In random matrix theory, large random covariance matrices for a class of stationary processes have been applied to many high-dimensional statistical problems. This paper provides a central limit theorem for the linear spectral statistics of these matrices, by proving, after noticing that the variables in the matrix of observations have a same covariance structure as that in sample covariance matrix formed by Gaussian random variables, that the mean and covariance functions of the Gaussian limits depends on the spectral density of the underlying process. Some examples of linear and nonlinear stationary processes to be applied to the result are given.
Testing for serial correlation is fundamental in regression analysis, yet methods for matrix-type data remain limited. This paper develops a test for serial correlation within the trace regression framework. By extending empirical likelihood to matrix data, we propose an empirical log-likelihood ratio statistic. Its asymptotic distribution is derived under both the null and local alternatives. Some simulations are conducted to demonstrate its finite-sample performance. Finally, the proposed test is applied to straw-burning fire-point data, confirming its practical utility.
The development of new models using specialized forms of reliability functions has been a topic of interest among researchers for the past five decades. A significant amount of research also exists in the bivariate case, where various bivariate distributions have been constructed using different failure patterns. As far as bivariate quantile-based reliability analysis is concerned, research in this direction is in its early stages. In this paper, we propose a new distribution by assuming a linear form of quantile-based hazard gradients. This is a novel approach for developing bivariate distributions. We extensively study the properties and applications of this distribution and present several well-known bivariate distributions as members of this family. The marginal and joint density functions of the distribution exhibit various shapes, which enable it to model various types of bivariate lifetime data. The distributional characteristics based on ordinary moments and L-moments are analyzed, and the reliability properties of the distribution are examined. Despite being constructed using a linear form of quantile-based hazard gradients, the hazard gradients derived from the distribution function exhibit non-monotonic shapes, which further enhance the distribution's applicability in modeling bivariate data. The utility of the distribution in modeling is illustrated using real data examples.
In this paper, we study the distance between zeros of the second derivative and roots of the polynomial p(n)(z) = Pi(s)(l=1)(z - xi(l)) Pi(n+1-s)(j=1)(z - X-j), where X-1, ... , Xn+1-s are i.i.d. complex-valued random variables, and xi(1), ... , xi(s) are distinct, fixed deterministic values. By giving a study which is about Hoeffding's inequality for cross-product sum, we reveal that under certain conditions, with probability at least 1 - o(1), p(n)(z) has s zeros of the second derivative, w(1)((n)), ... , w(s)((n)), such that for 1 < l < s, w(l)((n)) is the unique zero of the second derivative of p(n) that is within a distance of O(n(-1)) of xi(l).
Quantiles are used to measure risk or return in risk management and investment decision-making, while the data available in finance are often of high-frequency. In this paper, we study the joint confidence regions for a finite number of quantiles under strong mixing high-frequency data by employing the blockwise empirical likelihood (EL) method and the blockwise adjust empirical likelihood (AEL) method. We firstly construct the blockwise EL ratio statistic and the blockwise AEL ratio statistic for a finite number of quantiles and prove their asymptotic properties. Then we conduct simulations which show that the confidence regions based on the AEL method perform better than the EL method and the normal approximation method. As an application of our theoretic results, we also give an empirical analysis for real data.
This study explores the investment-proportional reinsurance control policies within a Stackelberg game framework, with the preferences based on the alpha-maxmin mean-variance criterion. In our model, two competing insurers act as followers, and their competitive relationship is governed by a non-zero-sum game. The two insurers can mitigate their risk exposure by purchasing proportional reinsurance. Serving as the leader, the reinsurer modifies the reinsurance premium based on the insurer's response strategy. Both the reinsurer and the two insurers are permitted to allocate their wealth between a risky asset, governed by geometric Brownian motion, and a risk-free asset, and they exhibit a preference for avoiding ambiguity. Unlike the prevalent assumption in the literature that the insurer or reinsurer exhibits extreme ambiguity aversion, this study accommodates a spectrum of ambiguity aversion levels among all parties involved. Each insurer aims to enhance its alpha-maxmin mean-variance utility of terminal relative performance, whereas the reinsurer aims to maximize the alpha-maxmin mean-variance utility of its terminal wealth. The equilibrium reinsurance premium, retention levels, and investment strategies, along with their corresponding equilibrium value functions, are derived by solving the extended Hamilton-Jacobi-Bellman (HJB) equations. Moreover, we also provide some numerical examples to illustrate our findings.
In this article, we introduce and study new kernel-based regression estimators constructed from a sample drawn from a q-distribution. Our approach is nonparametric and is based on kernel methods. The analysis is carried out using operations that exploit the properties of q-calculus. Under mild assumptions, we compute the bias and variance of these estimators. We also establish the weak convergence and strong consistency not only of the regression estimators, but also of the density estimators based on the same type of sample.
