Statistics 101, 201, and 202 are three open-source interactive web applications built with R \citep{R} and Shiny \citep{shiny} to support the teaching of introductory statistics and probability. The apps help students carry out common statistical computations -- computing probabilities from standard probability distributions, constructing confidence intervals, conducting hypothesis tests, and fitting simple linear regression models -- without requiring prior knowledge of R or any other programming language. Each app provides numerical results, plots rendered with \texttt{ggplot2} \citep{ggplot2}, and inline mathematical derivations typeset with MathJax \citep{cervone2012mathjax}, so that computation and statistical reasoning appear side by side in a single interface. The suite is organised around a broad pedagogical progression: Statistics~101 introduces probability distributions and their properties; Statistics~201 addresses confidence intervals and hypothesis tests; and Statistics~202 covers the simple linear model. All three apps are freely accessible online and their source code is released under a CC-BY-4.0 license.
Artificial intelligence has recently experienced remarkable advances, fueled by large models, vast datasets, accelerated hardware, and, last but not least, the transformative power of differentiable programming. This new programming paradigm enables end-to-end differentiation of complex computer programs (including those with control flows and data structures), making gradient-based optimization of program parameters possible. As an emerging paradigm, differentiable programming builds upon several areas of computer science and applied mathematics, including automatic differentiation, graphical models, optimization and statistics. This book presents a comprehensive review of the fundamental concepts useful for differentiable programming. We adopt two main perspectives, that of optimization and that of probability, with clear analogies between the two. Differentiable programming is not merely the differentiation of programs, but also the thoughtful design of programs intended for differentiation. By making programs differentiable, we inherently introduce probability distributions over their execution, providing a means to quantify the uncertainty associated with program outputs.
This paper investigates the volatility dynamics and underlying long memory features of four major cryptocurrencies—Bitcoin, Ethereum, Litecoin, and Ripple—which were selected due to their high liquidity, large trading volumes, and historical significance in the digital asset market. The long-range dependence exhibited in cryptocurrency markets is often overlooked. However, based on the strong evidence of persistent dependence in the return series, we adopt advanced volatility models that are capable of accommodating high volatility and heavy-tails, as well as the long memory properties of cryptocurrencies. Specifically, we employ long-memory extensions of the GAS (Long memory GAS) and GARCH (Fractionally Integrated Asymmetric Power ARCH) models, integrating heavy-tailed innovation distributions: the Generalized Hyperbolic Distribution (GHD) and Generalized Lambda Distribution (GLD). Standard GARCH and GAS models are included as benchmarks. The performance of the models are assessed using Value-at-Risk (VaR) estimation, backtesting (in-sample and out-of-sample) and volatility forecasting metrics. The results indicate that long memory models, particularly the FIAPARCH model, consistently outperforms the standard GAS and GARCH models in capturing tail risk and the volatility persistence. These findings emphasize the critical role of long memory in modeling the risk of cryptocurrencies, indicating that accounting for volatility persistence can significantly enhance the accuracy of risk estimates and strengthen risk management practices.
The problem of recursive filtering in linear state-space models is considered. The solution to this problem is the classical Kalman filter which is optimal in the sense that it minimizes the variance of the estimated states, if the error processes of the state and observation equations are both Gaussian. However, the Kalman filter is well known to be sensitive to outliers, so robustness is an issue. Two approximate conditional-mean (ACM) type filters for vector-valued observations are proposed that generalize existing univariate filters of similar type to the multivariate case.
These new ACM-type filters are compared by simulations in a multivariate setting with additive outliers to the classical Kalman filter and the robust least squares (rLS) filter, another approach robustifying the Kalman filter. Additionally, different settings of tuning parameters and their impact are investigated. The results of the simulation experiments show that in the presence of additive outliers the multivariate ACM-type filters not only outperform the classical Kalman filter, as expected, but they also outperform the rLS filter.
Muhammad Ammar Naufal, Ja'faruddin Ja'faruddin, Satrina Satrina
et al.
This study explores the application of ethnomathematics in the We Tenri Olle Tomb, located in Pancana Village, Barru Regency, South Sulawesi. The study aims to identify mathematical concepts, particularly geometry, embedded within the tomb’s architectural elements. Using a qualitative descriptive method, data were collected through direct observation and literature review. The results show that the tomb's design reflects the application of planar geometry concepts, such as rectangular walls, and spatial geometry, such as a semi-spherical dome. Geometric transformations, including reflections on walls, translations in floral patterns, and rotations in window designs, were also identified. The tomb design embodies the dynamic cultural acculturation between the local Bugis culture and European colonial influences while symbolizing the diplomatic relationship between the Tanete Kingdom and the Netherlands. This study contributes to ethnomathematics by connecting cultural heritage with mathematical concepts and providing insights for developing culture-based mathematics education to enhance students’ understanding of geometry and appreciation for local traditions.
The paper focuses on the Vasicek model driven by a tempered fractional Brownian motion.
We derive the asymptotic distributions of the least-squares estimators (based on continuous-time observations)
for the unknown drift parameters. This work continues the investigation by Mishura and Ralchenko (Fractal
and Fractional, 8(2:79), 2024), where these estimators were introduced and their strong consistency
was proved.
