Maximum Likelihood Estimation Procedure Research Articles

Damage‐informed ground motion and semi‐empirical fragility assessment

AbstractCalibrating parametric fragility curves via empirical damage data is one of the standard approaches to derive seismic structural vulnerability models. Fragilities based on empirical data require the characterization of the ground motion (GM) intensity at the building sites in the area affected by the earthquake producing the observed damages. This is commonly conducted via ShakeMap, that is, a map of the expected values of a Gaussian random field (GRF) of the logarithms of a GM intensity measure conditional to magnitude, location, and possibly a set of recordings of the earthquake. Once that intensity and damage data at the same sites are available, the typical approach calibrates a two‐parameter fragility model. However, ShakeMap estimates are affected by uncertainty deriving from that of the GM model used to characterize it. Furthermore, such an uncertainty can be reduced by building damage data, which provide information on the shaking intensity at the sites where damage is observed. It is shown herein that if this uncertainty is not addressed, also considering the shaking information provided by damage, the estimates of the fragility parameters obtained using a median ShakeMap only can be biased, and a recommended maximum likelihood estimation procedure – which exploits the expectation maximization algorithm – is provided. These arguments are illustrated via an application considering damage data from the 2009 L'Aquila earthquake in central Italy.

Lomax tangent generalized family of distributions: Characteristics, simulations, and applications on hydrological-strength data

This article proposes and discusses a novel approach for generating trigonometric G-families using hybrid generalizers of distributions. The proposed generalizer is constructed by utilizing the tangent trigonometric function and distribution function of base model G(x). The newly proposed family of uni-variate continuous distributions is named the “Lomax Tangent Generalized Family of Distributions (LT-G)” and structural-mathematical-statistical properties are derived. Some special and sub-models of the proposed family are also presented.A Weibull-based model, ‘The Lomax Tangent Weibull (LT-W) Distribution,” is discussed and the plots of density (pdf) and hazard (hrf) functions are also explained. Model parameter estimates are estimated by employing the maximum likelihood estimation (MLE) procedure. The accuracy of the MLEs is evaluated through Monte Carlo simulation. Last but not least, to demonstrate the flexibility and potential of the proposed distribution, two actual hydrological and strength data sets are analyzed. The obtained results are compared with well-known, competitive, and related existing distributions.

Non-proportional hazards model with a PVF frailty term: application with a melanoma dataset

Survival data analysis often uses the Cox proportional hazards (PH) model. This model is widely applied due to its straightforward interpretation of the hazard ratio under the assumption that the hazard rates for two subjects remain constant over time. However, in several randomized clinical trials with long-term survival data comparing two new treatments, it is frequently observed that Kaplan-Meier plots exhibit crossing survival curves. This violation of the PH assumption of the Cox PH model can not be applied to evaluate the treatment's effect on survival. This paper introduces a novel long-term survival model with non-PH that incorporates a frailty term into the hazard function. This model allows us to examine the effect of prognostic factors on survival and quantify the degree of unobservable heterogeneity. The model parameters are estimated using the maximum likelihood estimation procedure, and we evaluate the performance of the proposed models through simulation studies. Additionally, we demonstrate the applicability of our approach by fitting the models to a real skin cancer dataset.

Constrained inference in scaled mixed effects models with applications to London Junior School Project data

Multiple outcomes arise frequently in many different settings. Mixed effect models are a useful tool for the analyses of such data. However, when outcomes are measured on different scales, analyses based on any one scale are misleading. Often parameters of the model are subject to known order restrictions. To incorporate heterogeneity across different outcomes, we propose a scaled linear mixed effect model. To estimate parameters, we propose a maximum likelihood estimation procedure based on a restricted version of the expectation-conditional maximization either algorithm. Constrained hypotheses testing procedures are developed using likelihood ratio tests. The empirical significance levels and powers are studied using simulations. This article shows that incorporating the restrictions improves the mean squared errors of the estimates and the power of the tests. The methodology is applied on the London Junior School Project data incorporating known restrictions of patterns of scores. Journal of Statistical Research 2023, Vol 57, No.1-2, pp.81-94

Regression analysis of longitudinal data with random change point.

