A Comparison of Maximum Likelihood Estimations for Normal Mean under Right Censoring

2020 ◽  
Vol 72 (2) ◽  
pp. 122-132
Author(s):  
Junfeng Liu ◽  
Xiaoxia Zhang
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Expectation Maximization ◽  
Asymptotic Variance ◽  
Likelihood Estimation ◽  
Right Censoring ◽  
Log Likelihood ◽  
Framework Approach ◽  
Normal Mean ◽  
Maximum Likelihood Estimations

For efficiently estimating the normal mean ([Formula: see text]) under right censoring (threshold =[Formula: see text], [Formula: see text] is known), we compare two approaches within the maximum likelihood estimation (MLE) framework. Approach I is a hierarchical MLE for which only the empirical censoring probability is utilized. Approach II is the direct MLE for which expectation-maximization (EM) algorithm is applied to all individual observations. We use discrete approximation to explain that the asymptotic variance of Approach II estimate equals the inverse Fisher information calculated from the full log-likelihood. We prove that Approach II gives a uniformly smaller asymptotic variance than Approach I and the variance ratio is a decreasing function of [Formula: see text]. We further prove some supportive results and graphically demonstrate that EM algorithm monotonically converges to the unique MLE.

scholarly journals Comparison of Recent Acceleration Techniques for the EM Algorithm in One- and Two-Parameter Logistic IRT Models

Psych ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 209-252
Author(s):  
Marie Beisemann ◽  
Ortrud Wartlick ◽  
Philipp Doebler
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Binary Data ◽  
Likelihood Estimation ◽  
Acceleration Techniques ◽  
Irt Models ◽  
The Em Algorithm ◽  
Log Likelihood ◽  
Two Parameter

The expectation–maximization (EM) algorithm is an important numerical method for maximum likelihood estimation in incomplete data problems. However, convergence of the EM algorithm can be slow, and for this reason, many EM acceleration techniques have been proposed. After a review of acceleration techniques in a unified notation with illustrations, three recently proposed EM acceleration techniques are compared in detail: quasi-Newton methods (QN), “squared” iterative methods (SQUAREM), and parabolic EM (PEM). These acceleration techniques are applied to marginal maximum likelihood estimation with the EM algorithm in one- and two-parameter logistic item response theory (IRT) models for binary data, and their performance is compared. QN and SQUAREM methods accelerate convergence of the EM algorithm for the two-parameter logistic model significantly in high-dimensional data problems. Compared to the standard EM, all three methods reduce the number of iterations, but increase the number of total marginal log-likelihood evaluations per iteration. Efficient approximations of the marginal log-likelihood are hence an important part of implementation.

scholarly journals Comparison of Recent Acceleration Techniques for the EM algorithm in One- and Two-Parameter Logistic IRT models

2020 ◽  
Author(s):  
Marie Beisemann ◽  
Ortrud Wartlick ◽  
Philipp Doebler
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Likelihood Estimation ◽  
Acceleration Techniques ◽  
Irt Models ◽  
The Em Algorithm ◽  
Expectation Maximisation ◽  
Log Likelihood ◽  
Two Parameter

The Expectation-Maximisation (EM) algorithm is an important numerical method for maximum likelihood estimation in incomplete-data problems. However, convergence of the EM algorithm can be slow, and for this reason, many EM acceleration techniques have been proposed. After a review of acceleration techniques in a unified notation with illustrations, three recently proposed EM acceleration techniques are compared in detail: quasi-Newton methods (QN) (Zhou et al., 2011), "squared" iterative methods (SQUAREM) (Varadhan & Roland, 2004; Roland et al., 2007; Varadhan & Roland, 2008) and parabolic EM (PEM) (Berlinet & Roland, 2009). These acceleration techniques are applied to marginal maximum likelihood estimation with the EM algorithm in one- and two-parameter logistic item response theory (IRT) models for binary data, and their performance is compared. QN and SQUAREM methods accelerate convergence of EM algorithm for the two parameter logistic model significantly in high-dimensional data problems. Compared to standard EM, all three methods reduce the number of iterations, but increase the number of total marginal log-likelihood evaluations per iteration. Efficient approximations of the marginal log-likelihood are hence an important part ofimplementations.

scholarly journals Estimation of 2- and 3-parameter BURR type XII distribution using EM algorithm

2014 ◽  
Vol 10 (2) ◽  
Author(s):  
Nor Hidayah Ismail ◽  
Zarina Mohd Khalid
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Censored Data ◽  
Simulation Study ◽  
Expectation Maximization ◽  
Likelihood Estimation ◽  
Model Parameters ◽  
Gamma Distributions ◽  
Burr Type Xii

