Properties Of Least Squares Estimators Research Articles

Estimation of the Elementary Chirp Model Parameters

In this paper, we propose some estimation techniques to estimate the elementary chirp model parameters, which are encountered in sonar, radar, acoustics, and other areas. We derive asymptotic theoretical properties of least squares estimators and approximate least squares estimators for the one component elementary chirp model. It is proved that the proposed estimators are strongly consistent and follow the normal distribution asymptotically. We also suggest how to obtain proper initial values for these methods. The problem of finding initial values is a difficult problem when the number of components in the model is large, or when the signal-to-noise ratio is low, or when two frequency rates are close to each other. We propose sequential procedures to estimate the multiple component elementary chirp model parameters. We prove that the theoretical properties of sequential least squares estimators and sequential approximate least squares estimators coincide with those of least squares estimators and approximate least squares estimators, respectively. Further, the asymptotic variances of the proposed estimators attain the Cramér-Rao lower bounds asymptotically when errors are normal random variables and independently and identically distributed. To evaluate the performance of the proposed estimators, numerical experiments are performed. It is observed that the proposed sequential estimators perform well even in situations where least squares estimators do not perform well. We illustrate the performance of the proposed sequential algorithm on a bat data.

Correction to: Asymptotic Properties of Least Squares Estimators and Sequential Least Squares Estimators of a Chirp-like Signal Model Parameters

Correction to: Asymptotic Properties of Least Squares Estimators and Sequential Least Squares Estimators of a Chirp-like Signal Model Parameters

Statistical inference for time series with non-precise data

For linear time series models, the asymptotic properties of least squares estimation are known. An analogous result for the stationary time series with non-precise data is considered in this paper. The least squares estimation for the general autoregressive model with fuzzy data is investigated with a suitable fuzzy metric. The fuzzy-type least squares approach is defined, and analogues of the conventional normal equations and fuzzy least squares estimators (FLSEs) are also derived. A numerical example for computing FLSEs and forecasting the future values of fuzzy series is given with the financial data. Asymptotic normality and consistency are established for the FLSEs. A confidence region based on a class of FLSEs is constructed. The asymptotic relative efficiency of FLSEs with respect to the crisp least squares estimators is provided. Some simulation results are also presented to illustrate the small sample behavior of FLSEs.

Estimation of hydrodynamic parameters for underwater systems using a simple off-line regression method: a case study

Estimation of hydrodynamic model parameters such as added mass parameter and drag coefficients is very crucial while mathematically modelling any underwater system. Owing to the nature of the model being coupled and its high non-linearity, estimation of these parameters becomes complicated. Various methods of parameter estimation have been currently employed, involving experiments, computational fluid analysis, and simulations, each having its own advantages and disadvantages. Most of the existing methods use sophisticated external instrumentation and experimentally estimate the parameters using computationally expensive adaptive algorithms, which may not be required or cannot be generalised to all the systems. In this paper, a simple off-line estimation method for estimating crucial hydrodynamic parameters using onboard sensors is presented. The error between the data from numerical simulations (using a mathematical model) and experiments is iteratively minimised using a gradient descent-based optimisation algorithm, by having the unknown model parameters as design variables of the error minimisation process. The method combines the properties of least squares estimation and the free decay tests, where the system can be excited with any known external inputs. A mathematical model with the unknown parameters, fully defining the behaviour of the system, is required and open loop experimental data from onboard sensors for a known input is sufficient for the estimation process, thereby eliminating the requirement of additional instrumentation. Non-linear mathematical models can be directly used in estimation, unlike few other methods which require linearisation and approximation. This method can be generalised to any system, provided sufficient information on experimental input and output, and the equivalent mathematical model of the system are available. The proposed method has been successfully implemented to estimate the added mass and drag coefficients of a standalone, single degree of freedom variable buoyancy module, ‘vBuoy’. A mathematical model defining the dynamics of the heave motion of vBuoy has been derived and the parameters involved in the model have been estimated with the proposed method. The proposed method, as well as the results, are validated by comparing the experiments and simulations at different conditions. The results showed that the proposed method was well suited for the estimation of hydrodynamic parameters of underwater systems.

A note on the asymptotic properties of least squares estimation in high dimensional constrained factor models

Constrained factor models proposed by Tsai and Tsay (2010) have wide potential applications. The existing asymptotic theory of the least squares estimator, however, falls short of asymptotic representations and limiting distributions, which limits the applicabilities. This paper fills this gap by explicitly giving the asymptotic representations and associated limiting distributions. Theoretical analysis indicates that the least square estimates are asymptotically biased. Bias-corrected estimators are therefore proposed. Monte Carlo simulations confirm our theoretical results.

