Approximation to the Distribution of the Least Squares Estimators in Two Dimensional Cosine Models by Randomly Weighted Bootstrap

Recently, Kundu and Gupta （Metrika, 48：83 C 97, 1998） established the asymptotic normality of the least squares estimators in the two dimensional cosine model. In this paper, we give the approximation to the general least squares estimators by using random weights which is called the Bayesian bootstrap or the random weighting method by Rubin （Annals of Statistics, 9：130 C 134, 1981） and Zheng （Acta Math. Appl. Sinica （in Chinese）, 10（2）： 247 C 253, 1987）. A simulation study shows that this approximation works very well.

LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION

In this article, we study a least squares estimator （LSE） of θ for the Ornstein- Uhlenbeck process X0=0,dXt=θXtdt＋dBt^ab, t ≥ 0 driven by weighted fractional Brownian motion B^a,b with parameters a, b. We obtain the consistency and the asymptotic distribution of the LSE based on the observation {Xs, s∈[0,t]} as t tends to infinity.

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes dY_(s)=(∑_(j=1)^(k)μ_(j)φ_(j)(s)-βY_(s))ds+dZ_(s)^(q,H),driven by the Hermite process Z_(s)^(q,H)with order q≥1 and a Hurst index H∈(1/2,1),where the periodic functionsφ_(j)(s),,j=1,...,κare bounded,and the real numbersμ_(j),,j=1,...,κtogether withβ>0 are unknown parameters.We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator.We also introduce alternative estimators,which can be looked upon as an application of the least squares estimator.In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean,our work can be regarded as its non-Gaussian extension.

ERRATUM TO： LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION （ACTA MATHEMATICA SCIENTIA 2016,36B （2） ：394-408）

We give a correction of Theorem 2.2 of Shen, Yin and Yan （2016）.

Asymptotic Properties of Estimators for Ornstein-Uhlenbeck Processes with Small Symmetricα-Stable Motions

The asymptotic behaviors for estimators of the drift parameters in the Ornstein-Uhlenbeck process driven by small symmetricα-stable motion are studied in this paper.Based on the discrete observations,the conditional least squares estimators(CLSEs)of all the parameters involved in the Ornstein–Uhlenbeck process are proposed.We establish the consistency and the asymptotic distributions of our estimators asεgoes to 0 and n goes to∞simultaneously.

Comparison of Two Time Series Decomposition Methods: Least Squares and Buys-Ballot Methods

This paper discusses comparison of two time series decomposition methods: The Least Squares Estimation (LSE) and Buys-Ballot Estimation (BBE) methods. As noted by Iwueze and Nwogu (2014), there exists a research gap for the choice of appropriate model for decomposition and detection of presence of seasonal effect in a series model. Estimates of trend parameters and seasonal indices are all that are needed to fill the research gap. However, these estimates are obtainable through the Least Squares Estimation (LSE) and Buys-Ballot Estimation (BBE) methods. Hence, there is need to compare estimates of the two methods and recommend. The comparison of the two methods is done using the Accuracy Measures (Mean Error (ME)), Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE). The results from simulated series show that for the additive model;the summary statistics (ME, MSE and MAE) for the two estimation methods and for all the selected trending curves are equal in all the simulations both in magnitude and direction. For the multiplicative model, results show that when a series is dominated by trend, the estimates of the parameters by both methods become less precise and differ more widely from each other. However, if conditions for successful transformation (using the logarithmic transform in linearizing the multiplicative model to additive model) are met, both of them give similar results.

Hepatic MR imaging using IDEAL-IQ sequence:Will Gd-EOB-DTPA interfere with reproductivity of fat fraction quantification?

