Maximum Likelihood Estimates Of Parameters Research Articles

MODELING THE BENEFITS OF A MARRIAGE REVERSE ANNUITY CONTRACT WITH DEPENDENCY ASSUMPTIONS USING ARCHIMEDEAN COPULA

Social security benefits may not be enough for retirement. Equity release products like marriage reverse annuities can boost retirement income for older couples. Marriage reverse annuity’s contract convert all or part of the real estate value of elderly spouses while they are living (joint life status) or after one partner dies (last survivor status). Since husband and wife face the same death risk, the chance of death between spouses is believed to be dependent for realism. Thus, copula models the future dependency model of a husband and wife. Sklar's theorem states that copulas link bivariate distribution and marginal cumulative functions. One of the most common copulas is Archimedean copula. Clayton, Gumbel, and Frank are Archimedean copula that will be used in this investigation. The Indonesian Mortality Table IV data is used to obtain the marginal distribution of the male and female which will then be used to construct copulas (Clayton, Gumbel, and Frank) that combine two marginal distributions into a joint distribution. The marginal distribution of Indonesian Mortality Table IV is uncertain, hence Canonical Maximum Likelihood parameter estimation is utilized to estimate the parameter of copulas. Multiple-state models depict the marriage reverse annuity model for joint life and last survivor status. The probability structure is based on Sklar's theorem and copula survival function. The contract benefits calculation utilizing copulas (Clayton, Gumbel, and Frank) shows that joint life status benefits are higher than last survivor status. Joint life status uses the dependence assumption with Frank's copula to calculate the smallest annual benefit value of a marriage reverse annuity contract, while last survivor status uses the independence assumption (without copula).

Scenario analysis of livestock-related PM2.5 pollution based on a new heteroskedastic spatiotemporal model

The air in the Lombardy Plain, Italy, is one of the most polluted in Europe due to limited atmosphere circulation and high emission levels. There is broad scientific consensus that ammonia (NH3) emissions have a primary impact on air quality, and in Lombardy, the agricultural sector and livestock activities are widely recognised as being responsible for approximately 97% of regional ammonia emissions due to the high density of livestock.In this paper, we quantify the relationship between ammonia emissions and PM2.5 concentrations in the Lombardy Plain and evaluate PM2.5 changes due to the reduction of ammonia emissions through a ‘what-if’ scenario analysis. The information in the data is exploited using a spatiotemporal statistical model capable of handling spatial and temporal correlation as well as missing data. To do this, we propose a new heteroskedastic extension of the well-established Hidden Dynamic Geostatistical Model. Maximum likelihood parameter estimates are obtained by the expectation–maximisation algorithm and implemented in a new version of the D-STEM software.Considering the years between 2016 and 2020, the scenario analysis is carried out on high-resolution PM2.5 maps of the Lombardy Plain. As a result, it is shown that a 26% reduction in NH3 emissions in the wintertime could reduce the PM2.5 average by 1.44 μg/m3 while a 50% reduction could reduce the PM2.5 average by 2.76 μg/m3 which corresponds to a reduction close to 3.6% and 7% respectively. Finally, results are detailed by province and land type.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Quartic Transmuted Exponential Distribution: Characteristics and Parameter Estimation

The scope for generating high-rank transmuted distributions has expanded beyond the cubic to achieve improved performance in baseline distributions such as those of the Gamma type. This paper develops a Quartic Rank Transmutation Distribution (QRTD), a new family of transmuted distributions with enhanced flexibility for modelling complex data problems, including those with multi-modal distributions. Application is carried out to obtain a transmuted exponential distribution (QTED). Various characteristics of the new exponential distribution are presented, including the cumulative distribution function, the reliability and hazard functions, moments, and relevant order statistics. These features support the legitimacy and robustness of the proposed QTED. Additionally, the paper identifies specific parameter ranges that exhibit notable behaviours in the new distribution and its survival quantities. The maximum likelihood estimates of parameters are described, with simulation studies indicating that their precision improves with larger sample sizes. The performance of the QTED is found to be superior to existing lower-rank cubic and quadratic transmuted exponential distributions based on information criteria using real lifetime data. The applications demonstrate that the high-rank transmutation map could be instrumental in obtaining new transmutations of other relevant distributions with improved performance. This development signifies a major advancement in the field of probability distributions, offering more sophisticated tools for statisticians and researchers to model and analyse complex data patterns more accurately and effectively. Thus, the QRTD and its applications hold significant promise for future research and practical implementations in various statistical and applied fields.

