scholarly journals On the Computation of the Maximum Likelihood Estimates of the Parameters in a Mixture Model

2007 ◽  
Vol 6 (2) ◽  
pp. 57-66 ◽  
Author(s):  
G. Nanjundan
Keyword(s):  
Maximum Likelihood ◽  
Poisson Distribution ◽  
Social Group ◽  
Numerical Study ◽  
Mixture Distribution ◽  
Maximum Likelihood Estimates ◽  
Number Of Children ◽  
The Em Algorithm ◽  
The Social ◽  
Stepwise Procedure

A social group may consist of sterile and fertile couples where sterile couples cannot reproduce. When the number of children for a fertile couple is distributed according to a Poisson distribution, the probability distribution of the number of children per couple in the social group is a mixture of a distribution singular at zero and a Poisson distribution. The estimation of the parameters in the mixture distribution is considered in this paper. Since the maximum likelihood (ML) metod does not provide estimates in closed forms, it is proposed to obtain the estimates using the EM algorithm. A stepwise procedure for computing the estimates is presented. A stepwise procedure for computing the estimates is presented. A numerical study is carried out to compare these estimates with the conditional ML estimates determined using Newton-Raphson iterative procedure.

scholarly journals Estimation of different types of entropies for the Kumaraswamy distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (3) ◽  
pp. e0249027
Author(s):  
Abdulhakim A. Al-Babtain ◽  
Ibrahim Elbatal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy
Keyword(s):  
Maximum Likelihood ◽  
Numerical Study ◽  
Beta Function ◽  
Simulated Data ◽  
Likelihood Estimation ◽  
Real Data ◽  
Maximum Likelihood Estimates ◽  
Kumaraswamy Distribution ◽  
Random System ◽  
Entropy Measures

The estimation of the entropy of a random system or process is of interest in many scientific applications. The aim of this article is the analysis of the entropy of the famous Kumaraswamy distribution, an aspect which has not been the subject of particular attention previously as surprising as it may seem. With this in mind, six different entropy measures are considered and expressed analytically via the beta function. A numerical study is performed to discuss the behavior of these measures. Subsequently, we investigate their estimation through a semi-parametric approach combining the obtained expressions and the maximum likelihood estimation approach. Maximum likelihood estimates for the considered entropy measures are thus derived. The convergence properties of these estimates are proved through a simulated data, showing their numerical efficiency. Concrete applications to two real data sets are provided.

scholarly journals Fast Computation of the EM Algorithm for Mixture Models

2021 ◽  
Author(s):  
Masahiro Kuroda
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Mixture Models ◽  
Computation Time ◽  
Likelihood Estimation ◽  
Heterogeneous Data ◽  
Maximum Likelihood Estimates ◽  
The Em Algorithm ◽  
Clustering And Classification ◽  
Number Of Iterations

Mixture models become increasingly popular due to their modeling flexibility and are applied to the clustering and classification of heterogeneous data. The EM algorithm is largely used for the maximum likelihood estimation of mixture models because the algorithm is stable in convergence and simple in implementation. Despite such advantages, it is pointed out that the EM algorithm is local and has slow convergence as the main drawback. To avoid the local convergence of the EM algorithm, multiple runs from several different initial values are usually used. Then the algorithm may take a large number of iterations and long computation time to find the maximum likelihood estimates. The speedup of computation of the EM algorithm is available for these problems. We give the algorithms to accelerate the convergence of the EM algorithm and apply them to mixture model estimation. Numerical experiments examine the performance of the acceleration algorithms in terms of the number of iterations and computation time.

Estimation of Two-Parameter Logistic Item Response Curves

1984 ◽  
Vol 9 (4) ◽  
pp. 263-276 ◽  
Author(s):  
Robert K. Tsutakawa
Keyword(s):  
Maximum Likelihood ◽  
Item Response ◽  
Information Matrix ◽  
Simulated Data ◽  
Maximum Likelihood Estimates ◽  
Response Curves ◽  
Numerical Work ◽  
Item Parameters ◽  
The Em Algorithm ◽  
Two Parameter

Under the assumption that ability parameters are sampled from a normal distribution, the EM algorithm is used to derive maximum likelihood estimates for item parameters of the two-parameter logistic item response curves. The observed information matrix is then used to approximate the covariance matrix of these estimates. Responses to a questionnaire on general arthritis knowledge are used to illustrate the procedure and simulated data are used to compare the estimated and actual item parameters. The resulting estimates are found to be very close to those obtained from LOGIST. A computational note is included to facilitate the extensive numerical work required to implement the procedure.

scholarly journals The The Topp Leone-G Power Series Class of Distributions with Applications

2021 ◽  
pp. 827-846
Author(s):  
Fastel Chipepa ◽  
Boikanyo Makubate ◽  
Broderick Oluyede ◽  
Kethamile Rannona
Keyword(s):  
Maximum Likelihood ◽  
Power Series ◽  
Poisson Distribution ◽  
Simulation Study ◽  
Real Data ◽  
Maximum Likelihood Estimates ◽  
Data Sets ◽  
New Class ◽  
Proposed Model ◽  
Special Case

We present a new class of distributions called the Topp-Leone-G Power Series (TL-GPS) class of distributions. This model is obtained by compounding the Topp-Leone-G distribution with the power series distribution. Statistical prop- erties of the TL-GPS class of distributions are obtained. Maximum likelihood estimates for the proposed model were obtained. A simulation study is carried out for the special case of Topp-Leone Log-Logistic Poisson distribution to assess the performance of the maximum likelihood estimates. Finally, we apply Topp-Leone-log-logistic Poisson distribution to real data sets to illustrate the usefulness and applicability of the proposed class of distributions.

