In this paper, we investigate the hypotheses testing of regression parameters in linear models with φ -mixing random errors. Based on the central limit theorem for weighted sums of φ -mixing sequences and Cram é r-Wold theorem, the limiting distribution of M-test statistics are obtained when there is a small deviation from a linear hypotheses. In addition, the weak consistent estimators of nuisance parameters which involved in the asymptotic distribution of M-test statistics are also established. Finally, we provide some numerical simulations to verify the finite sample performance of the theoretical results.
The generalized autoregressive conditional heteroscedasticity, GARCH, models are the most popular models for capturing time-varying symmetric volatility in financial and economic time series data. Most of the theoretical results on GARCH models assume a constant or linear conditional mean. The smooth transition autoregressive, STAR, models allow for nonlinear behavior in the time series. The STAR-GARCH model combines regime switching and volatility models, with the regime switching in the mean equation and volatility in the variance equation. The estimation of smooth transition autoregressive models with GARCH errors, STAR-GARCH, are not entirely straightforward. Therefore, likelihood functions are appratimed using numerical methods. It is worth mentioning that the convergence of the maximum likelihood estimator for STAR-GARCH models is sensitive to initial values. This paper investigates modified maximum likelihood estimators of the parameters of smooth transition autoregressive models with GARCH errors and asymptotic distribution of modified maximum likelihood estimators. As a result, we can select optimal models based on Vuong’s test. A simulation study leads strong support to the results presented in the paper. We studied Wilshire 5000 total market full cap index dataset and selected an optimal model for this data based on the theoretical results.
The underlying idea of equivalence tests is to validate approximate models rather than merely rejecting them. This paper studies a test designed to confirm whether an observed multinomial dataset can be modelled by a parametric family using a ϕ -divergence as a measure of dissimilarity. The critical value is determined from the asymptotic normality of the test statistic. The asymptotic correctness and consistency of the proposed method are theoretically proved. The test’s performance in finite samples is studied via simulation. Furthermore, to illustrate the usefulness of the test, a parametric multinomial model for the risk preferences of experimental subjects is validated.
In this paper, we investigate tests based on the characteristic function for assessing symmetry in the framework of circular data. We introduce new procedures that rely on the empirical characteristic function and are able to detect alternatives that classical tests often fail to capture. A Monte Carlo study confirms the nice power properties of our proposed tests, highlighting their effectiveness and robustness across different scenarios.
A method for the prediction of future values of α -stable, power GARCH models using auxiliary processes (exogenous predictor) is presented. This predictor is optimal in the sense of minimization of the conditional mean square error. Simulation studies demonstrate that, compared to a standard (endogenous) prediction method, this α -stable GARCH predictor is capable of reducing the relative mean absolute error by 1
This paper makes three contributions to the estimation of distribution functions by smoothed empirical distribution functions based on kernel smoothing. First, we provide some theoretical evidence that the order of the kernel plays a minor role. Second, we propose two new data-based bandwidth selectors that are easy to implement, analyse them, and compare them with related approaches known from the literature. Our numerical experiments show that our proposals perform very well and that one of them is particularly strong w.r.t. the integrated squared error (ISE) in direct comparison with the empirical distribution function. Third, we prove strong consistency of the smoothed empirical distribution function in the case where the kernel smoothing is based on our data-based bandwidth selectors.
In this paper, we first define the jointly restricted empirical likelihood (JREL) as the product of empirical likelihood. This likelihood possesses a more general structure compared to Owen’s empirical likelihood. By constructing this JREL based on various general assumptions derived from estimating equations, it offers greater flexibility than existing empirical likelihood functions, making it suitable for a wide range of nonparametric applications, such as nonparametric Bridge regression. To assess its performance, we conducted a regression simulation study and applied it to the riboflavin dataset using a Bridge regression model.
The excess over threshold (EOT) distribution is the conditional distribution of the excesses over a threshold, given that the threshold has been exceeded. We propose empirical and kernel-based plug-in estimators of the EOT distribution function and derive their asymptotic bias, variance, and the limiting distributions of the studentized estimators as the sample size increases. We compare the asymptotic efficiency of the proposed estimators and obtain the asymptotically optimal bandwidth for the kernel-based estimator. The kernel estimator with optimal bandwidth is found to be asymptotically more accurate than the empirical estimator. We also prove strong uniform and L_2 convergence of the proposed estimators. Simulations reveal that the kernel estimator outperforms both the empirical and the generalized pareto distribution approximations of the EOT distribution function, especially for sample size up to 500. Using two real data sets, we demonstrate the applications of the estimators in the fields of hydrology and seismology.
In this paper, we discuss stochastic comparisons of two finite mixture models of different heterogeneous geometric distributions with different mixing proportions, in terms of usual stochastic, hazard rate, likelihood ratio and mean residual life orders. Finally, an application to shock models is discussed to illustrate the significance of the results established here.
