A note on confidence intervals with extended least squares parameter estimates

1987 ◽  
Vol 15 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Lewis B. Sheiner ◽  
Stuart L. Beal
Keyword(s):  
Least Squares ◽  
Confidence Intervals ◽  
Parameter Estimates

scholarly journals A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model

2019 ◽  
Vol 9 (15) ◽  
pp. 3120
Author(s):  
Sandro Amador ◽  
Mahmoud El-Kafafy ◽  
Álvaro Cunha ◽  
Rune Brincker
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Confidence Intervals ◽  
Modal Parameters ◽  
Parameter Estimates ◽  
Modal Parameter ◽  
Likelihood Estimator ◽  
Pole Residue ◽  
Modal Model

Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.

Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

2011 ◽  
Vol 64 (S1) ◽  
pp. S3-S18 ◽  
Author(s):  
Yuanxi Yang ◽  
Jinlong Li ◽  
Junyi Xu ◽  
Jing Tang
Keyword(s):  
Least Squares ◽  
Stochastic Models ◽  
Influence Function ◽  
A Priori ◽  
Functional Model ◽  
Global Navigation Satellite Systems ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Integrated Navigation ◽  
Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

INFLUENCE OF MATRIX ERRORS ON PARAMETER ESTIMATES BY THE LEAST SQUARES METHOD

2021 ◽  
Author(s):  
V. A. Galanina ◽  
◽  
L. A. Reshetov ◽  
M. V. Sokolovskay ◽  
A. E. Farafonova ◽  
...  
Keyword(s):  
Least Squares ◽  
Linear Model ◽  
Least Squares Method ◽  
Statistical Characteristics ◽  
Parameter Estimates ◽  
Model Matrix ◽  
Least Squares Estimates

The paper investigates the effect of distorsions of the linear model matrix on the statistical characteristics of the least squares estimates.

Outlier Detection in Linear Regression

2011 ◽  
pp. 510-550 ◽  
Author(s):  
A. A. M. Nurunnabi ◽  
A. H. M. Rahmatullah Imon ◽  
A. B. M. Shawkat Ali ◽  
Mohammed Nasser
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Outlier Detection ◽  
Estimation Procedure ◽  
Support Vector ◽  
Parameter Estimates ◽  
Multivariate Statistical ◽  
Data Contamination ◽  
The People ◽  
Computational Simplicity

Regression analysis is one of the most important branches of multivariate statistical techniques. It is widely used in almost every field of research and application in multifactor data, which helps to investigate and to fit an unknown model for quantifying relations among observed variables. Nowadays, it has drawn a large attention to perform the tasks with neural networks, support vector machines, evolutionary algorithms, et cetera. Till today, least squares (LS) is the most popular parameter estimation technique to the practitioners, mainly because of its computational simplicity and underlying optimal properties. It is well-known by now that the method of least squares is a non-resistant fitting process; even a single outlier can spoil the whole estimation procedure. Data contamination by outlier is a practical problem which certainly cannot be avoided. It is very important to be able to detect these outliers. The authors are concerned about the effect outliers have on parameter estimates and on inferences about models and their suitability. In this chapter the authors have made a short discussion of the most well known and efficient outlier detection techniques with numerical demonstrations in linear regression. The chapter will help the people who are interested in exploring and investigating an effective mathematical model. The goal is to make the monograph self-contained maintaining its general accessibility.

Estimating an Autoregressive Current Effects Model of Sales Response when Observations are Aggregated over Time: Least Squares versus Maximum Likelihood

1988 ◽  
Vol 25 (3) ◽  
pp. 301-307
Author(s):  
Wilfried R. Vanhonacker
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Serial Correlation ◽  
Mean Squared Error ◽  
Generalized Least Squares ◽  
Parameter Estimates ◽  
Squared Error ◽  
Positive Serial Correlation ◽  
Serial Correlation Coefficient ◽  
Over Time

Estimating autoregressive current effects models is not straightforward when observations are aggregated over time. The author evaluates a familiar iterative generalized least squares (IGLS) approach and contrasts it to a maximum likelihood (ML) approach. Analytic and numerical results suggest that (1) IGLS and ML provide good estimates for the response parameters in instances of positive serial correlation, (2) ML provides superior (in mean squared error) estimates for the serial correlation coefficient, and (3) IGLS might have difficulty in deriving parameter estimates in instances of negative serial correlation.

scholarly journals A Monograph on Nonlinear Regression Models

2018 ◽  
Vol 7 (4.10) ◽  
pp. 543
Author(s):  
B. Mahaboob ◽  
B. Venkateswarlu ◽  
C. Narayana ◽  
J. Ravi sankar ◽  
P. Balasiddamuni
Keyword(s):  
Regression Model ◽  
Least Squares ◽  
Nonlinear Regression ◽  
Nonlinear Models ◽  
Nonlinear Least Squares ◽  
Error Variance ◽  
Ordinary Least Squares ◽  
Parameter Estimates ◽  
Nonlinear Regression Model ◽  
Nonlinear Regression Models

This research article uses Matrix Calculus techniques to study least squares application of nonlinear regression model, sampling distributions of nonlinear least squares estimators of regression parametric vector and error variance and testing of general nonlinear hypothesis on parameters of nonlinear regression model. Arthipova Irina et.al [1], in this paper, discussed some examples of different nonlinear models and the application of OLS (Ordinary Least Squares). MA Tabati et.al (2), proposed a robust alternative technique to OLS nonlinear regression method which provide accurate parameter estimates when outliers and/or influential observations are present. Xu Zheng et.al [3] presented new parametric tests for heteroscedasticity in nonlinear and nonparametric models.  

Parameter Estimation in Marketing Models in the Presence of Influential Response Data: Robust Regression and Applications

1984 ◽  
Vol 21 (3) ◽  
pp. 268-277 ◽  
Author(s):  
Vijay Mahajan ◽  
Subhash Sharma ◽  
Yoram Wind
Keyword(s):  
Parameter Estimation ◽  
Regression Analysis ◽  
Least Squares ◽  
Robust Regression ◽  
Ordinary Least Squares ◽  
Regression Coefficients ◽  
Parameter Estimates ◽  
Response Data ◽  
Marketing Models ◽  
Marketing Data

In marketing models, the presence of aberrant response values or outliers in data can distort the parameter estimates or regression coefficients obtained by means of ordinary least squares. The authors demonstrate the potential usefulness of the robust regression analysis in treating influential response values in marketing data.

scholarly journals Analysis of algebraic weighted least-squares estimators for enzyme parameters

1992 ◽  
Vol 288 (2) ◽  
pp. 533-538 ◽  
Author(s):  
M E Jones
Keyword(s):  
Least Squares ◽  
Coefficient Of Variation ◽  
Constant Coefficient ◽  
Weighted Least Squares ◽  
Simulated Data ◽  
Standard Errors ◽  
Parameter Estimates ◽  
Spreadsheet Program ◽  
Least Squares Estimators ◽  
Constant Coefficient Of Variation

An algorithm for the least-squares estimation of enzyme parameters Km and Vmax. is proposed and its performance analysed. The problem is non-linear, but the algorithm is algebraic and does not require initial parameter estimates. On a spreadsheet program such as MINITAB, it may be coded in as few as ten instructions. The algorithm derives an intermediate estimate of Km and Vmax. appropriate to data with a constant coefficient of variation and then applies a single reweighting. Its performance using simulated data with a variety of error structures is compared with that of the classical reciprocal transforms and to both appropriately and inappropriately weighted direct least-squares estimators. Three approaches to estimating the standard errors of the parameter estimates are discussed, and one suitable for spreadsheet implementation is illustrated.

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