BACKGROUND Iterative decomposition of water and fat with echo asymmetry and least squares estimation quantification sequence(IDEAL-IQ)is based on chemical shift-based water and fat separation technique to get proton density fat fraction.Multiple studies have shown that using IDEAL-IQ to test the stability and repeatability of liver fat is acceptable and has high accuracy.AIM To explore whether Gadoxetate Disodium(Gd-EOB-DTPA)interferes with the measurement of the hepatic fat content quantified with the IDEAL-IQ and to evaluate the robustness of this technique.METHODS IDEAL-IQ was used to quantify the liver fat content at 3.0T in 65 patients injected with Gd-EOB-DTPA contrast.After injection,IDEAL-IQ was estimated four times,and the fat fraction(FF)and R2* were measured at the following time points:Precontrast,between the portal phase(70 s)and the late phase(180 s),the delayed phase(5 min)and the hepatobiliary phase(20 min).One-way repeated-measures analysis was conducted to evaluate the difference in the FFs between the four time points.Bland-Altman plots were adopted to assess the FF changes before and after injection of the contrast agent.P<0.05 was considered statistically significant.RESULTS The assessment of the FF at the four time points in the liver,spleen and spine showed no significant differences,and the measurements of hepatic FF yielded good consistency between T1 and T2[95%confidence interval:-0.6768%,0.6658%],T1 and T3(-0.3900%,0.3178%),and T1 and T4(-0.3750%,0.2825%).R2* of the liver,spleen and spine increased significantly after injection(P<0.0001).CONCLUSION Using the IDEAL-IQ sequence to measure the FF,we can obtain results that will not be affected by Gd-EOB-DTPA.The high reproducibility of the IDEAL-IQ sequence makes it available in the scanning interval to save time during multiphase examinations.

The Consistency of LSE Estimators in Partial Linear Regression Models under Mixing Random Errors

In this paper,we consider the partial linear regression model y_(i)=x_(i)β^(*)+g(ti)+ε_(i),i=1,2,...,n,where(x_(i),ti)are known fixed design points,g(·)is an unknown function,andβ^(*)is an unknown parameter to be estimated,random errorsε_(i)are(α,β)-mix_(i)ng random variables.The p-th(p>1)mean consistency,strong consistency and complete consistency for least squares estimators ofβ^(*)and g(·)are investigated under some mild conditions.In addition,a numerical simulation is carried out to study the finite sample performance of the theoretical results.Finally,a real data analysis is provided to further verify the effect of the model.

Power Inverted Topp–Leone Distribution in Acceptance Sampling Plans

We introduce a new two-parameter model related to the inverted Topp–Leone distribution called the power inverted Topp–Leone(PITL)distribution.Major properties of the PITL distribution are stated;including;quantile measures,moments,moment generating function,probability weighted moments,Bonferroni and Lorenz curve,stochastic ordering,incomplete moments,residual life function,and entropy measure.Acceptance sampling plans are developed for the PITL distribution,when the life test is truncated at a pre-specified time.The truncation time is assumed to be the median lifetime of the PITL distribution with pre-specified factors.The minimum sample size necessary to ensure the specified life test is obtained under a given consumer’s risk.Numerical results for given consumer’s risk,parameters of the PITL distribution and the truncation time are obtained.The estimation of the model parameters is argued using maximum likelihood,least squares,weighted least squares,maximum product of spacing and Bayesian methods.A simulation study is confirmed to evaluate and compare the behavior of different estimates.Two real data applications are afforded in order to examine the flexibility of the proposed model compared with some others distributions.The results show that the power inverted Topp–Leone distribution is the best according to the model selection criteria than other competitive models.

Parameter identification of hysteretic model of rubber-bearing based on sequential nonlinear least-square estimation

In order to evaluate the nonlinear performance and the possible damage to rubber-bearings （RBs） during their normal operation or under strong earthquakes, a simplified Bouc-Wen model is used to describe the nonlinear hysteretic behavior of RBs in this paper, which has the advantages of being smooth-varying and physically motivated. Further, based on the results from experimental tests performed by using a particular type of RB （GZN 110） under different excitation scenarios, including white noise and several earthquakes, a new system identification method, referred to as the sequential nonlinear least- square estimation （SNLSE）, is introduced to identify the model parameters. It is shown that the proposed simplified Bouc- Wen model is capable of describing the nonlinear hysteretic behavior of RBs, and that the SNLSE approach is very effective in identifying the model parameters of RBs.