This article proposes an estimator under stochastic linear restrictions to handle the issue of multicollinearity in generalized linear models. Besides, a genetic algorithm is applied to choose the tuning parameters of the proposed estimator. To ascertain whether the prior and sample information is consistent, a test statistic is given. A comparison of the estimators via matrix mean square error is also provided. The efficacy of the estimators is assessed using three simulation experiments with response variables from binomial, negative binomial, and gamma distributions, respectively, and a numerical example with a response variable from a Poisson distribution is discussed. The results demonstrate that the proposed estimator performs better in a numerical example for suitably selected tuning parameter values, while it completely surpasses all of its counterparts in simulation experiments.
This paper is concerned with the investigation of the properties of the hazard rate of the system protected by a block. By examining the characteristics of the hazard rate of the system supported by a protection block, a comparison is made with the redundancy method, according to the hazard rate. It is shown that the lifetime of the system supported by the protection block is larger than the lifetime of the system equipped with redundant system asymptotically in the hazard rate order. The extension of the results to a system with multiple components is also discussed for a consecutive k-out-of-n system.
In this paper, we investigate the strong uniform consistency rate of the Nadaraya-Watson (NW) kernel regression estimator under a widely orthant dependence model. The performance of the proposed estimator is illustrated through some simulations. Furthermore, a large comparison study with other NW estimators is presented based on the selected bandwidth type, and the performance of these NW estimators is evaluated using the global mean squared error (GMSE) criterion. Moreover, a real data analysis is provided to support the theoretical results.
We give axiomatic characterisations of generalized psi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-estimators and (usual) psi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-estimators (also called Z-estimators), respectively. The key properties of estimators that come into play in the characterisation theorems are the symmetry, the (strong) internality and the asymptotic idempotency. In the proofs, a separation theorem for Abelian subsemigroups plays a crucial role.
The estimation of risk for aggregated random variables is explored using (risk) measures within a broad and flexible class, called meanimiles. By separating the estimation of marginal distributions from the estimation of the dependence structure, modeled through copulas, this approach can avoid restrictive assumptions on the choice of risk measure, copula, or margins. The separation of dependence structure and margins allows for highly adaptable estimation procedures, accommodating parametric and nonparametric methods for both components. The framework’s generality enables its application to diverse contexts. Its utility is demonstrated in two real-data applications: optimizing horticultural auction portfolios and assessing disaster risk across US climate regions.
Degradation models are more and more used in reliability analysis. Among the most classical models, one can cite the Wiener process with a linear drift and the gamma process. Such models can be used to define complex maintenance policies, for instance. However, up to now, no goodness-of-fit test has been proposed formally for a general sampling scheme. In this paper, we first derive a moments method estimator for independent and gamma distributed random variables with common rate parameter (the shape being possibly different for each random variables) for which we provide asymptotic properties. Next, we consider a goodness-of-fit test for such sequence of random variables. This is an extension of a test derived by Villaseñor and González-Estrada (Villaseñor and González-Estrada 2015). This result is used to propose a goodness-of-fit test for homogeneous gamma processes under a general sampling scheme. We have applied this test to a real data set and then we have studied numerically the performance of our test in terms of coverage probabilities and power.
Logistic regression is widely used to estimate the proportion of a population possessing sensitive characteristics. It is essential to validate the assumed model before making statistical inferences to avoid erroneous conclusions. We propose employing a suite of statistical tests to evaluate the goodness-of-fit of logistic regression when only probabilistic versions of the binary response variable are collected using randomized response techniques. We rigorously derive the asymptotic properties of the proposed test statistics under the null hypothesis and some assumptions. Through comprehensive simulation studies and real data analyses, we demonstrate the finite-sample performance of the tests in terms of size and power.
We propose a test for detecting distributional changes in a sequence of multivariate observations based on optimal measure transport. The test is formulated using the empirical characteristic functions of the transported observations and is therefore distribution-free. We provide a method for constructing an exact test and we derive the asymptotic distribution of the test statistic. The performance of the proposed procedure is illustrated through a Monte Carlo simulation study.
We consider the statistical estimation of the specific surface area of particle samples, i.e., the total surface area of all particles divided by the total volume. In the engineering literature of particle statistics, there exist various estimators, for which no rigorous mathematical derivation is given. They are based on assumptions on relations between particle size, volume and surface area, which are not or only weakly empirically verified. In this study, we give a rigorous derivation of the most important estimators employing methods for marked point processes. There the particle sizes play the role of points and volumes or surface areas are the marks. On this theoretical basis, we discuss the role of a shape factor for particles, the so-called sphericity. In a case study, we analyse statistically a sample of fine rock material. This includes the consideration of some basic statistical assumptions as well as the quality of the specific surface area estimators.
Uniform designs have excellent space-filling properties in whole experimental domain, but in practice, the projection uniformity of designs need be considered in low-dimensional space. In this paper, the average uniformity pattern of asymmetric designs is defined by the reproducing kernel function, and the minimum projection uniformity criterion is provided to screen the designs with better projection uniformity. Moreover, the analytical relationships between the average uniformity pattern and the generalized wordlength pattern, the orthogonal vector, the design efficiency respectively are built, an improved lower bound of the average uniformity pattern is obtained, which is used as a benchmark to measure the projection uniformity of asymmetric designs in different dimensions. Some numerical examples are provided to explain the theoretical results.