Demand Estimation Under the Multinomial Logit Model from Sales Transaction Data

2020 ◽  
Author(s):  
Tarek Abdallah ◽  
Gustavo Vulcano
Keyword(s):  
Maximum Likelihood ◽  
Logit Model ◽  
Likelihood Function ◽  
Numerical Study ◽  
Multinomial Logit Model ◽  
Multinomial Logit ◽  
Problem Definition ◽  
Maximum Likelihood Estimates ◽  
Computational Time ◽  
Mm Algorithm

Problem definition: A major task in retail operations is to optimize the assortments exhibited to consumers. To this end, retailers need to understand customers’ preferences for different products. Academic/practical relevance: This is particularly challenging when only sales and product-availability data are recorded, and not all products are displayed in all periods. Similarly, in revenue management contexts, firms (airlines, hotels, etc.) need to understand customers’ preferences for different options in order to optimize the menu of products to offer. Methodology: In this paper, we study the estimation of preferences under a multinomial logit model of demand when customers arrive over time in accordance with a nonhomogeneous Poisson process. This model has recently caught important attention in both academic and industrial practices. We formulate the problem as a maximum-likelihood estimation problem, which turns out to be nonconvex. Results: Our contribution is twofold: From a theoretical perspective, we characterize conditions under which the maximum-likelihood estimates are unique and the model is identifiable. From a practical perspective, we propose a minorization-maximization (MM) algorithm to ease the optimization of the likelihood function. Through an extensive numerical study, we show that our algorithm leads to better estimates in a noticeably short computational time compared with state-of-the-art benchmarks. Managerial implications: The theoretical results provide a solid foundation for the use of the model in terms of the quality of the derived estimates. At the same time, the fast MM algorithm allows the implementation of the model and the estimation procedure at large scale, compatible with real industrial applications.

scholarly journals Modelling epistatic effects of embryo and endosperm QTL on seed quality traits

2006 ◽  
Vol 87 (1) ◽  
pp. 61-71 ◽  
Author(s):  
YUEHUA CUI ◽  
JIANGUO WU ◽  
CHUNHAI SHI ◽  
RAMON C. LITTELL ◽  
RONGLING WU
Keyword(s):  
Maximum Likelihood ◽  
Seed Quality ◽  
Genetic Research ◽  
Maximum Likelihood Estimates ◽  
Endosperm Tissue ◽  
Significant Interaction Effect ◽  
Epistatic Effects ◽  
The Em Algorithm ◽  
Coordinated Expression ◽  
Gel Consistency

Coordinated expression of embryo and endosperm tissues is required for proper seed development. The coordination among these two tissues is controlled by the interaction between multiple genes expressed in the embryo and endosperm genomes. In this article, we present a statistical model for testing whether quantitative trait loci (QTL) active in different genomes, diploid embryo and triploid endosperm, epistatically affect a trait expressed on the endosperm tissue. The maximum likelihood approach, implemented with the EM algorithm, was derived to provide the maximum likelihood estimates of the locations of embryo- and endosperm-specific QTL and their main effects and epistatic effects. This model was used in a real example for rice in which two QTL, one from the embryo genome and the other from the endosperm genome, exert a significant interaction effect on gel consistency on the endosperm. Our model has successfully detected Waxy, a candidate gene in the embryo genome known to regulate one of the major steps of amylose biosynthesis in the endosperm. This model will have great implications for agricultural and evolutionary genetic research.

scholarly journals The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 650 ◽  
Author(s):  
Abdullah M. Almarashi ◽  
Mohammed Elgarhy ◽  
Farrukh Jamal ◽  
Christophe Chesneau
Keyword(s):  
Maximum Likelihood ◽  
Numerical Study ◽  
Stochastic Ordering ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Data Sets ◽  
Wide Sense ◽  
Rate Functions ◽  
Generated Family ◽  
Family Of Distributions

In this paper, we propose a generalization of the so-called truncated inverse Weibull-generated family of distributions by the use of the power transform, adding a new shape parameter. We motivate this generalization by presenting theoretical and practical gains, both consequences of new flexible symmetric/asymmetric properties in a wide sense. Our main mathematical results are about stochastic ordering, uni/multimodality analysis, series expansions of crucial probability functions, probability weighted moments, raw and central moments, order statistics, and the maximum likelihood method. The special member of the family defined with the inverse Weibull distribution as baseline is highlighted. It constitutes a new four-parameter lifetime distribution which brightensby the multitude of different shapes of the corresponding probability density and hazard rate functions. Then, we use it for modelling purposes. In particular, a complete numerical study is performed, showing the efficiency of the corresponding maximum likelihood estimates by simulation work, and fitting three practical data sets, with fair comparison to six notable models of the literature.

scholarly journals Uniqueness of Maximum Likelihood Estimators for a Backup System in a Condition-Based Maintenance

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Qihong Duan ◽  
Ying Wei ◽  
Xiang Chen
Keyword(s):  
Maximum Likelihood ◽  
Em Algorithm ◽  
Maximum Likelihood Estimators ◽  
Condition Based Maintenance ◽  
Maximum Likelihood Estimates ◽  
Model Parameters ◽  
Process Data ◽  
Parameter Estimation Problem ◽  
Backup System ◽  
The Em Algorithm

A parameter estimation problem for a backup system in a condition-based maintenance is considered. We model a backup system by a hidden, three-state continuous time Markov process. Data are obtained through condition monitoring at discrete time points. Maximum likelihood estimates of the model parameters are obtained using the EM algorithm. We establish conditions under which there is no more than one limitation in the parameter space for any sequence derived by the EM algorithm.

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