Maximum Likelihood Type Estimators Research Articles

An boosting business intelligent to customer lifetime value with robust M-estimation

When a business concentrates too much on acquiring new clients rather than retaining old ones, mistakes are sometimes made. Each customer has a different value. Customer lifetime value (CLV) is a metric used to assess longterm customer value. Customer value is a key concern in any commercial endeavor. When there are variations in customer behavior, CLV forecasts the value of total customer income when the data distribution is not normal, and outliers are present. Robust M-estimation, a maximum likelihood type estimator, is used in this study to enhance CLV data. Through the minimization of the regression parameter from the residual value, robust Mestimation eliminates data outliers in customer metric data. With an accuracy of 94.15%, R-square is used to gauge model performance. This research shows that CLV optimization can be used as a marketing and sales strategy by companies.

Estimating positive powers of the scale parameters under order restriction for two normal populations with a common mean

ABSTRACT The problem of estimating positive powers of scale parameters has been considered under order restriction when samples are available from two normal populations with a common mean μ. Maximum-likelihood estimators and several plug-in type estimators using some popular estimators of the common mean have been proposed. Sufficient conditions for improving equivariant estimators for the positive powers of the scale parameters under order restriction have been derived. Consequently, several improved estimators have been proposed. A numerical comparison among all the proposed estimators in the special cases (c = 1 and ) has been made in terms of risk values using the quadratic loss function, and recommendations are given for the use of the estimators. Finally, a real-life example has been considered to show the potential application of the model problem.

Benefits of noise in M-estimators: Optimal noise level and probability density

For the robust estimation of a location parameter, we consider a parallel array of maximum likelihood type estimators (M-estimators). We investigate the possibility of added noise as a design variable of the M-estimators, and characterize a nonzero optimal amount of added noise maximizing the efficiency for estimation. The added noise shows its benefits to the asymptotic efficiency of the M-estimator when the noise level and the noise probability density are optimally tuned. The optimal noise level can be theoretically derived by maximizing the asymptotic efficiency as the probability density of added noise is given. Based on the Parzen-window density estimation technique, we approximate the infinite-dimensional non-convex optimization of the optimal probability density of added noise as a simpler optimization problem with respect to a finite-dimensional vector under certain constraints. This approximate solution for the optimal probability density of added noise shows its feasibility for various M-estimators with an arbitrary array size, which is also validated by simulation results.

Heading Estimation for Pedestrian Dead Reckoning Based on Robust Adaptive Kalman Filtering.

Pedestrian dead reckoning (PDR) using smart phone-embedded micro-electro-mechanical system (MEMS) sensors plays a key role in ubiquitous localization indoors and outdoors. However, as a relative localization method, it suffers from the problem of error accumulation which prevents it from long term independent running. Heading estimation error is one of the main location error sources, and therefore, in order to improve the location tracking performance of the PDR method in complex environments, an approach based on robust adaptive Kalman filtering (RAKF) for estimating accurate headings is proposed. In our approach, outputs from gyroscope, accelerometer, and magnetometer sensors are fused using the solution of Kalman filtering (KF) that the heading measurements derived from accelerations and magnetic field data are used to correct the states integrated from angular rates. In order to identify and control measurement outliers, a maximum likelihood-type estimator (M-estimator)-based model is used. Moreover, an adaptive factor is applied to resist the negative effects of state model disturbances. Extensive experiments under static and dynamic conditions were conducted in indoor environments. The experimental results demonstrate the proposed approach provides more accurate heading estimates and supports more robust and dynamic adaptive location tracking, compared with methods based on conventional KF.

Hybrid estimators for small diffusion processes based on reduced data

We deal with the Bayes type estimators and the maximum likelihood type estimators of both drift and volatility parameters for small diffusion processes defined by stochastic differential equations with small perturbations from high frequency data. From the viewpoint of numerical analysis, initial Bayes type estimators for both drift and volatility parameters based on reduced data are required, and adaptive maximum likelihood type estimators with the initial Bayes type estimators, which are called hybrid estimators, are proposed. The asymptotic properties of the initial Bayes type estimators based on reduced data are derived and it is shown that the hybrid estimators have asymptotic normality and convergence of moments. Furthermore, a concrete example and simulation results are given.

Noise Enhancement in Robust Estimation of Location

In this paper, we investigate the noise benefits to maximum likelihood type estimators (M-estimator) for the robust estimation of a location parameter. Two distinct noise benefits are shown to be accessible under these conditions. With symmetric heavy-tailed noise distributions, the asymptotic efficiency of the estimation can be enhanced by injecting extra noise into the M-estimators. With an asymmetric contaminated noise model having a convex cumulative distribution function, we demonstrate that addition of noise can reduce the maximum bias of the median estimator. These findings extend the analysis of stochastic resonance effects for noise-enhanced signal and information processing.