This paper introduces and studies various modes of statistical convergence for sequences of complex uncertain random variables-a mathematical framework designed to model hybrid phenomena involving both randomness and uncertainty in complex-valued settings. The primary contribution is the formulation of statistical convergence concepts with respect to the chance measure, including almost sure convergence, convergence in measure, convergence in mean, and convergence in distribution. Key relationships among these convergence types are established through rigorous theorems, and their distinctions are illustrated with carefully constructed examples and counterexamples. Notably, we prove that statistical convergence in measure implies convergence in chance measure, but not vice versa, and demonstrate that convergence in distribution does not generally entail convergence in measure. The results extend and unify existing theories of statistical convergence for uncertain and random variables, offering a comprehensive analytical toolkit for sequences where both probabilistic and epistemic uncertainties coexist. This work lays a foundational basis for further research in double and triple sequences of complex uncertain random variables, with potential applications in signal processing, financial modeling, and complex systems analysis under hybrid uncertainty.
When collecting sensitive information, randomized response techniques are widely used to protect respondent privacy, however, existing scrambling response models repeatedly suffer from an inherent tradeoff between estimation efficiency and privacy protection. In this article, we proposed a novel hybrid scrambling response model that links additive and multiplicative noise through a tuning parameter, allowing flexible control over privacy and efficiency. Along the proposed model, a generalized class of estimators for the population mean is introduced by using auxiliary information. The unbiasedness, variance, privacy protection measure, and a unified privacy efficiency measure of the proposed estimator are derived analytically. Efficiency comparisons with several renowned estimators under prominent scrambling models demonstrate the superiority of the proposed estimator under realistic conditions. Extensive simulation experiments and three real world data applications confirm that the proposed estimator consistently achieves lower mean squared error while maintaining enhanced privacy protection compared to existing methods, particularly when strong auxiliary information is available.
In this paper, we consider a periodic random coefficients autoregressive model of order p, (PRCA(p)). Some probabilistic properties are obtained, namely, the necessary conditions for strict periodic stationarity and second-order periodic stationarity, the existence of small-order and higher-order moments, the covariance structure and the geometric ergodicity. Calculation of moments up to fourth order, for p = 1, is done. Consistency and asymptotic normality for quasi-maximum likelihood (QML) estimator and for least squares (LS) estimator of the periodic parameters are established. A test for testing periodic randomness in the coefficients is proposed. An intensive simulation study shows that the parameters are well estimated even when the distributions of the white noise and the perturbation are not Gaussian. The test appears to perform well. Real data sets are used for comparing the PRCA(p) to one existing model and also for detecting possible periodic randomness in the coefficients.
This study introduced MultiMiss2, a novel closed-form methodology for estimating two randomly missing observations in non-replicated two-way factorial experiments. By minimizing the squared error loss (L2-norm), the approach provides analytic estimators that consistently demonstrated reduced bias, variance, mean absolute error (MAE), and mean squared error (MSE) as replications increased. Sensitivity analyses confirmed its robustness, with numerical optimizations converging to analytic solutions across all scenarios. Comparative benchmarking against established imputation methods (Mice, Amelia, MissForest, and Hmisc) showed that, while these approaches performed adequately in multivariate contexts, they were less suited for factorial designs. In contrast, MultiMiss2 consistently yielded smaller centered squared error loss, SSE, offering more accurate and reliable estimates. Although limited by the inability to estimate interaction effects in non-replicated designs, this methodology marks a significant step forward in factorial data analysis. Future work will extend the framework to handle more missing values, replicated designs, and higher-order factorial structures.
A more cost-effective and time-efficient method for data collection is Ranked Set Sampling (RSS), which yields more accurate results compared to the classical Simple Random Sampling (SRS) method. The Unit Birnbaum-Saunders (U-BS) distribution, a newly introduced unit distribution, is particularly useful for analyzing bounded data within the range (0, 1). This article focuses examines various estimation techniques for unknown parameters of the U-BS distribution, applied in both RSS and SRS. These techniques include: Maximum Likelihood, Least Squares, Maximum Product of Spacing, Minimum distances and Kolmogorov method. The efficiency and comparison of these techniques are illustrated through a simulation study, where key metrics such as the average estimators, average bias, mean square errors (MSE), and mean relative errors are computed. Additionally, a real-world milk production dataset is analyzed to demonstrate the practical applicability of the proposed estimators. The results from both the simulation and data analysis indicate that RSS is more efficient than SRS for estimating the parameters of the U-BS distribution.
