Statistical Problems Research Articles

Analyzing the State-of-the-Art in Descriptive Statistics, Storytelling, and Performance Parameter Confirmation for IoT Applications

This paper gives thorough look at the most up-to-date methods in describing statistics, telling stories, and confirming performance parameters for Internet of Things (IoT) apps. It is very important to use descriptive statistics to summarize and explain the features of IoT data. We talk about different statistical measures, like mean, median, mode, variance, and standard deviation, and stress how important they are for understanding how IoT data is distributed. In the Internet of Things (IoT), storytelling means using stories to share lessons gained from analyzing data. We talk about how important storytelling methods are for helping people understand and use complex IoT data. We also look at different ways to tell stories, such as data visualization, story frameworks, and ways to get people involved. Confirming performance parameters is important for making sure that IoT applications are reliable and accurate. We look at different ways to check performance factors, like computer modeling, hypothesis testing, and confidence ranges. We also talk about the problems that can come up and the best ways to handle them when confirming performance parameters in IoT settings. We find important trends and problems in summary statistics, telling stories, and confirming performance parameters for IoT applications through this research. We stress the need for advanced analysis methods to handle the huge amount, speed, and range of IoT data that is being collected. We also stress how important it is to have good communication strategies when turning data ideas into choices that IoT users can act on. This paper gives researchers, practitioners, and decision-makers who work with IoT systems useful information by giving a full picture of the most recent progress in descriptive statistics, stories, and confirming performance parameters for IoT.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Number of runs of ones of length exceeding a threshold in a modified binary sequence with locks

Let us consider a sequence of binary (zero-one) trials. A counter registers ones but once an one is registered the mechanism is locked for a fixed number of trials following each registration. On such a sequence of a fixed length we define a random variable enumerating runs of ones of length exceeding a threshold. Exact recursive expressions are obtained for the probability mass function, generating functions and moments, as well as an exact closed formula for the mean of this random variable. A simulation algorithm is also provided for approximating values of this random variable on such a modified binary sequence. Statistical inference problems associated with the probability of ones are examined by numerical techniques and simulation. An application connected to the chance that a stochastic process remains or not in statistical control is discussed. Illustrative numerical examples exemplify further the theoretical results.

Enhanced Multimodal Nonparametric Probabilistic Method for Model-Form Uncertainty Quantification and Digital Twinning

The nonparametric probabilistic method (NPM) is a physics-based machine learning approach for model-form (MF) uncertainty quantification (UQ), model updating, and digital twinning. It extracts from reference data information not captured by a real-time computational model and infuses it into a “hyperparameterized,” stochastic version of this model by solving an inverse statistical problem formulated using a loss function and a few hyperparameters. The loss function is designed such that the mean value and statistical fluctuations of some quantities of interest predicted using the real-time stochastic model match target values obtained from the reference data. NPM’s performance hinges upon the efficient minimization of the loss function, which is unfortunately stochastic and nonconvex and thus prone to getting the optimization procedure trapped in suboptimal local minima. The latter scenario is exacerbated when the reference data is scarce. The paper addresses these issues by adopting the squared quadratic Wasserstein distance as the measure of dissimilarity between two different sets of data and by reformulating NPM’s inverse statistical problem as a multimodal data-assimilation problem. The potential of the resulting enhanced NPM for MF-UQ, model updating, and digital twinning is demonstrated using numerical simulations relevant to applications in structural dynamics, including structural health monitoring.

Epigenetics, Microbiome and Personalized Medicine: Focus on Kidney Disease.

