Computational Statistics最新文献

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered (alpha)-stable process (X_t). The upper bounds of all moments of the first exit position (left| X_{tau _D}right|) and the first exit time (tau _D) are explicitly obtained. It is found that the probability density function of (left| X_{tau _D}right|) or (tau _D) exponentially decays with the increase of (left| X_{tau _D}right|) or (tau _D), and (mathrm{E}left[ tau _Dright] sim mathrm{E}left[ left| X_{tau _D}-mathrm{E}left[ X_{tau _D}right] right| ^2right]), (mathrm{E}left[ tau _Dright] sim left| mathrm{E}left[ X_{tau _D}right] right|). Next, we obtain the Feynman–Kac representation of the Dirichlet problem by employing the semigroup theory. Furthermore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.

The Benford law is used world-wide for detecting non-conformance or data fraud of numerical data. It says that the significand of a data set from the universe is not uniformly, but logarithmically distributed. Especially, the first non-zero digit is One with an approximate probability of 0.3. There are several tests available for testing Benford, the best known are Pearson’s (chi ^2)-test, the Kolmogorov–Smirnov test and a modified version of the MAD-test. In the present paper we propose some tests, three of the four invariant sum tests are new and they are motivated by the sum invariance property of the Benford law. Two distance measures are investigated, Euclidean and Mahalanobis distance of the standardized sums to the orign. We use the significands corresponding to the first significant digit as well as the second significant digit, respectively. Moreover, we suggest inproved versions of the MAD-test and obtain critical values that are independent of the sample sizes. For illustration the tests are applied to specifically selected data sets where prior knowledge is available about being or not being Benford. Furthermore we discuss the role of truncation of distributions.

Let ({X_{i},1le ile n}) be a sequence of linear process based on dependent random variables with random coefficients, which has a mean shift at an unknown location. The cumulative sum (CUSUM, for short) estimator of the change point is studied. The strong convergence, (L_{r}) convergence, complete convergence and the rate of strong convergence are established for the CUSUM estimator under some mild conditions. These results improve and extend the corresponding ones in the literature. Simulation studies and two real data examples are also provided to support the theoretical results.

Abstract

Semi-supervised learning approaches have been successfully applied in a wide range of engineering and scientific fields. This paper investigates the generative model framework with a missingness mechanism for unclassified observations, as introduced by Ahfock and McLachlan (Stat Comput 30:1–12, 2020). We show that in a partially classified sample, a classifier using Bayes’ rule of allocation with a missing-data mechanism can surpass a fully supervised classifier in a two-class normal homoscedastic model, especially with moderate to low overlap and proportion of missing class labels, or with large overlap but few missing labels. It also outperforms a classifier with no missing-data mechanism regardless of the overlap region or the proportion of missing class labels. Our exploration of two- and three-component normal mixture models with unequal covariances through simulations further corroborates our findings. Finally, we illustrate the use of the proposed classifier with a missing-data mechanism on interneuronal and skin lesion datasets.

Finite mixture models have been widely used to model and analyze data from heterogeneous populations. In practical scenarios, these types of data often confront upper and/or lower detection limits due to the constraints imposed by experimental apparatuses. Additional complexity arises when measures of each mixture component significantly deviate from the normal distribution, manifesting characteristics such as multimodality, asymmetry, and heavy-tailed behavior, simultaneously. This paper introduces a flexible model tailored for censored data to address these intricacies, leveraging the finite mixture of skew-t distributions. An Expectation Conditional Maximization Either (ECME) algorithm, is developed to efficiently derive parameter estimates by iteratively maximizing the observed data log-likelihood function. The algorithm has closed-form expressions at the E-step that rely on formulas for the mean and variance of truncated skew-t distributions. Moreover, a method based on general information principles is presented for approximating the asymptotic covariance matrix of the estimators. Results obtained from the analysis of both simulated and real datasets demonstrate the proposed method’s effectiveness.

