In this paper, we consider parameter estimation and quasi-likelihood ratio tests for multidimensional ergodic jump-diffusion processes defined by stochastic differential equations. In general, simultaneous estimation faces some challenges such as an increase of computational time for optimization and instability of estimation accuracy as the dimensionality of parameters grows. To address these issues, we propose an adaptive quasi-log likelihood function based on the joint quasi-log likelihood function. We then show that the resulting adaptive estimators possess consistency and asymptotic normality. Furthermore, we extend the joint quasi-log likelihood function and construct a test statistic using the proposed adaptive estimators. We prove that the proposed test statistic converges in distribution to a χ ^2 -distribution under the null hypothesis and that the associated test is consistent. Finally, we conduct numerical simulations using a specific jump-diffusion process model to examine the asymptotic behavior of the proposed adaptive estimators and test statistics.
Accurate prediction of hourly precipitation is particularly important for the early warning systems of floods and storms. However, the intermittent nature of precipitation which is characterized by long dry periods with sudden bursts of heavy rainfall events, makes forecasting a challenging task for usual forecasting methods. This study proposes a zero-adjusted gamma generalized additive model for location, scale, and shape (ZAGA-GAMLSS) estimated using gamboostLSS for probabilistic hourly precipitation forecasting. Three models incorporating lagged precipitation, meteorological variables, and neighboring station information are evaluated using a rolling-origin validation strategy with a four-year training window and a 120-h forecast horizon. The proposed models are compared with persistence, ARIMA, Random Forest, and LSTM benchmarks using point and interval forecast measures. The results demonstrate improved overall forecasting performance while providing calibrated prediction intervals and notably improved prediction of heavy rainfall events. The proposed framework offers a flexible approach for probabilistic precipitation forecasting and uncertainty quantification in flood forecasting applications.
While univariate time series analysis is well established, capturing nonlinear dynamics and high-dimensional lag structure remains a significant challenge. Sufficient dimension reduction (SDR) provides a useful and powerful framework to address the curse of dimensionality in such contexts. In this study, we introduce a convolution transformation method (CM) for estimating SDR subspaces in time series, with particular focus on the time series central mean subspace (TS-CMS) and the time series central subspace (TS-CS). We develop explicit matrix-based estimators for these two subspaces and extend a Fourier transform-based method (FM), originally developed for TS-CMS estimation, to the broader TS-CS setting. The asymptotic properties of the proposed estimators are established. Through simulation studies and empirical applications, we compare the performance of the CM and FM estimators. The numerical results suggest that the CM estimator performs well for estimating the TS-CMS, while the Fourier transform-based procedure tends to perform better for estimating the TS-CS in the settings considered.
Estimating treatment effects in randomized controlled trials (RCTs) is often costly and time-consuming because large sample sizes are needed to ensure adequate statistical power. Covariate adjustment using statistical models, particularly analysis of covariance (ANCOVA), can improve efficiency and reduce required enrollment. We examined the use of super-covariates (SCs), defined as participant-specific predicted outcomes under control treatment derived from a prognostic model, as adjustment variables in ANCOVA. An important feature of this approach is an ensemble of multiple predictive learners (typically machine learning methods) trained on historical control data. Although SCs from well-trained prognostic models improve the efficiency of ANCOVA estimators, their performance remains insufficiently characterized in practice, especially when (i) baseline covariate distributions differ between the historical data and the new RCT and (ii) the new RCT has a relatively small sample size. Using statistical simulations, we evaluated ANCOVA estimators incorporating a simplified SC under these scenarios and provided practical guidance. SC-based adjustment improves efficiency when abundant historical data are available. To optimize prognostic modeling, investigators should leverage all historical data rather than limiting the training set to subsets matched to the new RCT population. For treatment effect estimation, an ANCOVA adjusting for both an SC and baseline covariates can be more efficient than adjusting for either alone. However, simulations reveal small-sample bias under extreme covariate–treatment interactions. Caution is advised against preemptive sample size reductions even under high correlations between the SC and historical control outcomes, as such correlations are not necessarily reproduced in a new RCT.
