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This paper proposes a semiparametric spatial lag model and develops a Bayesian estimation method for this model. In the estimation of the model, the paper combines Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm, random walk Metropolis sampler, and Gibbs sampling techniques to sample all the parameters. The paper conducts numerical simulations to validate the proposed Bayesian estimation theory using a numerical example. The simulation results demonstrate satisfactory estimation performance of the parameter part and the fitting performance of the nonparametric function under different spatial weight matrix settings. Furthermore, the paper applies the constructed model and its estimation method to an empirical study on the relationship between economic growth and carbon emissions in China, illustrating the practical application value of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a semiparametric spatial lag model and develops a Bayesian estimation method for this model. In the estimation of the model, the paper combines Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm, random walk Metropolis sampler, and Gibbs sampling techniques to sample all the parameters. The paper conducts numerical simulations to validate the proposed Bayesian estimation theory using a numerical example. The simulation results demonstrate satisfactory estimation performance of the parameter part and the fitting performance of the nonparametric function under different spatial weight matrix settings. Furthermore, the paper applies the constructed model and its estimation method to an empirical study on the relationship between economic growth and carbon emissions in China, illustrating the practical application value of the theoretical results.

Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.

We consider the estimation of the boundary of a set when it is known to be sufficiently smooth, to satisfy certain shape constraints and to have an additive structure. Our proposed method is based on spline estimation of a conditional quantile regression and is resistant to outliers and/or extreme values in the data. This work is a desirable extension of existing works in the literature and can also be viewed as an alternative to existing estimators that have been used in empirical analysis. The results of a Monte Carlo study show that the new method outperforms the existing methods when outliers or heterogeneity are present. Our theoretical analysis indicates that our proposed boundary estimator is uniformly consistent under a set of standard assumptions. We illustrate practical use of our method by estimating two production functions using real-world datasets. [ABSTRACT FROM AUTHOR]

In this research, smoothing spline was used as a suitable nonparametric method for estimating one of the most important time series models, which is the Nonlinear Additive Autoregressive Model (NAAR) due to the importance of this model in terms of its representation of many phenomena and trying to control these phenomena through using accurate statistical models and how to estimate them in an advanced numerical method. This method is one of the smoothing techniques in finding the best fit for the curve, as the main idea is to find a smoothing function to reduce the sum of the squares of the residuals of the model related to the research topic. This type of method was used widely in nonparametric regression and gave good results in matching the curve of functions, but the expansion here was by using this method in functions that are nonparametric and non-linear in time series. The simulation method was used with three non-linear models of different formulas using four types of sample sizes for identifying the performance of the estimation method and presenting it through drawing to ensure the operation of the method. As for the real data, the monthly dollar sales data of the Central Bank of Iraq were modeled for years 2015-2017 then estimating the model curve in the proposed method, as the results indicated matching the curve of the Central Bank of Iraq's sales of dollars for the period studied. The results showed that the proposed method for matching the NAAR model curve was the best when using non-linear functions of various formulas. [ABSTRACT FROM AUTHOR]

In this paper, we consider composite quantile regression for partial functional linear regression model with polynomial spline approximation. Under some mild conditions, the convergence rates of the estimators and mean squared prediction error, and asymptotic normality of parameter vector are obtained. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least-squares based method when there are outliers in the dataset or the random error follows heavy-tailed distributions. Finally, we apply the proposed methodology to a spectroscopic data sets to illustrate its usefulness in practice. [ABSTRACT FROM AUTHOR]

As ultra high‐dimensional longitudinal data are becoming ever more apparent in fields such as public health and bioinformatics, developing flexible methods with a sparse model is of high interest. In this setting, the dimension of the covariates can potentially grow exponentially as exp(n1/2) with respect to the number of clusters n. We consider a flexible semiparametric approach, namely, partially linear single‐index models, for ultra high‐dimensional longitudinal data. Most importantly, we allow not only the partially linear covariates but also the single‐index covariates within the unknown flexible function estimated nonparametrically to be ultra high dimensional. Using penalized generalized estimating equations, this approach can capture correlation within subjects, can perform simultaneous variable selection and estimation with a smoothly clipped absolute deviation penalty, and can capture nonlinearity and potentially some interactions among predictors. We establish asymptotic theory for the estimators including the oracle property in ultra high dimension for both the partially linear and nonparametric components, and we present an efficient algorithm to handle the computational challenges. We show the effectiveness of our method and algorithm via a simulation study and a yeast cell cycle gene expression data. [ABSTRACT FROM AUTHOR]

Proportional rates models are frequently used for the analysis of recurrent event data with multiple event categories. When some of the event categories are missing, a conventional approach is to either exclude the missing data for a complete-case analysis or employ a parametric model for the missing event type. It is well known that the complete-case analysis is inconsistent when the missingness depends on covariates, and the parametric approach may incur bias when the model is misspecified. In this paper, we aim to provide a more robust approach using a rate proportion method for the imputation of missing event types. We show that the log-odds of the event type can be written as a semiparametric generalized linear model, facilitating a theoretically justified estimation framework. Comprehensive simulation studies were conducted demonstrating the improved performance of the semiparametric method over parametric procedures. Multiple types of Pseudomonas aeruginosa infections of young cystic fibrosis patients were analyzed to demonstrate the feasibility of our proposed approach. [ABSTRACT FROM AUTHOR]