Noise benefits to robust M-estimation of location in dependent observations

In the presence of weakly dependent noise, we analyze the asymptotic efficiency of robust maximum likelihood type estimators (M-estimators) for location estimation. We observe the noise-enhanced asymptotic efficiency effect by tuning the background noise level for a single M-estimator. We also theoretically derive the asymptotic efficiency of a parallel array of M-estimators, and illustrate the possibility of the noise enhancement of the asymptotic efficiency by injecting extra additive noise components. These obtained results enrich the research of noise benefits for the robust estimation of location from the dependent observations.

Variable bandwidth local maximum likelihood type estimation for diffusion processes

The method of robust approach is applied to estimate drift function and diffusion function of diffusion processes with discrete-time observations. The proposed method combines the ideas of local linear regression technique and maximum likelihood type estimation technique, so the advantages of local linear estimators persist and overcome the disadvantages of least-squares estimator. Moreover, a variable bandwidth instead of a constant bandwidth is considered in the local maximum likelihood type estimators. The consistency and asymptotic normality of the local maximum likelihood type estimators for drift and diffusion functions are developed under some given conditions. We perform a simulation study to evaluate the robust performances of the proposed estimators.

Robust cardiac motion estimation using ultrafast ultrasound data: a low-rank topology-preserving approach

Cardiac motion estimation is an important diagnostic tool for detecting heart diseases and it has been explored with modalities such as MRI and conventional ultrasound (US) sequences. US cardiac motion estimation still presents challenges because of complex motion patterns and the presence of noise. In this work, we propose a novel approach to estimate cardiac motion using ultrafast ultrasound data. Our solution is based on a variational formulation characterized by the L2-regularized class. Displacement is represented by a lattice of b-splines and we ensure robustness, in the sense of eliminating outliers, by applying a maximum likelihood type estimator. While this is an important part of our solution, the main object of this work is to combine low-rank data representation with topology preservation. Low-rank data representation (achieved by finding the k-dominant singular values of a Casorati matrix arranged from the data sequence) speeds up the global solution and achieves noise reduction. On the other hand, topology preservation (achieved by monitoring the Jacobian determinant) allows one to radically rule out distortions while carefully controlling the size of allowed expansions and contractions. Our variational approach is carried out on a realistic dataset as well as on a simulated one. We demonstrate how our proposed variational solution deals with complex deformations through careful numerical experiments. The low-rank constraint speeds up the convergence of the optimization problem while topology preservation ensures a more accurate displacement. Beyond cardiac motion estimation, our approach is promising for the analysis of other organs that exhibit motion.

Estimation for stochastic differential equations with mixed effects

We consider the long-term behaviour of a one-dimensional mixed effects diffusion process with a multivariate random effect φ in the drift coefficient. We first study the estimation of the random variable φ based on the observation of one sample path on the time interval as T tends to infinity. The process is not Markov and we characterize its invariant distributions. We build moments and maximum likelihood-type estimators of the random variable φ which are consistent and asymptotically mixed normal with rate . Moreover, we obtain non-asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from N i.i.d. processes is investigated. Estimators are built and studied as both N and tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is whereas for deterministic components, the rate is . Illustrative examples are given.

A statistically robust EEG re-referencing procedure to mitigate reference effect.

The electroencephalogram (EEG) remains the primary tool for diagnosis of abnormal brain activity in clinical neurology and for in vivo recordings of human neurophysiology in neuroscience research. In EEG data acquisition, voltage is measured at positions on the scalp with respect to a reference electrode. When this reference electrode responds to electrical activity or artifact all electrodes are affected. Successful analysis of EEG data often involves re-referencing procedures that modify the recorded traces and seek to minimize the impact of reference electrode activity upon functions of the original EEG recordings. We provide a novel, statistically robust procedure that adapts a robust maximum-likelihood type estimator to the problem of reference estimation, reduces the influence of neural activity from the re-referencing operation, and maintains good performance in a wide variety of empirical scenarios. The performance of the proposed and existing re-referencing procedures are validated in simulation and with examples of EEG recordings. To facilitate this comparison, channel-to-channel correlations are investigated theoretically and in simulation. The proposed procedure avoids using data contaminated by neural signal and remains unbiased in recording scenarios where physical references, the common average reference (CAR) and the reference estimation standardization technique (REST) are not optimal. The proposed procedure is simple, fast, and avoids the potential for substantial bias when analyzing low-density EEG data.

