LEAST SQUARES ESTIMATION FOR NONLINEAR REGRESSION MODELS WITH HETEROSCEDASTICITY

This paper develops an asymptotic theory of nonlinear least squares estimation by establishing a new framework that can be easily applied to various nonlinear regression models with heteroscedasticity. As an illustration, we explore an application of the framework to nonlinear regression models with nonstationarity and heteroscedasticity. In addition to these main results, this paper provides a maximum inequality for a class of martingales, which is of interest in its own right.

Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders

Fractal-fractional (FF) differential and integral operators having the capability to subsume features of retaining memory and self-similarities are used in the present research analysis to design a mathematical model for the rubella epidemic while taking care of dimensional consistency among the model equations. Infectious diseases have history in their transmission dynamics and thus non-local operators such as FF play a vital role in modeling dynamics of such epidemics. Monthly actual rubella incidence cases in Pakistan for the years 2017 and 2018 have been used to validate the FF rubella model and such a data set also helps for parameter estimation. Using nonlinear least-squares estimation with MATLAB function lsqcurvefit, some parameters for the classical and the FF model are obtained. Upon comparison of error norms for both models (classical and FF), it is found that the FF produces the smaller error. Locally asymptotically stable points (rubella-free and rubella-present) of the model are computed when the basic reproduction number { {\mathcal R} }_{0} is less and greater than unity and the sensitivity is investigated. Moreover, solution of the FF rubella system is shown to exist. A new iterative method is proposed to carry out numerical simulations which resulted in getting insights for the transmission dynamics of the rubella epidemic.

A New Reliability Model Based on Lindley Distribution with Application to Failure Data

Software reliability is an important feature that influences systems’ reliability. Software reliability models are a common tool to evaluate software reliability quantitatively. Various reliability models have been suggested based on the NHPP (nonhomogeneous Poisson process). In this article, a new NHPP model based on the Lindley distribution is proposed. The mathematical formulas for its measures of reliability are obtained and graphically illustrated. The proposed model’s parameters are estimated using both the NLSE (nonlinear least squares estimation) and the WNLSE (weighted nonlinear least squares estimation) methods. The model is then validated based on several different reliability datasets. The methods of estimation are evaluated and compared using three different criteria. The performance of the new model is also evaluated and compared, both objectively and subjectively, with three previously suggested models. The application results show that our new model demonstrates good performance in our selected failure data.

Optimal Design of Experiments for Hybrid Nonlinear Models, with Applications to Extended Michaelis–Menten Kinetics

Biochemical mechanism studies often assume statistical models derived from Michaelis–Menten kinetics, which are used to approximate initial reaction rate data given the concentration level of a single substrate. In experiments dealing with industrial applications, however, there are typically a wide range of kinetic profiles where more than one factor is controlled. We focus on optimal design of such experiments requiring the use of multifactor hybrid nonlinear models, which presents a considerable computational challenge. We examine three different candidate models and search for tailor-made D- or weighted-A-optimal designs that can ensure the efficiency of nonlinear least squares estimation. We also study a compound design criterion for discriminating between two candidate models, which we recommend for design of advanced kinetic studies. Supplementary materials accompanying this paper appear on-line

The application of nonlinear least-squares estimation algorithms in atmospheric density model calibration

Purpose The purpose of this paper is to focus on the performance of three typical nonlinear least-squares estimation algorithms in atmospheric density model calibration. Design/methodology/approach The error of Jacchia-Roberts atmospheric density model is expressed as an objective function about temperature parameters. The estimation of parameter corrections is a typical nonlinear least-squares problem. Three algorithms for nonlinear least-squares problems, Gauss–Newton (G-N), damped Gauss–Newton (damped G-N) and Levenberg–Marquardt (L-M) algorithms, are adopted to estimate temperature parameter corrections of Jacchia-Roberts for model calibration. Findings The results show that G-N algorithm is not convergent at some sampling points. The main reason is the nonlinear relationship between Jacchia-Roberts and its temperature parameters. Damped G-N and L-M algorithms are both convergent at all sampling points. G-N, damped G-N and L-M algorithms reduce the root mean square error of Jacchia-Roberts from 20.4% to 9.3%, 9.4% and 9.4%, respectively. The average iterations of G-N, damped G-N and L-M algorithms are 3.0, 2.8 and 2.9, respectively. Practical implications This study is expected to provide a guidance for the selection of nonlinear least-squares estimation methods in atmospheric density model calibration. Originality/value The study analyses the performance of three typical nonlinear least-squares estimation methods in the calibration of atmospheric density model. The non-convergent phenomenon of G-N algorithm is discovered and explained. Damped G-N and L-M algorithms are more suitable for the nonlinear least-squares problems in model calibration than G-N algorithm and the first two algorithms have slightly fewer iterations.

Robust estimation and inference for general varying coefficient models with missing observations

This paper considers estimation and inference for a class of varying coefficient models in which some of the responses and some of the covariates are missing at random and outliers are present. The paper proposes two general estimators—and a computationally attractive and asymptotically equivalent one-step version of them—that combine inverse probability weighting and robust local linear estimation. The paper also considers inference for the unknown infinite-dimensional parameter and proposes two Wald statistics that are shown to have power under a sequence of local Pitman drifts and are consistent as the drifts diverge. The results of the paper are illustrated with three examples: robust local generalized estimating equations, robust local quasi-likelihood and robust local nonlinear least squares estimation. A simulation study shows that the proposed estimators and test statistics have competitive finite sample properties, whereas two empirical examples illustrate the applicability of the proposed estimation and testing methods.