On approximate least squares estimators of parameters of one-dimensional chirp signal

ABSTRACTChirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.

Using information about technologies, markets and firm behaviour to decompose a proper productivity index

This paper uses distance functions to define new output and input quantity indexes that satisfy important axioms from index number theory (e.g., identity, transitivity, proportionality and time-space reversal). Dividing the output index by the input index yields a new productivity index that can be decomposed into a measure of technical change, a measure of environmental change, and several measures of efficiency change. A problem with this new index is that it cannot be computed without estimating the production frontier. The paper shows how assumptions concerning technologies, markets and firm behaviour can be used to inform the estimation process. The focus is on the asymptotic properties of least squares estimators when the explanatory variables in the production frontier model are endogenous. In this case, the ordinary least squares estimator is usually inconsistent. However, there is one situation where it is super-consistent. A fully-modified ordinary least squares estimator is also available in this case. To illustrate the main ideas, the paper uses US state-level farm data to estimate a stochastic production frontier. The parameter estimates are then used to obtain estimates of the economically-relevant components of productivity change.

Spurious Regressions in Time Series with Long Memory

This article studies the asymptotic properties of least squares estimators and related test statistics in some spurious regression models that are generated by stationary or nonstationary fractionally integrated processes. We show that even when the fractionally integrated processes are long-range dependent, the asymptotic distributions of the least squares statistics, after appropriately rescaling and normalizing, are functionals of standard Brownian motions rather than of fractional Brownian motions. Some Monte Carlo simulations are provided to indirectly illustrate the performance of the asymptotic results for finite samples.

Asymptotic properties of least squares estimation for a new fuzzy autoregressive model

In this paper, we extend the standard autoregressive model to the case where the explanatory and response variables are random fuzzy variables. The fuzzy least-squares estimators (FLSE) of the model parameters are derived and their asymptotic properties are established. A simulation is conducted to evaluate our method, and it is found that the proposed method provides a better performance. AMS Subject Classification:94D05, 62F12.

Asymptotic properties of least squares estimation with fuzzy observations

This paper deals with the asymptotic properties of the least squares estimators for fuzzy linear regression models with fuzzy triangular input–output and random error terms. The asymptotic normality and strong consistency of the fuzzy least squares estimator (FLSE) are investigated; a confidence region based on a class of FLSEs is proposed; the asymptotic relative efficiency of FLSEs with respect to the crisp least squares estimators is also provided and a numerical example is given. Some simulation results are also presented to illustrate the behavior of FLSEs.

Nonparametric estimate of almost periodic signals

The asymptotic properties of least-squares estimates of almost periodic signals under random noise are investigated. The consistency and strong consistency of the estimates are proved.

On Least Squares Fitting for Stationary Spatial Point Processes

SummaryThe K-function is a popular tool for fitting spatial point process models owing to its simplicity and wide applicability. In this work we study the properties of least squares estimators of model parameters and propose a new method of model fitting via the K-function by using subsampling. We demonstrate consistency and asymptotic normality of our estimators of model parameters and compare the efficiency of our procedure with existing procedures. This is done through asymptotic theory, simulation experiments and an application to a data set on long leaf pine-trees.

Asymptotic properties of least-squares estimates of Hammerstein-Wiener models

This paper investigates the asymptotic properties of least squares estimates of Hammerstein-Wiener model structures, and in doing so establishes consistency and asymptotic normality under fairly mild conditions on the additive noise process, the inputs and the static non-linearities. In relation to the asymptotic distributional results, a consistent procedure for the estimation of the asymptotic variance of the parameter estimates is provided. A key theme of this paper is to demonstrate how recent results from the econometrics literature may be employed in an engineering setting. In this respect the Hammerstein-Wiener model structure serves as a demonstration example. A simulation study complements the theoretical findings.

Asymptotic Properties of Least Squares Estimators for the Parameters in Undamped Exponential Signals

SYNOPTIC ABSTRACTThe superimposed exponential signals model plays an important role in Statistical Signal Processing. In this paper we study the properties of the least squares estimators and the estimators obtained by maximizing the periodogram. The asymptotic equivalence of these two estimators can be obtained. The consistency and the asymptotic distribution of the least squares estimators under the assumption of finite variance of the error random variables are established. Some simulation results are reported comparing the performances of the two different estimators.