On inference for fractional differential equations

Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter $$H>1/2$$ . Rates of convergence for the approximation task are provided, and numerical experiments show that our procedure leads to good results in terms of estimation.

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.

Limit of detection determination for censored samples

The problem of setting the limit of detection is considered for censored samples and heterogeneous errors. After formal definitions of the critical level and of the method detection limit, we obtain simplified maximum likelihood-type estimators. The estimation problem of the truncation parameter and its uncertainty are reviewed. Upper confidence bounds for the limit of detection are derived and some simulation results are given.

Estimation for misspecified ergodic diffusion processes from discrete observations

The joint estimation of both drift and diffusion coefficient parameters is treated under the situation where the data are discretely observed from an ergodic diffusion process and where the statistical model may or may not include the true diffusion process. We consider the minimum contrast estimator, which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density. The asymptotic normality of the minimum contrast estimator is proved. In particular, the rate of convergence for the minimum contrast estimator of diffusion coefficient parameter in a misspecified model is different from the one in the correctly specified parametric model.

PARAMETER ESTIMATION FOR SPDEs WITH MULTIPLICATIVE FRACTIONAL NOISE

We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter H ∈ (0, 1). Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general results are applied to stochastic heat equation driven by a fractional Brownian motion.

Adaptive collocation with application in height system transformation

In collocation applications, the prior covariance matrices or weight matrices between the signals and the observations should be consistent to their uncertainties; otherwise, the solution of collocation will be distorted. To balance the covariance matrices of the signals and the observations, a new adaptive collocation estimator is thus derived in which the corresponding adaptive factor is constructed by the ratio of the variance components of the signals and the observations. A maximum likelihood estimator of the variance components is thus derived based on the collocation functional model and stochastic model. A simplified Helmert type estimator of the variance components for the collocation is also introduced and compared to the derived maximum likelihood type estimator. Reasonable and consistent covariance matrices of the signals and the observations are arrived through the adjustment of the adaptive factor. The new adaptive collocation with related adaptive factor constructed by the derived variance components is applied in a transformation between the geodetic height derived by GPS and orthometric height. It is shown that the adaptive collocation is not only simple in calculation but also effective in balancing the contribution of observations and the signals in the collocation model.

Statistical estimation of optimal portfolios for non‐Gaussian dependent returns of assets

AbstractThis paper discusses the asymptotic efficiency of estimators for optimal portfolios when returns are vector‐valued non‐Gaussian stationary processes. We give the asymptotic distribution of portfolio estimators ĝ for non‐Gaussian dependent return processes. Next we address the problem of asymptotic efficiency for the class of estimators ĝ. First, it is shown that there are some cases when the asymptotic variance of ĝ under non‐Gaussianity can be smaller than that under Gaussianity. The result shows that non‐Gaussianity of the returns does not always affect the efficiency badly. Second, we give a necessary and sufficient condition for ĝ to be asymptotically efficient when the return process is Gaussian, which shows that ĝ is not asymptotically efficient generally. From this point of view we propose to use maximum likelihood type estimators for g, which are asymptotically efficient. Furthermore, we investigate the problem of predicting the one‐step‐ahead optimal portfolio return by the estimated portfolio based on ĝ and examine the mean squares prediction error. Copyright © 2008 John Wiley & Sons, Ltd.

Robust Parameter Estimation for Uncertain Gross-Error Models

Many assumptions commonly made in science and engineering problems are at most approximations to reality and they do not always hold unfortunately. Recognizing this fact, the concept of robustness has been attracted and robust procedures have been developed to cover this fact. One of the most important ideas on this direction is Huber's M-estimator (maximum likelihood type estimator). Identifying the neighbourhoods of stochastic models in terms of the class of ε-contaminated probability distribution, he derived an estimator that minimizes the maximum degradation of performance possible for an ॉ-deviation from the assumption. This idea, however, is not applicable in practice since the exact value of the gross error ε is not known. In this paper, an M-estimator applicable in such situations is derived. A numerical example is presented for illustrating the proposed idea.

Outlier Detection in Regression Models with ARIMA Errors using Robust Estimates

AbstractA diagnostic procedure for detecting additive and innovation outliers as well as level shifts in a regression model with ARIMA errors is introduced. The procedure is based on a robust estimate of the model parameters and on innovation residuals computed by means of robust filtering. A Monte Carlo study shows that, when there is a large proportion of outliers, this procedure is more powerful than the classical methods based on maximum likelihood type estimates and Kalman filtering. Copyright © 2001 John Wiley & Sons, Ltd.