This paper derives the exact transition density and cumulative distribution function of a linear combination of two independent Cox–Ingersoll–Ross (CIR) processes. By combining the Poisson–Gamma mixture representation of the non-central chi–square law with the Kummer-type convolution of Gamma densities, it derives a closed-form analytical expression involving confluent hypergeometric functions. This result extends the classical single-factor CIR transition law to a multifactor framework, providing the first explicit analytical characterization of the sum of two independent CIR diffusions. The proposed density admits stable numerical evaluation and facilitates exact likelihood computation, enabling rigorous parameter estimation in multifactor affine term–structure, stochastic volatility, and credit risk models. Numerical experiments confirm that the analytical density and CDF closely match Monte Carlo simulations across various parameter regimes, demonstrating high accuracy and computational efficiency.
We consider an explicit expression of the EM algorithm for Cauchy and its related distributions. The main idea is developed by making use of the exponential distribution structure, from which a general estimator function is then derived. The explicit forms of the EM algorithm for the Cauchy, log-Cauchy and another Cauchy distributions are derived as examples. We also derive explicit EM update equations for a skew-t distribution, as well as its finite mixture and regression models. An efficient acceleration for the EM algorithm is proposed. Illustrative examples of the mixture and regression models are also given.
This study introduces a new bounded probability distribution, termed the unit exponential odd exponentiated distribution, and investigates its important statistical properties. The parameters of the proposed model were estimated using the maximum likelihood estimation (MLE) method, with a simulation study conducted to evaluate the performance of the estimation process. To demonstrate its applicability, the model was applied to two real-world datasets: the human development index and maximum flood level. The analysis employed both classical and Bayesian approaches. The classical approach evaluated various model selection criteria and goodness-of-fit statistics. Results indicated that the proposed model outperformed nine competing models across multiple criteria. The study provides a theoretical derivation and practical implementation of Wald’s Sequential Probability Ratio Test (SPRT). The integration of the SPRT with the proposed distribution effectively captured critical patterns in the datasets, enhancing the model’s utility in real-world scenarios. A unit-interval quantile regression model is also proposed, demonstrating superior fit compared to conventional beta regression and Kumaraswamy regression models for outcomes confined within the unit interval exhibiting skewness. These findings highlight the significant contribution of the proposed model to the field of statistical modeling and data analysis.
We consider the problem of estimation of the CTE and CRTE. Based on the Kantorovich metric, we prove that the CTE and CRTE are preserved. The concepts of empirical distribution and empirical survival functions are used to obtain estimators for the CTE and CRTE. The asymptotic properties of the estimators are studied. Based on the kernel density estimator, we further propose estimators of the CTE and CRTE, and examine their consistency. We also construct estimators of the Q-CTE and Q-CRTE using the QDF estimators. It is shown that the estimators are consistent. Furthermore, the asymptotic distributions of the estimators are obtained. In both cases, we carry out simulation study and compute the AB and MSE values of the estimators. Finally, real data sets are considered for the purpose of illustration.
This paper proposes a novel threshold diffusion model in which the drift and diffusion components undergo regime shifts at distinct thresholds. Specifically, the drift and diffusion terms are governed by separate threshold values. This doubly threshold specification offers greater flexibility for capturing asymmetric dynamics in both the conditional mean and volatility. To estimate the model, we develop an approximate maximum likelihood estimation (AMLE) procedure that remains computationally tractable despite the model’s structural complexity. Monte Carlo simulations demonstrate that the proposed estimator is both consistent and efficient, and that standard information criteria—the Akaike information criterion (AIC), Bayesian information criterion (BIC), and Hannan–Quinn information criterion (HQIC)—are useful for distinguishing between models with shared versus separate thresholds. Empirical applications to United States (U.S.) Treasury interest rate data—including the 3-Month Treasury Bill and the 10-Year/3-Month yield spread–further support the proposed framework. The results reveal structural asymmetries that are better captured by allowing distinct threshold mechanisms in the drift and diffusion terms.
This study demonstrates the distinction between causality and predictability using a model in which the impulse response function converges to zero while the Granger causality measure remains positive in the limit of a parameter. This demonstration gives a perspective that goes beyond the exact relationship between a zero impulse response function and Granger non-causality. In addition, it is shown that this distinction is reflected in different causality measures. An empirical illustration demonstrates that the distinction is non-trivial even in practical applications.