Personalized medicine, which involves modifying treatment strategies/drug dosages based on massive laboratory/imaging data, faces large statistical and study design problems. The authors believe that the use of continuous multidimensional data, such as those regarding gut microbiota, or binary multidimensional systems properly transformed into a continuous variable, such as the epigenetic clock, offer an advantageous scenario for the design of trials of personalized medicine. We will discuss examples focusing on kidney diseases, specifically on IgA nephropathy. While gut dysbiosis can provide a treatment strategy to restore the standard gut microbiota using probiotics, transforming epigenetic omics data into epigenetic clocks offers a promising tool for personalized acute and chronic kidney disease care. Epigenetic clocks involve a complex transformation of DNA methylome data into estimated biological age. These clocks can identify people at high risk of developing kidney problems even before symptoms appear. Some of the effects of both the epigenetic clock and microbiota on kidney diseases seem to be mediated by endothelial dysfunction. These "big data" (epigenetic clocks and microbiota) can help tailor treatment plans by pinpointing patients likely to experience rapid declines or those who might not need overly aggressive therapies.

Parameter estimation for Gibbs distributions

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems coming from Gibbs distributions , which are families of probability distributions over a discrete space \(\Omega\) with probability mass function of the form \(\mu^{\Omega}_{\beta}(\omega)\propto e^{\beta H(\omega)}\) for \(\beta\) in an interval \([\beta_{\min},\beta_{\max}]\) and \(H(\omega)\in\{0\}\cup[1,n]\) . Two important parameters are the partition function , which is the normalization factor \(Z(\beta)=\sum_{\omega\in\Omega}e^{\beta H(\omega)}\) , and the vector of preimage counts \(c_{x}=|H^{-1}(x)|\) . We develop black-box sampling algorithms to estimate the counts using roughly \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions, where \(q=\log\frac{Z(\beta_{\max})}{Z(\beta_{\min})}\) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate \(Z(\beta_{\max})\) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of \(n\) and \(q\) up to logarithmic terms.

Fisher information contained in incomplete observations

The accuracy of statistical estimates of unknown distribution parameters depends not only on the bulk of sampling data but also on the method of data acquisition. The information content of experimental data is one of the basic requirements. Problems of mathematical statistics, in particular parametric estimation based on censored observations, have specific features. Typical representatives of models of incomplete observations on a straight line are models of random censoring, competing risks of (single, multiple) random censoring. The purpose of this study is to show that censoring does not always lead to loss of (Fisher) information. It is shown that if censoring is informative, i.e., the distribution of censoring random variables depends on the same parameter, it is possible to specify a model where information can be preserved due to censoring. On the contrary, if the censoring is not informative, then the loss of information is inevitable. The Cramer – Rao efficiency was taken as a criterion for the quality of the assessment, whereas the Fisher information was taken as the criterion for information about the unknown parameter.

Exponentially tighter bounds on limitations of quantum error mitigation

Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing without the heavy resource overheads required by fault-tolerant schemes. Recently, error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively ‘undone’ for larger system sizes. Our framework rigorously captures large classes of error-mitigation schemes in use today. By relating error mitigation to a statistical inference problem, we show that even at shallow circuit depths comparable to those of current experiments, a superpolynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. Noise also impacts other near-term applications by constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

Association between Prenatal Control and the Incidence of Preeclampsia in Ecuadorian Pregnant Women: A Cross-Sectional Study

Hypertensive disorders are one of the main complications that affect pregnancy, which makes them one of the most important causes of high maternal and fetal morbimortality. Approximately 80% of cases occur at term gestational age, while the remaining percentage begin at earlier gestational ages. Most cases of preeclampsia occur in healthy pregnant women. Therefore, it is important to establish the risk factors in prenatal control visits. Prenatal control is fundamental in which comprehensive surveillance of the pregnant woman is performed. It is known that hypertension in pregnancy is a frequent and potentially dangerous complication for the mother, fetus and newborn. For this reason, one of the objectives of prenatal care is to detect pre-pregnancy hypertension and pregnancy-induced hypertension syndrome (preeclampsia). A nationally representative sample of 20648 mothers from the National Health and Nutrition Survey 2018 (ENSANUT) was used. We used multicollinearity tests to rule out possible statistical modeling problems and a binary logistic linear regression model where Odds Ratio (OR) with their 95% confidence intervals (95% CI) were estimated for each of the independent variables. In addition, we used specificity tests to test the fit between our dependent and independent variables. Our results reveal that prenatal control reduced 2 times (OR= -2.0005; CI=-1.981;-2.055) the probability of suffering complications during pregnancy such as preeclampsia. This result is statistically significant (p<0.05). It was also demonstrated that women from rural areas, with lower income and low schooling are more susceptible to suffer complications during pregnancy compared to the rest of the population. Preeclampsia-eclampsia continues to be one of the pathologies that has the greatest impact on maternal and perinatal morbidity and mortality, so work should continue to find ways to prevent its onset or modify its evolution, avoiding eclampsia or other severe forms of the disease. The results found in this study showed that adequate prenatal care (more than five visits starting in the first trimester of pregnancy) has an impact on the prevention of pregnancy complications such as preeclampsia, since identifying and avoiding it is one of the main purposes of prenatal care.