A great deal of literature has been established for regression analysis of longitudinal data and in particular, many methods have been proposed for the situation where there exist some change points. However, most of these methods only apply to continuous response and focus on the situations where the change point only occurs on the response or the trend of the individual trajectory. In this article, we propose a new joint modeling approach that allows not only the change point to vary for different subjects or be subject-specific but also the effect heterogeneity of the covariates before and after the change point. The method combines a generalized linear mixed effect model with a random change point for the longitudinal response and a log-linear regression model for the random change point. For inference, a maximum likelihood estimation procedure is developed and the asymptotic properties of the resulting estimators, which differ from the standard asymptotic results, are established. A simulation study is conducted and suggests that the proposed method works well for practical situations. An application to a set of real data on COVID-19 is provided.

Efficient estimation of Cox model with random change point.

In clinical studies, the risk of a disease may dramatically change when some biological indexes of the human body exceed some thresholds. Furthermore, the differences in individual characteristics of patients such as physical and psychological experience may lead to subject-specific thresholds or change points. Although a large literature has been established for regression analysis of failure time data with change points, most of the existing methods assume the same, fixed change point for all study subjects. In this paper, we consider the situation where there exists a subject-specific change point and two Cox type models are presented. The proposed models also offer a framework for subgroup analysis. For inference, a sieve maximum likelihood estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. An extensive simulation study is conducted to assess the empirical performance of the proposed method and indicates that it works well in practical situations. Finally the proposed approach is applied to a set of breast cancer data.

Poisson XRani Distribution: An Alternative Discrete Distribution for Overdispersed Count Data

In this article, a one‐parameter probability mass function called Poisson XRani is proposed by combining the Poisson and the XRani distributions. A wide range of distributional properties such as the shape of the probability mass function, probability generating function, factorial moments, moment generating function, characteristic function, raw moments, dispersion index, mean, variance, coefficient of skewness, and coefficient of kurtosis are investigated. It is proven that the proposed probability mass function can easily handle overdispersed and right‐skewed count observations with heavy tails. Both maximum likelihood and Bayesian estimation techniques are used to estimate the unknown parameter of the proposed distribution and a simulation study was conducted to examine and analyze the performance of the maximum likelihood estimation procedure based on sample size, absolute bias, relative absolute bias, and root mean square error. The usefulness of the proposed distribution is assessed using two distinctive real datasets. These applications reveal that the new distribution provides an adequate fit compared to the other eight discrete distributions.

Generalized inverted Kumaraswamy-Rayleigh Distribution: Properties and Application

The paper proposes a new four parameter distribution called Generalized inverted Kumaraswamy- Rayleigh Distribution. Various properties of the proposed distribution including moments, mean deviation about mean and median, entropy, L-moments, stress-strength reliability etc. are obtained. The parameter estimation is carried out using Maximum likelihood estimation procedure. Finally, two real life data sets are incorporated to illustrate the usefulness and flexibility of the proposed model.

Correcting for bias effects due to exposure uncertainty in community noise exposure-response analyses.

A method for correcting the relationship-distorting effects of sound level uncertainty on community noise exposure-response investigations is presented. The method employs a "calibration model," which describes how the true exposure-response relationship becomes distorted (shallower) in the presence of sound level uncertainty and other factors. This model was incorporated in a maximum likelihood estimation procedure that solves for the coefficient values that would have been observed in the absence of sound level measurement error. Three interacting situational variables known to attenuate the slopes of fitting functions were incorporated in the model: the sample population sound level uncertainty, the sound level range, and the distribution of sound levels over that range. The method provided unbiased estimates of the slope and intercept when the calibration parameter values were known exactly. Sensitivity to inexact knowledge of the values was also examined. Although the bias effect could be removed, uncertainty in the solved regression coefficients increased with increasing predictor variable (sound level) uncertainty and with data set displacement from the asymptote.