The Burr Type XII distribution is one of the systems of continuous distributions and is widely known because the distribution includes the characteristics of various well known distributions such as Weibull and gamma distributions. Maximum likelihood estimation (MLE) has been a common method in estimating model parameters. In this paper, we present an alternative method that is expectation-maximization (EM) algorithm to estimate the two- and three- parameter Burr Type XII distributions in the presence of complete and censored data. Furthermore, simulation study is conducted to compare the efficiency and accuracy of MLE and EM algorithm. We discover that EM estimation is more efficient and accurate than those estimates obtained via MLE approach.________________________________________GRAPHICAL ABSTRACT

Maximum-likelihood estimation in linkage heterogeneity models including additional information via the EM algorithm

1995 ◽  
Vol 12 (5) ◽  
pp. 515-527 ◽  
Author(s):  
Jeanine J. Houwing-Duistermaat ◽  
Lodewijk A. Sandkuijl ◽  
Arthur A. B. Bergen ◽  
Hans C. van Houwelingen
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Likelihood Estimation ◽  
Additional Information ◽  
The Em Algorithm

scholarly journals Maximum likelihood estimation of the Latent Class Model through model boundary decomposition

2019 ◽  
Vol 10 (1) ◽  
pp. 51-84 ◽  
Author(s):  
Elizabeth Allman ◽  
Hector Banos Cervantes ◽  
Serkan Hosten ◽  
Kaie Kubjas ◽  
Daniel Lemke ◽  
...  
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Em Algorithm ◽  
Maximum Likelihood Estimator ◽  
Latent Class ◽  
Latent Class Model ◽  
Likelihood Estimation ◽  
Likelihood Estimator ◽  
Class Model ◽  
Small Models

The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.

scholarly journals A New Generalized t Distribution Based on a Distribution Construction Method

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2413
Author(s):  
Ruijie Guan ◽  
Xu Zhao ◽  
Weihu Cheng ◽  
Yaohua Rong
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Iterative Algorithm ◽  
Likelihood Estimation ◽  
Estimation Methods ◽  
Modified Method ◽  
The Em Algorithm ◽  
Likelihood Approach ◽  
T Distribution ◽  
Better Than

In this paper, a new generalized t (new Gt) distribution based on a distribution construction approach is proposed and proved to be suitable for fitting both the data with high kurtosis and heavy tail. The main innovation of this article consists of four parts. First of all, the main characteristics and properties of this new distribution are outined. Secondly, we derive the explicit expression for the moments of order statistics as well as its corresponding variance–covariance matrix. Thirdly, we focus on the parameter estimation of this new Gt distribution and introduce several estimation methods, such as a modified method of moments (MMOM), a maximum likelihood estimation (MLE) using the EM algorithm, a novel iterative algorithm to acquire MLE, and improved probability weighted moments (IPWM). Through simulation studies, it can be concluded that the IPWM estimation performs better than the MLE using the EM algorithm and the MMOM in general. The newly-proposed iterative algorithm has better performance than the EM algorithm when the sample kurtosis is greater than 2.7. For four parameters of the new Gt distribution, a profile maximum likelihood approach using the EM algorithm is developed to deal with the estimation problem and obtain acceptable.

scholarly journals Estimating the relative proportions of SARS-CoV-2 strains from wastewater samples

2022 ◽  
Author(s):  
Lenore Pipes ◽  
Zihao Chen ◽  
Svetlana Afanaseva ◽  
Rasmus Nielsen
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Expectation Maximization ◽  
Initial Step ◽  
Sequencing Depth ◽  
Maximum Likelihood Estimates ◽  
Reference Database ◽  
Wastewater Samples ◽  
Different Strains

Wastewater surveillance has become essential for monitoring the spread of SARS-CoV-2. The quantification of SARS-CoV-2 RNA in wastewater correlates with the Covid-19 caseload in a community. However, estimating the proportions of different SARS-CoV-2 strains has remained technically difficult. We present a method for estimating the relative proportions of SARS-CoV-2 strains from wastewater samples. The method uses an initial step to remove unlikely strains, imputation of missing nucleotides using the global SARS-CoV-2 phylogeny, and an Expectation-Maximization (EM) algorithm for obtaining maximum likelihood estimates of the proportions of different strains in a sample. Using simulations with a reference database of >3 million SARS-CoV-2 genomes, we show that the estimated proportions accurately reflect the true proportions given sufficiently high sequencing depth and that the phylogenetic imputation is highly accurate and substantially improves the reference database.

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