Maximum Likelihood Estimate

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Ying Zhang - One of the best experts on this subject based on the ideXlab platform.

  • weighted nonparametric Maximum Likelihood Estimate of a mixing distribution in nonrandomized clinical trials
    Statistics in Medicine, 2007
    Co-Authors: Ying Zhang
    Abstract:

    Hierarchical models have a variety of applications, including multi-center clinical trials, local estimation of disease rates, longitudinal studies, risk assessment, and meta-analysis. In a hierarchical model, observations are sampled conditional on individual unit-specific parameters and these parameters are sampled from a mixing distribution. In observational studies or nonrandomized clinical trails, observations may be biased samples from a population and heterogeneous with respect to some confounding factors. Without controlling the heterogeneity in the sample, the standard estimation of the mixing distribution may lead to inaccurate statistical inferences. In this article, we propose a weighted nonparametric Maximum Likelihood Estimate (NPMLE) of the mixing distribution and its smoothed version via weighted smoothing by roughening. The proposed estimator reduces bias by assigning a weight to each subject in the sample. The weighted NPMLE is shown to be weighted self-consistent and therefore can be easily calculated through a recursive approach. Simulation studies were conducted to evaluate the performance of the proposed estimator. We applied this method to clinical trial data evaluating a new treatment for stress urinary incontinence. Copyright © 2007 John Wiley & Sons, Ltd.

  • weighted nonparametric Maximum Likelihood Estimate of a mixing distribution in nonrandomized clinical trials
    Statistics in Medicine, 2007
    Co-Authors: Chaofeng Liu, Jun Xie, Ying Zhang
    Abstract:

    Hierarchical models have a variety of applications, including multi-center clinical trials, local estimation of disease rates, longitudinal studies, risk assessment, and meta-analysis. In a hierarchical model, observations are sampled conditional on individual unit-specific parameters and these parameters are sampled from a mixing distribution. In observational studies or nonrandomized clinical trails, observations may be biased samples from a population and heterogeneous with respect to some confounding factors. Without controlling the heterogeneity in the sample, the standard estimation of the mixing distribution may lead to inaccurate statistical inferences. In this article, we propose a weighted nonparametric Maximum Likelihood Estimate (NPMLE) of the mixing distribution and its smoothed version via weighted smoothing by roughening. The proposed estimator reduces bias by assigning a weight to each subject in the sample. The weighted NPMLE is shown to be weighted self-consistent and therefore can be easily calculated through a recursive approach. Simulation studies were conducted to evaluate the performance of the proposed estimator. We applied this method to clinical trial data evaluating a new treatment for stress urinary incontinence.

Snezhana I Abarzhi - One of the best experts on this subject based on the ideXlab platform.

  • whittle Maximum Likelihood Estimate of spectral properties of rayleigh taylor interfacial mixing using hot wire anemometry experimental data
    Physical Review E, 2020
    Co-Authors: D Pfefferle, Snezhana I Abarzhi
    Abstract:

    Investigating the power density spectrum of fluctuations in Rayleigh-Taylor (RT) interfacial mixing is a means of studying characteristic length, timescales, anisotropies, and anomalous processes. Guided by group theory, analyzing the invariance-based properties of the fluctuations, our paper examines raw time series from hot-wire anemometry measurements in the experiment by Akula et al. [J. Fluid Mech. 816, 619 (2017)JFLSA70022-112010.1017/jfm.2017.95]. The results suggest that the power density spectrum can be modeled as a compound function presented as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a Maximum Likelihood estimator of the parameters. Those that maximize the log-Likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-Likelihood is used to evaluate the Fisher information matrix and provide an Estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is applied to analyze the goodness of fit, by verifying the hypothesis that the ratio between the observed periodogram and the Estimated power density spectrum follows a χ^{2} probability distribution. The dependence of the parameters of the compound function is investigated on the range of mode numbers over which the fit is performed. In the domain where the relative errors of the power-law exponent and the exponential decay rate are small and the goodness of fit is excellent, the parameters of the compound function are clearly defined, in agreement with the theory developed in the paper. The study of the power-law spectra in RT mixing data suggests that rigorous physics-based statistical methods can help researchers to see beyond visual inspection.