Generalized double Lindley distribution: A new model for weather and financial data

Abstract In this paper, we introduce a generalization of the two-parameter double Lindley distribution (TPDLD) of Kumar and Jose [C. S. Kumar and R. Jose, A new generalization to Laplace distribution, J. Math. Comput. 31 2020, 8–32], namely the generalized double Lindley distribution (GDLD) along with its location-scale extension (EGDLD). Then we discuss the estimation of parameters of the EGDLD by the maximum likelihood estimation procedure. Next, we illustrate this estimation procedure with the help of certain real life data sets, and a simulation study is carried out to examine the performance of various estimators of the parameters of the distribution.

Effect of ewe birth litter size and estimation of genetic parameters on ewe reproductive life traits.

Ewe lifetime productivity has economic implications for producers because shorter lifetime productivity results in less profit. Productive years of ewes from extensive, range-based systems of the United States West are generally less than ewes from more temperate regions of the United States. Accordingly, ewes from range-based systems, especially those employing shed-lambing strategies, have been selected for increased litter size to offset decreased lifetime productivity. However, the relationship of the ewe's birth litter size (ELSB) has not been considered a potential contributor to lifetime productivity. Longevity (number of productive years, n=1 per population) and stayability (probability to survive to the next age; ages 2-7years, n=6 per population) were investigated to understand ELSB effects on productive life. Columbia, Polypay, Rambouillet, and Targhee breeds were used in this study. Across-breed (n=11550) and within-breed (Columbia, n=4398; Polypay, n=4534; Rambouillet, n=5922; Targhee, n=6482) analyses were used. Depending on the population, records spanned from 1950 to 2008, where ewe's birth year was included as a fixed effect in the animal model using restricted maximum likelihood estimation procedures. Fixed effects investigated included ELSB (single-, twin-, or triplet-born) and ewe breed (across-breed analyses only). Regardless of trait or population used, heritability ranged from 0.06±0.02 to 0.34±0.03, where stayability at younger ages had the highest estimates. The breed effect was significant in all across-breed analyses (0.0001≤P≤0.038; n=7), where Polypay, a breed selected for accelerated lambing and increased fertility, averaged shorter productive life or had a lower probability of survival to the next age compared with other breeds (longevity: 0.009≤P≤0.223; stayability: 0.000≤P≤0.842). The ELSB was significant in 60% (n=5) and 37% (n=30) of longevity and stayability analyses, respectively. Except for Targhee, all analyses showed ewes born in smaller litter sizes were associated with longer productive lives or higher probability of surviving to the next age, particularly in across-breed analyses (e.g., longevity: single- vs twin-born ewes, P=0.004; vs triplet-born ewes, P=0.003). This study provides evidence that increasing prolificacy in ewes from extensive, range-based production systems may impact productive life. Due to the low heritability of these traits, additional investigation into modeling these traits with dominance effects and litter size needs to be conducted.

Describing complex disease progression using joint latent class models for multivariate longitudinal markers and clinical endpoints.

Neurodegenerative diseases are characterized by numerous markers of progression and clinical endpoints. For instance, multiple system atrophy (MSA), a rare neurodegenerative synucleinopathy, is characterized by various combinations of progressive autonomic failure and motor dysfunction, and a very poor prognosis. Describing the progression of such complex and multi-dimensional diseases is particularly difficult. One has to simultaneously account for the assessment of multivariate markers over time, the occurrence of clinical endpoints, and a highly suspected heterogeneity between patients. Yet, such description is crucial for understanding the natural history of the disease, staging patients diagnosed with the disease, unravelling subphenotypes, and predicting the prognosis. Through the example of MSA progression, we show how a latent class approach modeling multiple repeated markers and clinical endpoints can help describe complex disease progression and identify subphenotypes for exploring new pathological hypotheses. The proposed joint latent class model includes class-specific multivariate mixed models to handle multivariate repeated biomarkers possibly summarized into latent dimensions and class-and-cause-specific proportional hazard models to handle time-to-event data. Maximum likelihood estimation procedure, validated through simulations is available in the lcmm R package. In the French MSA cohort comprising data of 598 patients during up to 13 years, five subphenotypes of MSA were identified that differ by the sequence and shape of biomarkers degradation, and the associated risk of death. In posterior analyses, the five subphenotypes were used to explore the association between clinical progression and external imaging and fluid biomarkers, while properly accounting for the uncertainty in the subphenotypes membership.

