search, where uniform convergence rates of density functions are important for nonparametric and semiparametric estimation. Existin…

"Do countries with higher GDP per capita also have longer life expectancy?" I built a tool that lets you explore questions like that across 48 countries by picking any two of five metrics as scatter-plot axes. Two implementation hinges: (1) metrics that span orders of magnitude (population: Singapore 5.6M to India 1,417M, a 250× range) must be plotted and correlated on a log scale or every point …

I am trying to test Hurst exponent in different time lag range. However, i got negative values in some time lag range which is weird, because the Hurst exponent should have values within the range from 0 to 1. This is the Python code to calculate the Hurst exponent: *calculate Hurst* lag1 = 2 lags = range(lag1, 20) tau = [sqrt(std(subtract(ts[lag:], ts[:-lag]))) for lag in lags] plot(log(lags),…

Nature, Published online: 12 June 2026; doi:10.1038/d41586-026-01888-9 A new benchmark pitting AI against previously unseen maths problems shows systems still fall short of top human expertise.

Let's say I have a dataset library(tidyverse) set.seed(123) df = tibble( x = rnorm(1000), y = -0.1 * x + rnorm(1000, sd = 0.1), truncate = (y > (0.25*x -0.4)) & (y < 0.25*x+0.4)) ...

Just a question out of curiosity. I am reviewing a manuscript which used (forward) stepwise regression for selection of models for the prediction of metastases in a certain type of cancer. I am fully ...

I have activity data from treatment and control cohorts measured in biological samples, where each sample is recorded across multiple timepoints (different days). In my current analysis, each box in ...

IntroductionRecent perspectives in mathematics education highlight the importance of integrating cognitive and affective dimensions in understanding students’ reasoning. Accordingly, the present study investigates the interrelations of performance on geometric problem-solving and general cognitive skills, fluid intelligence, cognitive reflection, and mathematics-related beliefs.MethodsThe study e…

I understand the data summarization aspect of PCA thanks to the great answer here, but could you help me understand the dimensionality reduction aspect? Not the mathematics but the interpretation. I ...

Years ago I wrote about correcting covariate shift by reweighting your data. Your features come from the wrong distribution \(q\), you care about a target \(p\), so you weight every observation by \(\beta_i = p(x_i)/q(x_i)\) and your estimates are unbiased again. I ended that post by admitting the weights “can be quite a bit off,” and waved at fixing it another day. Here is the more basic questio…

One number in the array appears twice, while another number from the range [1, n] is missing. Our task is to find both: Repeating Number (A) Missing Number (B) Example Input: nums = [ 1 , 2 , 2 , 4 ] Output: [ 2 , 3 ] Here, 2 appears twice and 3 is missing. Brute Force Approach Intuition For every number from 1 to n , count how many times it appears in the array. If frequency is 2 , it is the rep…

The function (1 + cos(x))/2 gives a fair approximation to the Gaussian density exp(−x²) You can make the approximation much better by raising it to a power. The function ((1 + cos(x))/2)4 gives a good lower bound and ((1 + cos(x))/2)3.5597 gives a good upper bound. More on that here. There are other ways of […] The post Another Gaussian approximation first appeared on John D. Cook .

_Sitting with a Mad Mind : A Working Thesis_. forthcomingCaptured Resonance as Structural Law advances three falsifiable cross-scale predictions. The first — that the kinetics of attractor reconstitution following substrate-directed intervention, normalized for substrate-regeneration capacity, show comparable structural signatures across scales — is the prediction on which the cross-scale identit…

Statistical modeling of claim severity distributions is central to actuarial science and risk management, where parameter estimation must balance efficiency and robustness. Maximum likelihood estimation (MLE) is asymptotically efficient under correct model specification but sensitive to extreme observations and perturbations from the assumed distributional form. Robust L-estimators, including the…

Model created by researchers shows better outcomes are often more likely when people are not too ambitious It is the end of an idiom for motivational speakers. Instead of shooting for the moon when pursuing life’s goals, researchers say people should be advised to aim a little lower if they want the best outcome. The tip may lack the punch of uncompromising drive, but aiming for merely above aver…

Kullback-Leibler divergence Kullback-Leibler divergence is defined for two random variables X and Y by K-L divergence is non-negative, and it’s zero if and only if X and Y have the same distribution. But it is not a metric, for reasons explained here. For one thing, it’s not symmetric. Jeffreys divergence We can fix the symmetry problem by […] The post Turning K-L divergence into a metric first a…