This article aims to establish an analogy between engineering systems and biological systems by focusing on some reliability and statistical models and concepts. Since reliability is by definition concerned with the future health and behavior of a system, reliability models are also considered to have potential applications for the human organism. In fact, this potential already exists through the overlap and intersection between reliability theory and survival analysis. In this context, in this article, some reliability models and concepts that have not been discussed before will be adapted to a human organism, taking into account today's data-driven processes and artificial intelligence-based developments.
We introduce the Exponentially Truncated Centered (ETC) kernel, a new discrete associated kernel for nonparametric estimation of probability mass functions on count data. The ETC kernel combines exponentially decaying weights with a finite, locally adaptive truncation radius, thereby reducing boundary effects at zero while maintaining second-order accuracy at interior points. We derive closed-form expressions for the bias, variance, and mean integrated squared error, and propose an explicit normalization to enforce the summation-to-one constraint in finite samples. Bandwidth and truncation radius are selected jointly via a least-squares cross-validation criterion. We conduct an extensive simulation study comparing ETC with a broad range of competitors, including classical kernels (Binomial, discrete triangular) and more recent flexible proposals (Conway-Maxwell-Poisson, double Poisson, Gamma-Count, and optimal symmetric). The results indicate that ETC is consistently competitive in integrated squared error, with notable gains in multimodal settings, while its closed-form normalization provides a tangible computational advantage over kernels relying on infinite-series approximations. Real-data applications further illustrate its practical usefulness.
This paper investigates the effect of heterogeneity in the shape parameter beta and the scale parameter lambda on stochastic comparisons among samples drawn from independent Power-Lomax distributions. Under the assumption of independent components, the study focuses on the comparison of series and parallel systems with respect to various stochastic orders, including the usual stochastic tail order, the tail hazard rate order, and the tail reversed hazard rate order. Based on the theory of multivariate majorization, the paper systematically reveals how heterogeneity in beta and lambda influences stochastic comparisons. By introducing matrix majorization and analyzing lifetime variables under different system configurations, the work demonstrates that differences in component parameters significantly affect the overall reliability performance of the system.
This paper investigates the asymptotic properties of quantile estimators for a stationary S-mixing process. Moreover, the asymptotic normality and the theoretical exponential bounds are proven as the number of terms in the sequence tends to infinity. To demonstrate the validity of the obtained results, some simulations are provided.
This brief note provides a simple representation for the limiting likelihood ratio process in the case of a density with singularities. Ibragimov and Khasminskii show the limiting likelihood ratio process is related to a Poisson integral in this singular estimation problem.
Statistical process control (SPC) is a field of Statistics that uses statistical methods (or tools) to improve the quality of products and/or services. Among the tools used in SPC, control charts are the most popular. Most of the existing charts are designed based on the assumption of normality. These charts are known as parametric charts. When information about the type or nature of the underlying process distribution is not available, researchers recommend the use of nonparametric charts. This paper introduces new distribution-free group runs Shewhart, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts by combining the existing one-sided Shewhart, CUSUM or EWMA Mann-Whitney (MW) U sub-chart with an extended conforming run-length sub-chart. The performances of the resulting charts are investigated in terms of the characteristics of the distribution of the time to signal. The new charts are also compared to the existing basic and synthetic one-sided Shewhart, CUSUM and EWMA MW U charts. It was found that the new charts have very interesting properties as compared to the competing charts considered in this paper. A real-world silica concentrate data from a mining ore process is used to demonstrate the application and implementation of the new charts.
In the analysis of recurrent event data, time-dependent covariates were usually observed intermittently, which leads to missing covariates. At the same time, certain covariates have complex non-linear effect. To accommodate the potential non-linear covariate effect and the appearance of sparse covariate, a partial linear regression model with time-dependent covariates was proposed. For time-dependent covariates that cannot be continuously observed, a kernel imputation method was utilized. For the nonparametric covariate function, local polynomial kernel estimation was adopted, and the profile likelihood method was used to obtain parametric estimates. Some simulations were conducted to evaluate the estimation method, which demonstrated that our method outperformed other existing methods and a real data example was illustrated.
The present manuscript concerns component-wise estimation of the positive powers of the ordered restricted standard deviations for two normal populations under certain restrictions on the means. We obtain sufficient conditions to prove the dominance of equivariant estimators under scale-invariant strictly convex loss functions. Consequently, we propose various estimators that dominate the best affine equivariant estimator (BAEE). Furthermore, we derive a generalized Bayes estimator. As an application, we derive improved estimators for four special loss functions: quadratic, entropy, symmetric, and linex. An extensive Monte Carlo simulation is conducted to compare the risk performance of the proposed estimators. Finally, we provide a real data analysis for implementation purposes.