An Imperfect-debugging Fault-detection Dependent-parameter Software

Software reliability growth models （SRGMs） incorporating the imperfect debugging and learning phenomenon of developers have recently been developed by many researchers to estimate software reliability measures such as the number of remaining faults and software reliability. However, the model parameters of both the fault content rate function and fault detection rate function of the SRGMs are often considered to be independent from each other. In practice, this assumption may not be the case and it is worth to investigate what if it is not. In this paper, we aim for such study and propose a software reliability model connecting the imperfect debugging and learning phenomenon by a common parameter among the two functions, called the imperfect-debugging fault-detection dependent-parameter model. Software testing data collected from real applications are utilized to illustrate the proposed model for both the descriptive and predictive power by determining the non-zero initial debugging process.

Estimation for nearly unit root processes with GARCH errors

In this paper the limiting distribution of the least square estimate for the autoregressive coefficient of a nearly unit root model with GARCH errors is derived. Since the limiting distribution depends on the unknown variance of the errors, an empirical likelihood ratio statistic is proposed from which confidence intervals can be constructed for the nearly unit root model without knowing the variance. To gain an intuitive sense for the empirical likelihood ratio, a small simulation for the asymptotic distribution is given.

Asymptotic Behavior of the Drift Coefficient Estimator of Stochastic Differential Equations Driven by Small Noises

The parametric estimation problem for diffusion processes with small white noise based on continuous time observations is well developed. However,in parametric inference,it is more realistic and interesting to consider asymptotic estimation for diffusion processes based on discrete observations. The least squares method is used to obtain the estimator of the drift parameter for stochastic differential equations( SDEs) driven by general Lévy noises when the process is observed discretely. Its strong consistency and the rate of convergence of the squares estimator are studied under some regularity conditions.

Predictive Models for Cumulative Confirmed COVID-19 Cases by Day in Southeast Asia

Coronavirus disease 2019 outbreak has spread as a pandemic since the end of year 2019.This situation has been causing a lot of problems of human beings such as economic problems,health problems.The forecasting of the number of infectious people is required by the authorities of all countries including Southeast Asian countries to make a decision and control the outbreak.This research is to investigate the suitable forecasting model for the number of infectious people in Southeast Asian countries.A comparison of forecasting models between logistic growth curve which is symmetric and Gompertz growth curve which is asymmetric based on the maximumof Coefficient of Determination and theminimumof RootMean Squared Percentage Error is also proposed.The estimation of parameters of the forecasting models is evaluated by the least square method.In addition,spreading of the outbreak is estimated by the derivative of the number of cumulative cases.The findings show that Gompertz growth curve is a suitable forecasting model for Indonesia,Philippines,andMalaysia and logistic growth curve suits the other countries in South Asia.

Pattern synthesis optimization of 3-D ODAR based on improved GA using LSFE method

Pattern synthesis in 3-D opportunistic digital array radar(ODAR) becomes complex when a multitude of antennas are considered to be randomly distributed in a three dimensional space.In order to obtain an optimal pattern,several freedoms must be constrained.A new pattern synthesis approach based on the improved genetic algorithm(GA) using the least square fitness estimation(LSFE) method is proposed.Parameters optimized by this method include antenna locations,stimulus states and phase weights.The new algorithm demonstrates that the fitness variation tendency of GA can be effectively predicted after several "eras" by the LSFE method.It is shown that by comparing the variation of LSFE curve slope,the GA operator can be adaptively modified to avoid premature convergence of the algorithm.The validity of the algorithm is verified using computer implementation.

THE SUPERIORITY OF EMPIRICAL BAYES ESTIMATION OF PARAMETERS IN PARTITIONED NORMAL LINEAR MODEL

In this article,the empirical Bayes（EB）estimators are constructed for the estimable functions of the parameters in partitioned normal linear model.The superiorities of the EB estimators over ordinary least-squares（LS）estimator are investigated under mean square error matrix（MSEM）criterion.

