Traditional multivariate control charts typically assume that the in-control process mean vector is fixed at a target point and determine whether the process is out of control by monitoring deviations from that point. In many practical applications, however, the in-control state is more appropriately represented by an allowable region, where the process remains normal as long as the mean vector stays within its boundaries. To address such region-based monitoring problems, this paper proposes a multivariate control chart based on a weighted generalized likelihood ratio statistic. The proposed method integrates the exponentially weighted moving average (EWMA) smoothing scheme with the generalized likelihood ratio principle under regional constraints. It measures the minimum weighted distance from the smoothed vector to the in-control closed convex region through a quadratic programming formulation, thereby enabling dynamic monitoring. Theoretically, the steady-state limiting distribution of the proposed statistic is derived for general closed convex regions, together with an equivalent standardized-distance representation. For conical regions, the limiting distribution reduces to a scaled Chi-bar-squared distribution. Numerical studies show that the proposed chart performs well under different regional structures and shift scenarios, with favorable detection performance for small shifts and under independence, negative correlation, and complex boundary constraints. Two real-data examples further demonstrate its practical effectiveness. Compared with conventional one-sided and fixed-center two-sided control charts, the proposed method can more accurately determine whether the process mean has left or re-entered the in-control region, making it highly suitable for region-based monitoring.
Spatial regression models with autocorrelation in the regressand, and possibly also the regressors or the disturbance, depending on the situation, have been adopted in various branches of social science, including economics, geography, and sociology. In the applications of such models, maximum likelihood is commonly used by assuming the availability of data at all the locations in a space and taking the weight matrix as a given. We compare several approaches to estimating unknown parameters and predicting missing regressand data in the spatial lag model containing a parametric weight matrix. Each of the approaches involves using maximum likelihood estimators derived in the presence of missing data and the best linear unbiased predictor formulated by plugging in the estimators, which makes it possible to estimate the impacts of changes in the values of the regressors on the values of the regressand. In order to construct impact estimators, we apply the bootstrap method, as a standard procedure, in processing data generated in Monte Carlo experiments and the delta method, as an expedient procedure, in processing data observed in the real world. The results of the parameter estimations, the data prediction, and the impact estimations demonstrate that one particular approach performs well, on the whole, compared with other considered approaches comparable in computational requirements. Our comparison is informative not only for choosing the appropriate approach for conducting a regression analysis using the spatial lag model but also for future discussions of alternative approaches to estimation and prediction in other models, including the spatial Durbin model.
We propose a simple, adaptive, and computationally efficient estimator for a univariate distribution function that combines ordered-sample spacings with local nearest-neighbor density estimation. The method naturally assigns probability based on both the spacing between sample points and the local density, ensuring monotonicity and proper normalization. We establish theoretical guarantees, including uniform consistency and pointwise asymptotic normality, and provide practical guidance on finite-sample behavior, bias–variance trade-offs, and boundary adjustments. The estimator supports accurate quantile estimation and demonstrates clear advantages in applications such as goodness-of-fit testing, survival and reliability analysis, adaptive tail estimation, and monotone pre-smoothing of empirical distributions in small samples. By blending the simplicity of order-statistic approaches with the flexibility of local density estimation, our method offers a versatile tool for both theoretical and applied statistical work.
This paper derives the necessary and sufficient conditions that guarantee the existence of maximum likelihood estimators (MLEs) for multinomial choice models subject to positivity constraints. Such constraints arise intrinsically from scale normalization, probabilistic validity, and economic monotonicity requirements in frameworks including discrete choice analysis and item response theory. In contrast to classical results on the existence of MLEs without any constraints, which only require the overlapping condition, we derive an additional monotonicity condition that may fail as the sample size grows. For the binary choice setting with a single scalar covariate, this monotonicity condition simplifies to a tractable data-dependent inequality. For general multinomial specifications with multiple alternatives, where direct verification becomes intractable, we propose an operational, “almost” necessary and sufficient criterion. A concise numerical study illustrates its finite-iteration behavior, its performance under increasing augmentation dimensions, and the distinction between an interior MLE and boundary solutions to the corresponding closed constrained optimization problem.
As the combination and generalization of the ordinary regression model and the autoregressive model, the autoregressive model with explanatory variables can better capture the dynamic characterization for the economic variables and thus has been given considerable attention. To describe the influence of the covariates more effectively and comprehensively, in this paper, we propose a new semiparametric model, the single-index-driven autoregressive model with varying-coefficient explanatory variables. Firstly, the link function and the coefficient function in the new model are approximated locally by linear functions. Secondly, the quasi-initial estimators for unknown functions can be derived by using the local linear method and the average method. Finally, the local linear method and the least squares method, together with the back-fitting algorithm are employed to get the estimators for unknown functions and unknown parameter. Subsequently, the iterative algorithm to perform the estimation procedure is given in detail. Moreover, the asymptotic normality for the estimators of the nonparametric part and the parametric part are established. Numerical studies demonstrate that the estimation method is workable.