Estimating the trend component of an economic time series is equivalent to estimating its cycle component. This is because the cycle component can be estimated as trend residuals. Consequently, considerable attention has been devoted to estimating the trend component. Modeling trend as a linear function of time is the most primitive but still widely used trend estimation method. The Hodrick–Prescott (HP) filter is one of the most common methods for trend estimation. The HP filter provides a more flexible time trend than a linear trend. In 2021, there was a new development in the popular method. Phillips and Shi (2021) developed a new filter by applying L_2 -boosting to the HP filter and they called it the boosted HP (bHP) filter. Interest in this new trend estimation method has grown since its development. Recently, Yamada (2025a) showed that the three trends mentioned above, i.e., the linear trend, the HP trend, and the bHP trend, can be understood in a unified manner. This paper advances research in this direction. In this paper, we provide a further unified perspective on the three trends. We also present several results based on the perspective.
This study develops a comparative forecasting framework that integrates daily weather information with quarterly electricity generation, used here as a proxy for electricity demand in New Zealand, through mixed-frequency modelling approaches. The analysis progresses from baseline univariate time-series models to classical mixed data sampling regressions, advanced regularised and autoregressive mixed-frequency models, and machine learning-based mixed-frequency methods. The forecasting results show that mixed-frequency models can improve upon traditional univariate benchmarks by incorporating higher-frequency weather information. Among the advanced approaches, autoregressive mixed-frequency models deliver strong forecasting performance, particularly over shorter recent evaluation windows, while seasonal time-series benchmarks such as SARIMA remain highly competitive and achieve the lowest RMSE in the main eight-quarter evaluation period. Machine learning-based mixed-frequency models show mixed performance, likely reflecting the challenges posed by data dimensionality and limited sample size. The proposed framework provides interpretable forecasts and offers practical insights for electricity system planning in renewable-dominated energy systems.
An alternative dependent counting (ADC) nonnegative integer-valued autoregressive (INAR) process of the higher-order is introduced for modeling the count time series. The condition for the strictly stationarity (with the second moment) and ergodicity is given by embedding the proposed ADCINAR process into a random coefficient INAR process of the higher-order. Then, the estimation of parameters is discussed in details. The easiest methods are the conditional least squares (CLS) method and the Yule–Walker method by which only the INAR parameter and the innovation mean are estimable. For estimating the new parameter as well as the innovation variance, the two-step CLS (2CLS) method is separately applied on the basis of the estimated squared residuals, by plugging a suitable estimator for the parameter in the conditional mean function. To avoid unnecessary (unconditional) higher moments conditions that ensure desirable properties of the CLS/2CLS estimators, the Gaussian quasi maximum likelihood (QML) method is employed, together with its asymptotic properties under mild conditions. Some non-standard testing problems are mentioned for the proposed ADCINAR process. Monte Carlo simulations are carried out to assess the performance of the Gaussian QML estimator. Also, practical effectiveness of the proposed higher-order ADCINAR process over the widely used higher-order INAR process is illustrated by two real-world examples.
We propose a non-linear extension of the Linear Non-Gaussian Acyclic Model (LiNGAM) for causal discovery. For each variable, the method fits a penalized additive regression on the other variables and uses the Hilbert–Schmidt Independence Criterion (HSIC) to test whether the residuals depend on a predictor omitted from the fit. Residual dependence on the omitted predictor indicates an edge between the two variables, and the larger of the two HSIC scores identifies the parent. Pairwise HSIC tests therefore recover the causal edges under Benjamini–Hochberg control of the false discovery rate. Under smooth additive structural equations with mutually independent, non-Gaussian noise, the method consistently recovers the causal graph in polynomial time. In the linear case, its residual diagnostics match those of LiNGAM. For each selected edge, the method also yields a fitted partial effect curve, an interpretable summary of how the parent influences the child. Simulations confirm the recovery guarantee at finite samples, and on a wine chemistry dataset the recovered curves agree with the wine-chemistry literature without assuming a specific functional form.