An Exponentiated Skew-Elliptic Nonlinear Extension to the Log–Linear Birnbaum–Saunders Model with Diagnostic and Residual Analysis

In this paper, we propose a nonlinear regression model with exponentiated skew-elliptical errors distributed, which can be fitted to datasets with high levels of asymmetry and kurtosis. Maximum likelihood estimation procedures in finite samples are discussed and the information matrix is deduced. We carried out a diagnosis of the influence for the nonlinear model. To analyze the sensitivity of the maximum likelihood estimators of the model’s parameters to small perturbations in distribution assumptions and parameter estimation, we studied the perturbation schemes, the case weight, and the explanatory and response variables of perturbations; we also carried out a residual analysis of the deviance components. Simulation studies were performed to assess some properties of the estimators, showing the good performance of the proposed estimation procedure in finite samples. Finally, an application to a real dataset is presented.

Multilevel analysis of matching behavior: A comparison of maximum likelihood and Bayesian estimation.

While trying to infer laws of behavior, accounting for both within-subjects and between-subjects variance is often overlooked. It has been advocated recently to use multilevel modeling to analyze matching behavior. Using multilevel modeling within behavior analysis has its own challenges though. Adequate sample sizes are required (at both levels) for unbiased parameter estimates. The purpose of the current study is to compare parameter recovery and hypothesis rejection rates of maximum likelihood (ML) estimation and Bayesian estimation (BE) of multilevel models for matching behavior studies. Four factors were investigated through simulations: number of subjects, number of measurements by subject, sensitivity (slope), and variance of the random effect. Results showed that both ML estimation and BE with flat priors yielded acceptable statistical properties for intercept and slope fixed effects. The ML estimation procedure generally had less bias, lower RMSE, more power, and false-positive rates closer to the nominal rate. Thus, we recommend ML estimation over BE with uninformative priors, considering our results. The BE procedure requires more informative priors to be used in multilevel modeling of matching behavior, which will require further studies.

A new approach for semi-parametric regression analysis of bivariate interval-censored outcomes from case-cohort studies

Interval-censored failure time data frequently occur in many areas and a great deal of literature on their analyses has been established. In this article, we discuss the situation where one faces bivariate interval-censored data arising from case-cohort studies, which are commonly used as a tool to save costs when disease incidence is low and covariates are difficult to obtain. For this problem, a class of copula-based semi-parametric models is presented and for estimation, a sieve weighted maximum likelihood estimation procedure is developed. The resulting estimators of regression parameters are shown to be strongly consistent and asymptotically normal. Furthermore, the proposed method is generalized to the situation of non rare diseases. A simulation study is conducted to assess the finite sample performance of the proposed method and suggests that it performs well in practice.

Group variable selection for the Cox model with interval-censored failure time data.

Group variable selection is often required in many areas, and for this many methods have been developed under various situations. Unlike the individual variable selection, the group variable selection can select the variables in groups, and it is more efficient to identify both important and unimportant variables or factors by taking into account the existing group structure. In this paper, we consider the situation where one only observes interval-censored failure time data arising from the Cox model, for which there does not seem to exist an established method. More specifically, a penalized sieve maximum likelihood variable selection and estimation procedure is proposed and the oracle property of the proposed method is established. Also, an extensive simulation study is performed and suggests that the proposed approach works well in practical situations. An application of the method to a set of real data isprovided.