The main goal in this study was to determine confidence intervals for average age, average seniority, and average money-savings, for faculty members in a university retirement system using a simulation model. The simulation—built-in Arena—considers age, seniority, and the probability of continuing in the institution as the main input random variables in the model. An annual interest rate of 7% and an average annual salary increase of 3% were considered. The scenario simulated consisted of the teacher and the university making contributions, the faculty 5% of his salary, and the university 5% of the teacher’s salary. Since the base salaries with which teachers join to university are variable, we considered a monthly salary of MXN 23 181.2, corresponding to full-time teachers with middle salaries. The results obtained by a simulation of 30 replicates showed that the confidence intervals for the average age at retirement were (55.0, 55.2) years, for the average seniority (22.1, 22.3) years, and for the average savings amount (329 795.2, 341 287.0) MXN. Moreover, the risk that a retiree of 62 years of age and more of 25 years of work, is alive after his savings runs out is approximately 98% and this happens at 64 years of age.

Abstract

Fitting concentric ellipses is a crucial yet challenging task in image processing, pattern recognition, and astronomy. To address this complexity, researchers have introduced simplified models by imposing geometric assumptions. These assumptions enable the linearization of the model through reparameterization, allowing for the extension of various fitting methods. However, these restrictive assumptions often fail to hold in real-world scenarios, limiting their practical applicability. In this work, we propose two novel estimators that relax these assumptions: the Least Squares method (LS) and the Gradient Algebraic Fit (GRAF). Since these methods are iterative, we provide numerical implementations and strategies for obtaining reliable initial guesses. Moreover, we employ perturbation theory to conduct a first-order analysis, deriving the leading terms of their Mean Squared Errors and their theoretical lower bounds. Our theoretical findings reveal that the GRAF is statistically efficient, while the LS method is not. We further validate our theoretical results and the performance of the proposed estimators through a series of numerical experiments on both real and synthetic data.

The increased predictive power of machine learning models comes at the cost of increased complexity and loss of interpretability, particularly in comparison to parametric statistical models. This trade-off has led to the emergence of eXplainable AI (XAI) which provides methods, such as local explanations (LEs) and local variable attributions (LVAs), to shed light on how a model use predictors to arrive at a prediction. These provide a point estimate of the linear variable importance in the vicinity of a single observation. However, LVAs tend not to effectively handle association between predictors. To understand how the interaction between predictors affects the variable importance estimate, we can convert LVAs into linear projections and use the radial tour. This is also useful for learning how a model has made a mistake, or the effect of outliers, or the clustering of observations. The approach is illustrated with examples from categorical (penguin species, chocolate types) and quantitative (soccer/football salaries, house prices) response models. The methods are implemented in the R package cheem, available on CRAN.

The current status censoring takes place in survival analysis when the exact event times are not known, but each individual is monitored once for their survival status. The current status data often arise in medical research, from situations that involve multiple causes of failure. Examining current status competing risks data, commonly encountered in epidemiological studies and clinical trials, is more advantageous with Bayesian methods compared to conventional approaches. They excel in integrating prior knowledge with the observed data and delivering accurate results even with small samples. Inspired by these advantages, the present study is pioneering in introducing a Bayesian framework for both modelling and analysis of current status competing risks data together with covariates. By means of the proportional hazards model, estimation procedures for the regression parameters and cumulative incidence functions are established assuming appropriate prior distributions. The posterior computation is performed using an adaptive Metropolis–Hastings algorithm. Methods for comparing and validating models have been devised. An assessment of the finite sample characteristics of the estimators is conducted through simulation studies. Through the application of this Bayesian approach to prostate cancer clinical trial data, its practical efficacy is demonstrated.

We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix Q which depends on a parameter (theta ). Computing the probability distribution over states at time t requires the matrix exponential (exp ,left( tQright) ,), and inferring (theta ) from data requires its derivative (partial exp ,left( tQright) ,/partial theta ). Both are challenging to compute when the state space and hence the size of Q is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store Q. However, when Q can be written as a sum of tensor products, computing (exp ,left( tQright) ,) becomes feasible by the uniformization method, which does not require explicit storage of Q. Here we provide an analogous algorithm for computing (partial exp ,left( tQright) ,/partial theta ), the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that Q can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Implementation and data are available at https://github.com/spang-lab/TenSIR.