  • whittle Maximum Likelihood Estimate of spectral properties of rayleigh taylor interfacial mixing using hot wire anemometry experimental data
    arXiv: Fluid Dynamics, 2019
    Co-Authors: D Pfefferle, Devesh Ranjan, Snezhana I Abarzhi
    Abstract:

    The Rayleigh-Taylor instability (RTI) occurs in a broad range of processes in nature and technology. Analysing the power density spectrum of fluctuations in Rayleigh-Taylor (RT) flow is a means of highlighting characteristic length- and time-scales, anisotropies and anomalous processes. Raw time series from hot-wire anemometry measurements of Rayleigh-Taylor interfacial mixing experiment by Akula et al., JFM 816, 619-660 (2017) are considered as a sample case to adjust the parameters of a model power density spectrum. The results suggest that the power density spectrum of one of the flow components can be confidently modelled as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a Maximum Likelihood Estimator (MLE) of the parameters. Those that maximise the log-Likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-Likelihood is used to evaluate the Fisher information matrix and provide an Estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is used to verify the hypothesis that the ratio between the observed periodogram and the Estimated power density spectrum follows a chi-squared probability distribution. This step is performed to show goodness-of-fit. We also study the dependence of the model parameters on the range of mode numbers over which the fit is performed.

Raghuraman Mudumbai - One of the best experts on this subject based on the ideXlab platform.

  • CDC - The Maximum Likelihood Estimate for radiation source localization: Initializing an iterative search
    53rd IEEE Conference on Decision and Control, 2014
    Co-Authors: Kidane Yosief, Soura Dasgupta, Raghuraman Mudumbai
    Abstract:

    The Maximum Likelihood Estimate approach is adopted in this paper for finding the unknown radiation source location and strength. The problem is nonlinear and has to rely on iterative numerical algorithms. Since the problem has possibly multiple local maxima, the initial Estimate in those iterative algorithms plays a critical role in guaranteeing the global optimum. This paper proposes a way to generate such an initial Estimate which is easy to calculate. Besides some insights that justifies the proposed approach, it is shown that the proposed initial Estimate actually converges to the true but unknown Maximum Likelihood Estimate asymptotically thus ensuring that the initial Estimate is indeed in a neighborhood of the Maximum Likelihood Estimate and consequently the convergence to the global optimum by local iterative numerical algorithms.

  • The Maximum Likelihood Estimate for radiation source localization: Initializing an iterative search
    53rd IEEE Conference on Decision and Control, 2014
    Co-Authors: Kidane Yosief, Soura Dasgupta, Raghuraman Mudumbai
    Abstract:

    The Maximum Likelihood Estimate approach is adopted in this paper for finding the unknown radiation source location and strength. The problem is nonlinear and has to rely on iterative numerical algorithms. Since the problem has possibly multiple local maxima, the initial Estimate in those iterative algorithms plays a critical role in guaranteeing the global optimum. This paper proposes a way to generate such an initial Estimate which is easy to calculate. Besides some insights that justifies the proposed approach, it is shown that the proposed initial Estimate actually converges to the true but unknown Maximum Likelihood Estimate asymptotically thus ensuring that the initial Estimate is indeed in a neighborhood of the Maximum Likelihood Estimate and consequently the convergence to the global optimum by local iterative numerical algorithms.

D Pfefferle - One of the best experts on this subject based on the ideXlab platform.

  • whittle Maximum Likelihood Estimate of spectral properties of rayleigh taylor interfacial mixing using hot wire anemometry experimental data
    Physical Review E, 2020
    Co-Authors: D Pfefferle, Snezhana I Abarzhi
    Abstract:

    Investigating the power density spectrum of fluctuations in Rayleigh-Taylor (RT) interfacial mixing is a means of studying characteristic length, timescales, anisotropies, and anomalous processes. Guided by group theory, analyzing the invariance-based properties of the fluctuations, our paper examines raw time series from hot-wire anemometry measurements in the experiment by Akula et al. [J. Fluid Mech. 816, 619 (2017)JFLSA70022-112010.1017/jfm.2017.95]. The results suggest that the power density spectrum can be modeled as a compound function presented as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a Maximum Likelihood estimator of the parameters. Those that maximize the log-Likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-Likelihood is used to evaluate the Fisher information matrix and provide an Estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is applied to analyze the goodness of fit, by verifying the hypothesis that the ratio between the observed periodogram and the Estimated power density spectrum follows a χ^{2} probability distribution. The dependence of the parameters of the compound function is investigated on the range of mode numbers over which the fit is performed. In the domain where the relative errors of the power-law exponent and the exponential decay rate are small and the goodness of fit is excellent, the parameters of the compound function are clearly defined, in agreement with the theory developed in the paper. The study of the power-law spectra in RT mixing data suggests that rigorous physics-based statistical methods can help researchers to see beyond visual inspection.

  • whittle Maximum Likelihood Estimate of spectral properties of rayleigh taylor interfacial mixing using hot wire anemometry experimental data
    arXiv: Fluid Dynamics, 2019
    Co-Authors: D Pfefferle, Devesh Ranjan, Snezhana I Abarzhi
    Abstract:

    The Rayleigh-Taylor instability (RTI) occurs in a broad range of processes in nature and technology. Analysing the power density spectrum of fluctuations in Rayleigh-Taylor (RT) flow is a means of highlighting characteristic length- and time-scales, anisotropies and anomalous processes. Raw time series from hot-wire anemometry measurements of Rayleigh-Taylor interfacial mixing experiment by Akula et al., JFM 816, 619-660 (2017) are considered as a sample case to adjust the parameters of a model power density spectrum. The results suggest that the power density spectrum of one of the flow components can be confidently modelled as the product of a power law and an exponential. The data analysis is based on Whittle's approximation of the power density spectrum for independent zero-mean near-Gaussian signals to construct a Maximum Likelihood Estimator (MLE) of the parameters. Those that maximise the log-Likelihood are computed numerically through Newton-Raphson iteration. The Hessian of the log-Likelihood is used to evaluate the Fisher information matrix and provide an Estimate of the statistical error on the obtained parameters. The Kolmogorov-Smirnov test is used to verify the hypothesis that the ratio between the observed periodogram and the Estimated power density spectrum follows a chi-squared probability distribution. This step is performed to show goodness-of-fit. We also study the dependence of the model parameters on the range of mode numbers over which the fit is performed.

Kidane Yosief - One of the best experts on this subject based on the ideXlab platform.

  • CDC - The Maximum Likelihood Estimate for radiation source localization: Initializing an iterative search
    53rd IEEE Conference on Decision and Control, 2014
    Co-Authors: Kidane Yosief, Soura Dasgupta, Raghuraman Mudumbai
    Abstract:

    The Maximum Likelihood Estimate approach is adopted in this paper for finding the unknown radiation source location and strength. The problem is nonlinear and has to rely on iterative numerical algorithms. Since the problem has possibly multiple local maxima, the initial Estimate in those iterative algorithms plays a critical role in guaranteeing the global optimum. This paper proposes a way to generate such an initial Estimate which is easy to calculate. Besides some insights that justifies the proposed approach, it is shown that the proposed initial Estimate actually converges to the true but unknown Maximum Likelihood Estimate asymptotically thus ensuring that the initial Estimate is indeed in a neighborhood of the Maximum Likelihood Estimate and consequently the convergence to the global optimum by local iterative numerical algorithms.

  • The Maximum Likelihood Estimate for radiation source localization: Initializing an iterative search
    53rd IEEE Conference on Decision and Control, 2014
    Co-Authors: Kidane Yosief, Soura Dasgupta, Raghuraman Mudumbai
    Abstract:

    The Maximum Likelihood Estimate approach is adopted in this paper for finding the unknown radiation source location and strength. The problem is nonlinear and has to rely on iterative numerical algorithms. Since the problem has possibly multiple local maxima, the initial Estimate in those iterative algorithms plays a critical role in guaranteeing the global optimum. This paper proposes a way to generate such an initial Estimate which is easy to calculate. Besides some insights that justifies the proposed approach, it is shown that the proposed initial Estimate actually converges to the true but unknown Maximum Likelihood Estimate asymptotically thus ensuring that the initial Estimate is indeed in a neighborhood of the Maximum Likelihood Estimate and consequently the convergence to the global optimum by local iterative numerical algorithms.