Bias-Correction Methods for the Unit Exponential Distribution and Applications

The bias of the maximum likelihood estimator can cause a considerable estimation error if the sample size is small. To reduce the bias of the maximum likelihood estimator under the small sample situation, the maximum likelihood and parametric bootstrap bias-correction methods are proposed in this study to obtain more reliable maximum likelihood estimators of the unit exponential distribution parameters. The procedure to implement the bias-corrected maximum likelihood estimation method is derived analytically, and the steps to obtain the bias-corrected bootstrap estimators are presented. The simulation results show that the proposed maximum likelihood bootstrap bias-correction method can significantly reduce the bias and mean squared error of the maximum likelihood estimators for most of the parameter combinations in the simulation study. A soil moisture data set and a numerical example are used for illustration.

Review on Distributions Generated Using Trigonometric Functions

In the present scenario, researchers are interested in developing new distributions based on trigonometric functions to model data. Previously, data modelling was commonly done with standard and existing distribution functions. Recent research has demonstrated that trigonometrically transformed functions can be used to model a variety of data types, including data from the medical, psychiatric, industrial, and economic sectors as well as studies on life time data, observations on seasonal patterns, and data from other fields. This review article discusses some trigonometric distributions and their applications across a range of fields. Using an actual dataset, a comparison analysis of a few selected models is conducted. Maximum likelihood estimates of the model parameters and various information criteria measures are obtained, and goodness of fit tests are performed for comparison.

Reliability estimation of s-out-of-k system with Kumaraswamy distribution based on partially constant stress accelerated life tests

In reliability theory, the reliability inference for s-out-of-k systems holds significant importance. In this paper, we explore the estimation of reliability for s-out-of-k systems based on partially constant stress accelerated life tests. Assume that the latent failure times of the components follow the Kumaraswamy distribution. Maximum likelihood estimates for the unknown parameters are established, and their uniqueness is demonstrated. In addition, confidence intervals for the unknown parameters are constructed using the covariance matrix. Confidence intervals for the reliability functions are determined by the Delta method, while Bootstrap intervals are provided for comparison purposes. Subsequently, Bayesian point and interval estimates based on MCMC techniques considering different loss functions are discussed. Lastly, we conduct an extensive simulation study and analyse one real data set, which reveals that the Bayesian approach yields the best results.

Bayesian Parameter Estimation for Geometric Process with Rayleigh Distribution

The main purpose of this study is to deal with the parameter estimation problem for the geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Rayleigh. For this purpose, maximum likelihood and Bayesian parameter estimation methods are discussed. Lindley and MCMC approximation methods are used in Bayesian calculations. Additionally, a novel method called the Modified-Lindley approximation has been proposed as an alternative to the Lindley approximation. An extensive simulation study was conducted to compare the performances of the prediction methods. Finally, a real data set is analyzed for illustrative purposes.

Behaviour Analysis of Modeling and Model Evaluating Methods in System Identification for a Multiprocess Station

Systems are designed to perform specific task by giving certain input which produces the required output in an orderly manner known as process. The input, output, and the state variables should be known that will help in interacting with the system. The relation between these variables can be brought out by building a model that resembles or expresses the original performance of the system. The parameters of the model are estimated using the least squares approximation, maximum likelihood, maximum log-likelihood, and Bayesian parameter estimation methods by utilizing the experimental data from the multiprocess station. The selected parameters are converted to nine different transfer function models that represent the given dynamic system. The models framed are analyzed by the criterion curve technique using seven criterion functions evaluating the fitness of the model. Order of the model is found from Hankel matrix representation methods such as singular value decomposition and determinant method. Response of the models is compared with the original response to choose the best fit model by calculating ISE standard. All the above methods are used to model the system without physical and theoretical laws which is known as system identification.

Applications of nature-inspired metaheuristic algorithms for tackling optimization problems across disciplines

Nature-inspired metaheuristic algorithms are important components of artificial intelligence, and are increasingly used across disciplines to tackle various types of challenging optimization problems. This paper demonstrates the usefulness of such algorithms for solving a variety of challenging optimization problems in statistics using a nature-inspired metaheuristic algorithm called competitive swarm optimizer with mutated agents (CSO-MA). This algorithm was proposed by one of the authors and its superior performance relative to many of its competitors had been demonstrated in earlier work and again in this paper. The main goal of this paper is to show a typical nature-inspired metaheuristic algorithmi, like CSO-MA, is efficient for tackling many different types of optimization problems in statistics. Our applications are new and include finding maximum likelihood estimates of parameters in a single cell generalized trend model to study pseudotime in bioinformatics, estimating parameters in the commonly used Rasch model in education research, finding M-estimates for a Cox regression in a Markov renewal model, performing matrix completion tasks to impute missing data for a two compartment model, and selecting variables optimally in an ecology problem in China. To further demonstrate the flexibility of metaheuristics, we also find an optimal design for a car refueling experiment in the auto industry using a logistic model with multiple interacting factors. In addition, we show that metaheuristics can sometimes outperform optimization algorithms commonly used in statistics.