Nonstationary regression models with a lagged dependent variable

This paper studies regression models with a lagged dependent variable when both the dependent and independent variables are nonstationary, and the regression model is misspecified in some dimension. In particular, we discuss the limiting properties of leastsquares estimates of the parameters in such regression models, and the limiting distributions of their test statistics. We show that the estimate of the lagged dependent variable tends to unity asymptotically independent of its true value, while the estimates of the independent variables tend to zero. The limiting distributions of their test statistics are shown to diverge with sample size.

Properties of least squares estimates in set membership identification

The properties of the least square algorithm in the identification of linear, time-invariant, discrete-time systems, in the presence of unknown but bounded errors are investigated. It is known that the least squares algorithm enjoys strong optimality properties in the case of power bounded (l2) errors, while it may be far from optimal in the case of pointwise (l∞) bounded errors. Exact expressions of the local and global worst case errors of the least squares estimates are derived. Using these results, general conditions assuring convergence to zero of the global worst-case error are given. It is also shown that the least squares algorithm may be optimal in l1 and l∞ identification of FIR systems with pointwise bounded errors.

Properties of Least Squares Estimates in Set Membership Identification

In this paper we investigate the properties of the least square algorithm in the identification of linear, time invariant, discrete time systems, in presence of unknown but bounded errors. It is known that the least squares algorithm enjoys strong optimality properties in the case of power bounded (l2) errors, while it may be far from optimality in the case of pointwise (l∞) bounded errors. In the paper we derive exact expressions of the local and global worst case errors of the least squares estimates and we show that the global error may diverge as the number of measurements grows. Then, we give general conditions assuring convergence of the global worst-case error.

On finite-sample properties of adaptive least squares regression estimates

This paper is mainly concerned with deriving finite-sample properties of least squares estimators for the regression function in a nonparametric regression situation under some simplifying assumptions such as normally distributed errors with a common known variance. The selection of basis functions to be used for the construction of an estimator may be regarded as a smoothing problem, and will usually be done in a data-dependent way, A straightforward application of a result by P. J. Kernpthorne yields that, under a squared error loss, all selection procedures are admissible. Furthermore, the minimax approach provides an interpolating estimator, which is often impractical, Therefore, within a certain class of selection procedures an optimal one is determined using the minimax regret principle. It can be seen to behave similarly to the procedure minimizing either an unbiased risk estimator or, equivalently, the Cp-criterion.

Note on the Convergence to Normality of Quadratic forms in Independent Variables

Previous article Next article Note on the Convergence to Normality of Quadratic forms in Independent VariablesJ. M. Ten VregelaarJ. M. Ten Vregelaarhttps://doi.org/10.1137/1135021PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernstein, Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantitiés dépendantes, Math. Ann., 97 (1926), 1–59 52.0517.03 CrossrefGoogle Scholar[2] Robert J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons Inc., New York, 1980xiv+371 82a:62003 0538.62002 CrossrefGoogle Scholar[3] J. M. ten Vregelaar, Parameter estimation from noisy observations of inputs and outputs, Proceedings of the Third International Workshop on Statistical Modeling, Vienna, 1988, 253–279 Google Scholar[4] P. Whittle, Bounds for the moments of linear and quadratic forms in independent variables, Theory Probab. Appl., 3 (1960), 331–335 Google Scholar[5] P. Whittle, On the convergence to normality of quadratic forms in independent variables, Theory Probab. Appl., 9 (1964), 113–118 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails ASYMPTOTICS OF SPECTRAL DENSITY ESTIMATESEconometric Theory, Vol. 26, No. 4 | 4 November 2009 Cross Ref A LIMIT THEOREM FOR QUADRATIC FORMS AND ITS APPLICATIONSEconometric Theory, Vol. 23, No. 05 | 14 May 2007 Cross Ref Volume 35, Issue 1| 1991Theory of Probability & Its Applications1-204 History Submitted:10 August 1987Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135021Article page range:pp. 177-179ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

An Alternative Procedure for Estimating Unit Learning Curves

Abstract : The unit learning curve plays a prominent role in DOD cost analysis. In those cases where the model accurately describes the real-life situation, i.e., when the model is properly applied to the data, it can be a powerful tool for predicting unit production costs. There are, however, some unique estimation problems inherent in the model. The usual method of generating predicted unit production costs attempts to extend properties of least squares estimators to non-linear functions of these estimators. The result is biased estimates of unit production costs. Another problem common to many learning curve applications is estimating lot midpoints and slope coefficients when both estimates depend on each other and both quantities are unknown. This paper addresses the two problems discussed above and presents an alternative procedure for estimating unit learning curves.