Truncated correlation coefficient test for two random vectors

. Measuring the correlation between two random variables is a classic problem in statistics. The common textbook measures are Pearson r, Spearman ρ, and Kendall τ correlation coefficients. These three values are especially useful for two univariate variables, although they have been extended to two random vectors such as RV, which is a linear correlation coefficient for two random vectors. In this article, we propose a truncated correlation coefficient measure to test whether two random vectors are independent, together with some statistical properties. We perform extensive simulation studies to compare powers of RV and the proposed test. Results show that the proposed test is more powerful than RV to detect the independence. Finally, two real data analyses further show the performance of the proposed test.

Two-sample test for high-dimensional covariance matrices: A normal-reference approach

Testing the equality of the covariance matrices of two high-dimensional samples is a fundamental inference problem in statistics. Several tests have been proposed but they are either too liberal or too conservative when the required assumptions are not satisfied which attests that they are not always applicable in real data analysis. To overcome this difficulty, a normal-reference test is proposed and studied in this paper. It is shown that under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-squared-type mixture have the same limiting distribution. It is then justified to approximate the null distribution of the proposed test statistic using that of the chi-squared-type mixture. The distribution of the chi-squared-type mixture can be well approximated using a three-cumulant matched chi-squared-approximation with its approximation parameters consistently estimated from the data. The asymptotic power of the proposed test under a local alternative is also established. Simulation studies and a real data example demonstrate that the proposed test works well in general scenarios and outperforms the existing competitors substantially in terms of size control.

RELIGIOUS IDENTITY, (SELF-)IDENTIFICATION AND STATISTICS

The article examines the problems of religious statistics due to the multifaceted nature of religious identity and flexibility of religious identification and self-identifi cation. This is the cause of frequent diff erences in the estimated data on the number of followers of a particular confession. Population censuses could be a universal and comparable source, but not all countries ask about religious affiliation in the census. And even when there is a question on religion in the census, differences in wording (and differences in the meaning behind the wording) do not always allow for valid comparisons. Therefore, it is more common to find estimates of the number of followers of denominations based on various surveys. Religious identity is very multidimensional, and it often overlaps with other types of identity: ethnic, civil, local. A person can be associated with a certain faith by accepting the dogmatics, or it can be a set of practices and actions that define a way of life or certain milestones of the life cycle or moments/ periods of the religious calendar. Finally, the very feeling of belonging to cultural and historic tradition can bind a person to a particular faith. Therefore, in order to study quantitative indicators of religiosity, it is more convenient to operate with the procedural concepts of identifi cation and self-identifi cation. Religious statistics, or religious demography, was developed in the 1960s, i.e. in the period when the traditional forms of religiosity were being eroded, when religious dynamics in diff erent regions of the world were becoming more and more different, when it seemed that, at least in the West, religion was losing its position. Religious statistics require the adoption of some form of categorization. This article examines whether classifi cation should always be based on denominational principle alone. Religious statistics is a complex and delicate instrument, but nevertheless it draws some contours of religious structure, reveals the peculiarities of the religious situation and allows us to see the trends of its development.