Neuro-fuzzy system modeling based on automatic fuzzy clustering

A neuro-fuzzy system model based on automatic fuzzy dustering is proposed. A hybrid model identification algorithm is also developed to decide the model structure and model parameters. The algorithm mainly includes three parts：1） Automatic fuzzy C-means （AFCM）, which is applied to generate fuzzy rttles automatically, and then fix on the size of the neuro-fuzzy network, by which the complexity of system design is reducesd greatly at the price of the fitting capability; 2） R.ecursive least square estimation （RLSE）. It is used to update the parameters of Takagi-Sugeno model, which is employed to describe the behavior of the system;3） Gradient descent algorithm is also proposed for the fuzzy values according to the back propagation algorithm of neural network. Finally,modeling the dynamical equation of the two-link manipulator with the proposed approach is illustrated to validate the feasibility of the method.

Asymptotic inference for AR(1) panel data

A general asymptotic theory is given for the panel data AR(1) model with time series independent in different cross sections. The theory covers the cases of stationary process, local to unity process, unit root process, mildly integrated, mildly explosive and explosive processes. It is assumed that the cross-sectional dimension and time-series dimension are respectively N and T. The results in this paper illustrate that whichever the process is, with an appropriate regularization, the least squares estimator of the autoregressive coefficient converges in distribution to a normal distribution with rate at least O(N-1/3). Since the variance is the key to characterize the normal distribution, it is important to discuss the variance of the least squares estimator. We will show that when the autoregressive coefficient ρ satisfies |ρ| < 1, the variance declines at the rate O((NT)-1), while the rate changes to O(N^(-1) T^(-2)) when ρ = 1 and O(N^(-1)ρ^(-2 T+4)) when |ρ| > 1. ρ = 1 is the critical point where the convergence rate changes radically. The transition process is studied by assuming ρ depending on T and going to 1. An interesting phenomenon discovered in this paper is that, in the explosive case, the least squares estimator of the autoregressive coefficient has a standard normal limiting distribution in the panel data case while it may not has a limiting distribution in the univariate time series case.

Using self-location to calibrate the errors of observer positions for source localization

The uncertainty of observers＇ positions can lead to significantly degrading in source localization accuracy. This pa-per proposes a method of using self-location for calibrating the positions of observer stations in source localization to reduce the errors of the observer positions and improve the accuracy of the source localization. The relative distance measurements of the two coordinative observers are used for the linear minimum mean square error （LMMSE） estimator. The results of computer si-mulations prove the feasibility and effectiveness of the proposed method. With the general estimation errors of observers＇ positions, the MSE of the source localization with self-location calibration, which is significantly lower than that without self-location calibra-tion, is approximating to the Cramer-Rao lower bound （CRLB）.

A computational model of the human glucose-insulin regulatory system

Objective：A computational model of insulin secretion and glucose metabolism for assisting the diagnosis of diabetes mellitus in clinical research is introduced.The proposed method for the estimation of parameters for a system of ordinary differential equations（ODEs）that represent the time course of plasma glucose and insulin concentrations during glucose tolerance test（GTT）in physiological studies is presented.The aim of this study was to explore how to interpret those laboratory glucose and insulin data as well as enhance the Ackerman mathematical model.Methods：Parameters estimation for a system of ODEs was performed by minimizing the sum of squared residuals（SSR）function,which quantifies the difference between theoretical model predictions and GTT＇s experimental observations.Our proposed perturbation search and multiple-shooting methods were applied during the estimating process.Results：Based on the Ackerman＇s published data,we estimated the key parameters by applying R-based iterative computer programs.As a result,the theoretically simulated curves perfectly matched the experimental data points.Our model showed that the estimated parameters,computed frequency and period values,were proven a good indicator of diabetes.Conclusion：The present paper introduces a computational algorithm to biomedical problems,particularly to endocrinology and metabolism fields,which involves two coupled differential equations with four parameters describing the glucose-insulin regulatory system that Ackerman proposed earlier.The enhanced approach may provide clinicians in endocrinology and metabolism field insight into the transition nature of human metabolic mechanism from normal to impaired glucose tolerance.