Inference in semi-supervised (SS) settings has gained substantial attention in recent years due to increased relevance in modern big-data problems. In a typical SS setting, there is a much larger-sized unlabeled data, containing only observations of predictors, and a moderately sized labeled data containing observations for both an outcome and the set of predictors. Such data naturally arises when the outcome, unlike the predictors, is costly or difficult to obtain. One of the primary statistical objectives in SS settings is to explore whether parameter estimation can be improved by exploiting the unlabeled data. We propose a novel Bayesian method for estimating the population mean in SS settings. The approach yields estimators that are both efficient and optimal for estimation and inference. The method itself has several interesting artifacts. The central idea behind the method is to model certain summary statistics of the data in a targeted manner, rather than the entire raw data itself, along with a novel Bayesian notion of debiasing. Specifying appropriate summary statistics crucially relies on a debiased representation of the population mean that incorporates unlabeled data through a flexible nuisance function while also learning its estimation bias. Combined with careful usage of sample splitting, this debiasing approach mitigates the effect of bias due to slow rates or misspecification of the nuisance parameter from the posterior of the final parameter of interest, ensuring its robustness and efficiency. Concrete theoretical results, via Bernstein--von Mises theorems, are established, validating all claims, and are further supported through extensive numerical studies. To our knowledge, this is possibly the first work on Bayesian inference in SS settings, and its central ideas also apply more broadly to other Bayesian semi-parametric inference problems.
The main results in this paper concern large deviations for families of non-Gaussian processes obtained as suitable perturbations of continuous centered multivariate Gaussian processes which satisfy a large deviation principle. We present some corollaries and, as a consequence, we obtain logarithmic asymptotic estimates for exit probabilities from suitable halfspaces and quadrants.
Marica Manisera, Manlio Migliorati, Matteo Ventura
et al.
Ordered response scales are often used in questionnaires to measure individuals' attitudes or perceptions. Among different response scale formats, we focus on multi-point semantic differential scales, requiring the respondent to position himself/herself on a rating between two bipolar adjectives. The obtained rating data require appropriate statistical models. We resort to the CUM model (Combination of a discrete Uniform and a - linearly transformed - Multinomial random variable), recently proposed in the framework of the CUB (Combination of discrete Uniform and shifted Binomial random variables) class of models. CUM is also suited to all the ordinal response scales with a middle “indifference” option.
In the seminal paper on CUM, the methodological approach was developed for an odd number m of response categories, while simulations, case studies and implementation in R were limited to m = 7. The objective of this paper is to extend the original proposal and investigate the model performance in the case of m = 5, which often arises in real situations. The R functions for fitting a CUM model with m = 5 are implemented and made available; simulation studies are developed and compared with results obtained for m = 7 and a case study concerned with the evaluation of museums' visitor experience is proposed.
Joko Purwadi, Bagus Gumelar, Tri Widiantoro
et al.
The paper discusses the relationship between economic growth and the unemployment rate in Indonesia in each province in 2021. Both variables are considered as the dependent variable and there are 5 independent variables used in this research such as human index development, wage minimum region, poor citizens percentage, investment, and farmer rate value in each province. The method used to analyze is canonical correlation analysis, which is one of the dependent methods that are used for multivariate analysis. This method was used to determine which variable had the most significant relationship between dependent and independent variables, the data was taken from the Center of Statistics Bureau Indonesia in 2021. The result shows that among independent variables the human index development had the strongest relation it had 79%, while the correlation between the dependent and independent variable the unemployment rate gives the strongest influence it is 68%.
Multiple heterogeneous data sources are becoming increasingly available for statistical analyses in the era of big data. As an important example in finite-population inference, we develop a unified framework of the test-and-pool approach to general parameter estimation by combining gold-standard probability and non-probability samples. We focus on the case when the study variable is observed in both datasets for estimating the target parameters, and each contains other auxiliary variables. Utilizing the probability design, we conduct a pretest procedure to determine the comparability of the non-probability data with the probability data and decide whether or not to leverage the non-probability data in a pooled analysis. When the probability and non-probability data are comparable, our approach combines both data for efficient estimation. Otherwise, we retain only the probability data for estimation. We also characterize the asymptotic distribution of the proposed test-and-pool estimator under a local alternative and provide a data-adaptive procedure to select the critical tuning parameters that target the smallest mean square error of the test-and-pool estimator. Lastly, to deal with the non-regularity of the test-and-pool estimator, we construct a robust confidence interval that has a good finite-sample coverage property.
Estimation and evaluation of individualized treatment rules have been studied extensively, but real-world treatment resource constraints have received limited attention in existing methods. We investigate a setting in which treatment is intervened upon based on covariates to optimize the mean counterfactual outcome under treatment cost constraints when the treatment cost is random. In a particularly interesting special case, an instrumental variable corresponding to encouragement to treatment is intervened upon with constraints on the proportion receiving treatment. For such settings, we first develop a method to estimate optimal individualized treatment rules. We further construct an asymptotically efficient plug-in estimator of the corresponding average treatment effect relative to a given reference rule.