Compressed sensing: a discrete optimization approach

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label learning. We introduce an ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve small-scale instances of CS to certifiable optimality. When compared against solutions produced by three state of the art benchmark methods on synthetic data, our numerical results show that our approach produces solutions that are on average 6.22%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6.22\\%$$\\end{document} more sparse. When compared only against the experiment-wise best performing benchmark method on synthetic data, our approach produces solutions that are on average 3.10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.10\\%$$\\end{document} more sparse. On real world ECG data, for a given ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _2$$\\end{document} reconstruction error our approach produces solutions that are on average 9.95%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$9.95\\%$$\\end{document} more sparse than benchmark methods (3.88%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.88\\%$$\\end{document} more sparse if only compared against the best performing benchmark), while for a given sparsity level our approach produces solutions that have on average 10.77%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10.77\\%$$\\end{document} lower reconstruction error than benchmark methods (1.42%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.42\\%$$\\end{document} lower error if only compared against the best performing benchmark). When used as a component of a multi-label classification algorithm, our approach achieves greater classification accuracy than benchmark compressed sensing methods. This improved accuracy comes at the cost of an increase in computation time by several orders of magnitude. Thus, for applications where runtime is not of critical importance, leveraging integer optimization can yield sparser and lower error solutions to CS than existing benchmarks.

Complete integrability of the problem of full statistics of nonstationary mass transfer in the simple inclusion process.

The simple inclusion process (SIP) interpolates between two well-known lattice gas models: the independent random walkers and the Kipnis-Marchioro-Presutti model. Here we study large deviations of nonstationary mass transfer in the SIP at long times in one dimension. We suppose that N≫1 particles start from a single lattice site at the origin, and we are interested in the probability P(M,N,T) of observing M of the particles, 0≤M≤N, to the right of the origin at a specified time T≫1. At large times, the corresponding full probability distribution has a large-deviation behavior, -lnP(M,N,T)≃sqrt[T]s(M/N,N/sqrt[T]). We determine the rate function s exactly by uncovering and utilizing complete integrability, by the inverse scattering method, of the underlying equationsof the macroscopic fluctuation theory. We also analyze different asymptotic limits of the rate function s.

Financial analysis of rural tourism in Catalonia and Galicia pre‐ and post COVID‐19

AbstractThis article analyses the financial health of Catalan and Galician rural tourism businesses between 2019 and 2021 by examining their profitability and solvency, in order to assess their survivability over the period affected by the COVID‐19 pandemic. To this end, we classify firms into groups with similar financial health. The article uses accounting data from the SABI (Iberian Balance Sheet Analysis System) database and the Compositional Data (CoDa) methodology, which offers a reliable analysis of sectoral financial statements, and is immune to the statistical problems of classical ratios (skewness, non‐normality, non‐linearity, and outliers, among others). The classification with the k‐means method reveals three groups with distinct financial profiles. None have a positive return on equity. One cluster is under special financial distress, with high indebtedness, low liquidity and an extremely negative margin. The size of this cluster increases during the 2020 lockdown but returns to normal in 2021.

Statistics in Social Sciences

This work is the result of some research and documentation of the theoretical problems of statistics and its applications in the study of phenomena and processes in nature and society. Statistics has applications in the most diverse fields, such as: medicine, natural sciences, economics, politics, human behavior sciences, etc. Practically, in any field of human existence we witness operations of gathering, systematizing and interpreting information. That is why its role is very important. Therefore, this paper reviews some statistical methods that are used in social sciences. The focus is mainly on the applied aspect of the techniques and the author give insights about techniques that can be used to answer problems in the abovementioned areas of research. The article focuses on detailed explanations in the presentation of the consequences of the application of statistical procedures in the social sciences.