It has been shown that the error bound of the Lasso estimator attains the minimax optimal rate when the corresponding regression coefficients are sparse. The existing theory, however, is developed under fixed-design predictors with i.i.d. errors. Subsequent extensions to time-series predictors primarily focus on hard sparsity. This paper provides a unified treatment of the Lasso with time-series predictors under both hard and soft sparsity, with the latter allowing all regression coefficients to be nonzero. We first show that the key results continue to hold in the time-series setting. We then derive the convergence rate of the mean squared post-sample prediction error (MSPPE) for the Lasso predictor. Finally, for moderately dense coefficients, we show that the Lasso may require substantially fewer candidate variables to achieve the desired convergence rate. Monte Carlo simulations are conducted to evaluate the performance of the Lasso with time-series predictors.
This article introduces two matrix-variate asymmetric distributions for modeling data exhibiting skewness and latent scale variability: the matrix-variate shifted asymmetric Laplace (MVSAL) and the matrix-variate shifted generalized asymmetric Laplace (MVSGAL). Both distributions arise within the class of matrix-variate normal mean-variance mixtures, yielding a hierarchical representation that facilitates random generation and likelihood-based inference. Several structural properties are established, including closure under linear transformations and marginalization. An Expectation/Conditional Maximization (ECM) algorithm is developed for maximum likelihood estimation, and simulation results demonstrate accurate parameter recovery with improved stability as the sample size increases. An empirical application to seasonal climatic data from 100 municipalities in the state of São Paulo illustrates the practical use of the MVSGAL model in a multivariate matrix setting, capturing temporal persistence, cross-variable dependence, and asymmetric effects within a unified framework.
Identifying causal direction between two variables remainsś challenging when data exhibit strong nonlinearity, discreteness, or mixed continuous–discrete structures. Existing approaches often rely on restrictive functional assumptions, are limited to specific data types, or degrade substantially under complex noise and non-additive interactions. Motivated by recent advances in representation learning for sufficient dimension reduction, we propose a unified and highly flexible framework for bivariate causal direction discovery based on the belted and ensembled neural network (BENN). The key idea is to learn a low-dimensional sufficient predictor of the response through a bottleneck architecture that captures essential information in the conditional distribution, thereby simplifying the subsequent regression or classification task used to construct residuals. These residuals approximate the noise component of the underlying additive or latent-utility model, enabling reliable independence testing in both directions. The proposed framework naturally accommodates continuous, discrete, and mixed-type variable pairs without requiring any modification to the overall pipeline. Extensive synthetic experiments across a wide variety of nonlinear mechanisms, noise distributions, and response types, along with evaluations on real-world datasets, demonstrate that the BENN-based approach achieves state-of-the-art accuracy and stability. It performs favorably compared to existing information-theoretic methods, ANM variants, and recent models for mixed-type data. Owing to its modular structure, we discuss the method’s future potential to be integrated as a preprocessing step in multivariate causal discovery algorithms, such as RESIT-type procedures.
Indicator functions characterize properties of fractional factorial designs such as size and orthogonality via their coefficients. Solving systems of algebraic equations in these coefficients enables the enumeration of orthogonal designs, including classes of mixed-level designs. Counting functions generalize indicator functions by allowing point replication, thereby providing a formulation for the complete enumeration of orthogonal arrays with replicated points. However, the broader range of values for a counting function substantially increases the computational complexity of the resulting algebraic systems. To address this problem, we propose a sequential method that extends an array one factor at a time and solves a sequence of smaller algebraic systems in terms of the number of variables and the degrees of the polynomials based on the theory of projectivity. This approach decomposes the solution of a large zero-dimensional system of polynomial equations into tractable subproblems. We demonstrate the effectiveness of the proposed method by completely enumerating orthogonal arrays such as OA(24, 3^1 2^3, 2) and OA(32, 4^2 2^4, 3) .
The Cox regression is a popular model for analyzing survival data with predictors. However, its effectiveness can decline when multicollinearity is present, resulting in unreliable estimates from the standard maximum partial likelihood approach. In this study, we develop improved shrinkage estimators inspired by the Liu method to enhance the accuracy of coefficient estimation. Specifically, we develop several estimators, including linear shrinkage, Stein, positive Stein, pretest, and shrinkage pretest estimators, that leverage prior knowledge about the model parameters. We derive their asymptotic properties and assess their performance through extensive Monte Carlo simulations. We further illustrate how to evaluate the estimation strategies on survival data. Our findings reveal notable improvements, demonstrating the practical benefits of these estimators for researchers.