Shape constrained additive models are useful in estimating production functions or analysing disease risk where the relationship between predictors and response is known to be monotone or/and concave. We here consider the estimation of shape constrained additive models when the response is missing at random given missing data are common occurrence and problem in many real-life situations. To the best of our knowledge, no research has focused on this problem. Our paper nicely fills this gap and contributes to the literature by proposing a weighted constrained polynomial spline estimation method in a one-step backfitting procedure. The proposed method is not only easy to implement but also gives smooth estimators that satisfy shape constraints and accommodate missing data problem simultaneously. In theory, we show that the proposed estimator enjoys the optimal rate of convergence asymptotically. Both simulation studies and the application of our method to Norwegian farm data illustrate that the proposed method has superior performance due to the incorporation of weights and shape constraints. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of polynomial spline curve that can naturally interpolate the given data points. The proposed curve is C3 and can be adjusted by two free parameters when the data points are kept unchanged. In addition, we give three selection schemes of the free parameters for constructing the curve with approximate shortest arc length, the curve with approximate smallest strain energy and the curve with approximate shortest arc length and smallest strain energy. [ABSTRACT FROM AUTHOR]

Production frontier is an important concept in modern economics and has been widely used to measure production efficiency. Existing nonparametric frontier models often only allow one or low-dimensional input variables due to 'curse-of-dimensionality'. In this paper we propose a flexible additive frontier model which quantifies the effects of multiple input variables on the maximum output. In addition, we consider the estimation of the nonparametric frontier functions with shape restrictions. Economic theory often imposes shape constraints on production frontier, such as, monotonicity and concavity. A two-step constrained polynomial spline method is proposed to give smooth estimates that automatically satisfy such shape constraints. The proposed method is not only easy to compute, but also more robust to outliers. In theory, we established uniform consistency of the proposed method. We illustrate the proposed method by both simulation studies and an application to the Norwegian farm data. The numerical studies suggest that the proposed method has superior performance by incorporating shape constraints. [ABSTRACT FROM AUTHOR]

We propose time-varying coefficient model selection and estimation based on the spline approach, which is capable of capturing time-dependent covariate effects. The new penalty function utilizes local-region information for varying-coefficient estimation, in contrast to the traditional model selection approach focusing on the entire region. The proposed method is extremely useful when the signals associated with relevant predictors are time-dependent, and detecting relevant covariate effects in the local region is more scientifically relevant than those of the entire region. Our simulation studies indicate that the proposed model selection incorporating local features outperforms the global feature model selection approaches. The proposed method is also illustrated through a longitudinal growth and health study from National Heart, Lung, and Blood Institute. [ABSTRACT FROM AUTHOR]

In survival analysis, we may encounter the following three problems: nonlinear covariate effect, variable selection and measurement error. Existing studies only address one or two of these problems. The goal of this study is to fill the knowledge gap and develop a novel approach to simultaneously address all three problems. Specifically, a partially time-varying coefficient proportional hazards model is proposed to more flexibly describe covariate effects. Corrected score and conditional score approaches are employed to accommodate potential measurement error. For the selection of relevant variables and regularised estimation, a penalisation approach is adopted. It is shown that the proposed approach has satisfactory asymptotic properties. It can be effectively realised using an iterative algorithm. The performance of the proposed approach is assessed via simulation studies and further illustrated by application to data from an AIDS clinical trial. [ABSTRACT FROM AUTHOR]

في هذا البحث, استخدمنا طريقة الثلمة التكعيبية غير متعددة الحدود لحل منظومة مكونة من معادلتين فولتيرا التكاملية غير الخطية من النوع الثاني وقد استخدمنا برنامج matlab14 لحل النظام. وأخيرا، قدمنا العديد من الأمثلة التوضيحية لإظهار فعالية ودقة هذه الطريقة.

The best L1 approximation of the Heaviside function and the best ℓ1 approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L1 spline fits which are the best L1 approximations in an appropriate spline space obtained by the union of L1 interpolation splines. We prove here the existence of L1 spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes. [ABSTRACT FROM AUTHOR]

This paper presents the theory of Interpolation and its applications in numerical analysis. It specially focuses on cubic spline interpolation. Data interpolation is useful for scientific visualization for data interpretation. One of the efficient methods for data interpolation is cubic spline function. Cubic splines are the most popular. They produce an interpolated function that is continuous through to the second derivative; Splines tend to be stabler than fitting a polynomial through the N +1 points, with less possibility of wild oscillations between the tabulated points. [ABSTRACT FROM AUTHOR]