Inference for Dependent Data with Learned Clusters

Abstract This paper presents and analyzes an approach to cluster-based inference for dependent data. The primary setting considered here is with spatially indexed data in which the dependence structure of observed random variables is characterized by a known, observed dissimilarity measure over spatial indices. Observations are partitioned into clusters with the use of an unsupervised clustering algorithm applied to the dissimilarity measure. Once the partition into clusters is learned, a cluster-based inference procedure is applied to a statistical hypothesis testing procedure. The procedure proposed in the paper allows the number of clusters to depend on the data, which gives researchers a principled method for choosing an appropriate clustering level. The paper gives conditions under which the proposed procedure asymptotically attains correct size. A simulation study shows that the proposed procedure attains near nominal size in finite samples in a variety of statistical testing problems with dependent data.

Green-Kubo paralell tempering molecular dynamics determination of the thermal conductivity for the wurtzite ZnO: statistical problems of the method

The thermal conductivity for the wurtzite ZnO is determined in the temperature range from 300 to 1100 K by using parallel tempering molecular dynamics within the Green-Kubo approach and a classical Morse-Born-Mayer-Coulomb hybrid interaction potential. Compared to other previous calculations for the thermal conductivity of common crystals within the same Green-Kubo and molecular dynamics approach, the used parallel tempering scheme shows some appealing improvements in the calculation of the time self-correlation of the heat flux vector, although at the price of using a relatively large number of total computational steps. However, in spite of the found improvements for the calculation of the self-correlation of the heat flux vector, some statistical problems on this point remain on the particular application of the method. Finally, even with the presence of a clear statistical noise, the obtained values and temperature trend of the calculated thermal conductivity shows the classical 1/T decaying behavior reported in previous works for wurtzite ZnO and other related semiconductor systems using the alternative Boltzmann transport equation theory.

A generalizable method for estimating meteor shower false positives

Context. The determination of meteor shower or parent body associations is inherently a statistical problem. Traditional methods, primarily the similarity discriminants, have limitations, particularly in handling the increasing volume and complexity of meteoroid orbit data. Aims. We introduce a new more statistically robust and generalizable method for estimating false positive detections in meteor shower identification, leveraging kernel density estimation (KDE). The method is applied to fireball data from the European Fireball Network, a comprehensive photographic fireball observation network established in 1963 for the detailed monitoring and analysis of fireballs across central Europe Methods. Utilizing a dataset of 824 fireballs observed by the European Fireball Network, we applied a multivariate Gaussian kernel within KDE and Z-score data normalization. Our method analyzes the parameter space of meteoroid orbits and geocentric impact characteristics, focusing on four different similarity discriminants: DSH, D′, DH, and DN. Results. The KDE methodology consistently converges toward a true established shower-associated fireball rate within the EFN dataset of 18–25% for all criteria. This indicates that the approach provides a more statistically robust estimate of the shower-associated component. Conclusions. Our findings highlight the potential of KDE combined with appropriate data normalization in enhancing the accuracy and reliability of meteor shower analysis. This method addresses the existing challenges posed by traditional similarity discriminants and offers a versatile solution adaptable to varying datasets and parameters.

ChatGPT-generated help produces learning gains equivalent to human tutor-authored help on mathematics skills.

Authoring of help content within educational technologies is labor intensive, requiring many iterations of content creation, refining, and proofreading. In this paper, we conduct an efficacy evaluation of ChatGPT-generated help using a 3 x 4 study design (N = 274) to compare the learning gains of ChatGPT to human tutor-authored help across four mathematics problem subject areas. Participants are randomly assigned to one of three hint conditions (control, human tutor, or ChatGPT) paired with one of four randomly assigned subject areas (Elementary Algebra, Intermediate Algebra, College Algebra, or Statistics). We find that only the ChatGPT condition produces statistically significant learning gains compared to a no-help control, with no statistically significant differences in gains or time-on-task observed between learners receiving ChatGPT vs human tutor help. Notably, ChatGPT-generated help failed quality checks on 32% of problems. This was, however, reducible to nearly 0% for algebra problems and 13% for statistics problems after applying self-consistency, a "hallucination" mitigation technique for Large Language Models.