An accelerated failure time regression model for illness-death data: A frailty approach.

This work presents a new model and estimation procedure for the illness-death survival data where the hazard functions follow accelerated failure time (AFT) models. A shared frailty variate induces positive dependence among failure times of a subject for handling the unobserved dependency between the nonterminal and the terminal failure times given the observed covariates. The motivation behind the proposed modeling approach is to leverage the well-known interpretability advantage of AFT models with respect to the observed covariates, while also benefiting from the simple and intuitive interpretation of the hazard functions. A semiparametric maximum likelihood estimation procedure is developed via a kernel smoothed-aided expectation-maximization algorithm, and variances are estimated by weighted bootstrap. We consider existing frailty-based illness-death models and place particular emphasis on highlighting the contribution of our current research. The breast cancer data of the Rotterdam tumor bank are analyzed using the proposed as well as existing illness-death models. The results are contrasted and evaluated based on a new graphical goodness-of-fit procedure. Simulation results and data analysis nicely demonstrate the practical utility of the shared frailty variate with the AFT regression model under the illness-deathframework.

New hyperbolic sine-generator with an example of Rayleigh distribution: Simulation and data analysis in industry

This study focuses on a novel family of distributions inspired by the hyperbolic sine function. The Rayleigh distribution is the base model for the newly formed family of distributions known as the new hyperbolic Sine-Rayleigh distribution. The recommended distribution’s distinct structural traits have been examined. The behaviors of the distributional functions of the proposed model are depicted in several figures. The maximum likelihood estimation procedure is employed to estimate the specified distribution parameters. A simulation study was carried out to examine and evaluate the behavior of the estimators. Moreover, the efficacy of the specified distribution is supported by realistic data sets pertaining to engineering science.

Smallholder Tomato Production in Mwanza Region- A Technical Efficiency Analysis Approach.

This study measured technical efficiency and its determinants in maize production by small-scale producers in Mwanza region, using a stochastic frontier production function approach. A randomly selected sample of participants in the two districts was used. The Maximum Likelihood estimation procedure was followed to obtain the determinants of technical efficiency and technical efficiency levels of small-scale tomato producers. The minimum and maximum values of technical efficiency were between 20% and 91%, indicating that the least practices of specific producer operates at a minimum level of 20%, while the best practice producers operate at 91% technical efficiency level respectively. The summary results of the mean technical efficiency was 57%. The results indicate that farmers were not fully technically efficient with a mean technical efficiency score of 0.58. The study also revealed that most of the farmers irrespective of the size of the holdings have shown technical inefficiency problems. Experienced farmers were observed to have good measures of technical efficiency. The findings also indicated that significant variables of technical efficiency included household size, labour, fertilizer, seed, educational level, household size, farming experience, and access to extension services were statistically significant by positively impacted on producers’ technical efficiency in the study area.

Evaluation of maximum likelihood procedures in ranked set sampling under imperfect ranking

Ranked set sampling (RSS) is a cost-efficient alternative to simple random sampling (SRS) when ranking small sets of population units is much simpler than effectively measuring them. Maximum likelihood estimation (MLE) in RSS is usually based on the restrictive assumption of perfect ranking, assuming that the sample units lead to order statistics. However, this procedure is not flexible and often produces highly biased estimators when ranking errors are present. To remedy these constraints, in this work we suggest and evaluate alternative MLE procedures for RSS: (i) MLE based on the perfect ranking assumption; (ii) MLE based on the SRS assumptions; (iii) a weighted estimator derived from (i) and (ii); (iv) the choice of (i) or (ii) by testing the null hypothesis of perfect ranking. We carried out extensive simulations to investigate the efficiency of each strategy, which allowed us to conclude that although (i) performed the best under perfect ranking, (ii-iv) are preferable under imperfect ranking. We have also evaluated a parametric bootstrap algorithm to obtain confidence intervals in RSS based on the suggested estimators. Two applications using data from age determination of fishes and anthropometric measures of athletes illustrate and corroborate our findings.