COST AND RETURN ANALYSIS OF IRRIGATED UPLAND TOMATO (Lycopersicon Esculentum) PRODUCTION IN GOMBE STATE, NIGERIA

The study examined the cost and return analysis of irrigated upland tomato (Lycopersicon Esculentum) production, in Gombe State, Nigeria. Multi-stage random sampling procedure was used to select 459 respondents. Data were collected using structured questionnaire and were analysed using descriptive and inferential statistics. The result revealed that Most of the types of farms implements use for dry season tomato production in the study areas were tractors (32.4%), hand tools (37.6%) and animal traction (30.0%). Capital for dry season tomato production, the respondents in Dukku, Nafada, funakaye and Yamaltu Deba LGA respondents had a mean capital of N212,747.36, N183,662.55, N261,540.30, and N204,785.956 respectively. The findings also revealed the result of maximum likelihood estimate of parameters of the stochastic frontier production function of dry season tomato farming in the study area reveal that farm size (P< 0.001), labour (P< 0.001), farming experience (P< 0.001), and capital (P< 0.001) were the major factors statistically affecting the respondents rice production at 1%. The analysis of dry season maximum likelihood estimate of the parameter of stochastic frontier cost of tomato production showed that ploughing (P< 0.001), transplanting (P< 0.001), harvesting (P< 0.001) and capital (P< 0.001) were the determinant of cost of output of the respondents. The result further reveals the costs and return of dry season tomato farming in the study area depicts that the total variable cost of production was ₦2,384,894 with a total cost of production which is the total variable plus the fixed cost of ₦3,266,031. The revenue realized was ₦4,070,558. The net farm income for the dry season tomato farmers for one hectare in the study area was ₦804,529.23. The profit index of the respondents was ₦1.7. The major constraints to the irrigated upland tomato production were inadequate water for irrigation (23.0%), lack of irrigation facilities (20.7%), heat stress and drought (27.2%) preventing maximum productions in the study area. In addition, constraints to irrigated upland tomato farming in the study area includes high cost of irrigation services (26.0%), high-cost fertilizer (19.3%), high cost of pesticide (20.9%) and high cost of insecticide (18.2%). It was concluded that the respondents are operating at a lost and again the labour cost looking at the high cost incurred from production which clearly shows that there was over utilization of resources especially in the area of labour cost. The study made the following recommendations; efforts should be made to improve access to adequate water for irrigation purposes, there is a need to provide modern irrigation facilities to farmers to improve water management and efficiency, there is a need to promote value addition in tomato farming through processing and packaging to increase the shelf life of tomatoes and reduce post-harvest losses and farmers should be encouraged to formed cooperatives to be united in one voice to have a bargaining power over labour cost and can also make request to government on their needs and access inputs and training from policy support programme to achieve optimal yield.

Functional mixtures-of-experts

AbstractWe consider the statistical analysis of heterogeneous data for prediction, in situations where the observations include functions, typically time series. We extend the modeling with mixtures-of-experts (ME), as a framework of choice in modeling heterogeneity in data for prediction with vectorial observations, to this functional data analysis context. We first present a new family of ME models, named functional ME (FME), in which the predictors are potentially noisy observations, from entire functions. Furthermore, the data generating process of the predictor and the real response, is governed by a hidden discrete variable representing an unknown partition. Second, by imposing sparsity on derivatives of the underlying functional parameters via Lasso-like regularizations, we provide sparse and interpretable functional representations of the FME models called iFME. We develop dedicated expectation–maximization algorithms for Lasso-like regularized maximum-likelihood parameter estimation strategies to fit the models. The proposed models and algorithms are studied in simulated scenarios and in applications to two real data sets, and the obtained results demonstrate their performance in accurately capturing complex nonlinear relationships and in clustering the heterogeneous regression data.

Mixtures of regressions using matrix-variate heavy-tailed distributions

AbstractFinite mixtures of regressions (FMRs) are powerful clustering devices used in many regression-type analyses. Unfortunately, real data often present atypical observations that make the commonly adopted normality assumption of the mixture components inadequate. Thus, to robustify the FMR approach in a matrix-variate framework, we introduce ten FMRs based on the matrix-variate t and contaminated normal distributions. Furthermore, once one of our models is estimated and the observations are assigned to the groups, different procedures can be used for the detection of the atypical points in the data. An ECM algorithm is outlined for maximum likelihood parameter estimation. By using simulated data, we show the negative consequences (in terms of parameter estimates and inferred classification) of the wrong normality assumption in the presence of heavy-tailed clusters or noisy matrices. Such issues are properly addressed by our models instead. Additionally, over the same data, the atypical points detection procedures are also investigated. A real-data analysis concerning the relationship between greenhouse gas emissions and their determinants is conducted, and the behavior of our models in the presence of heterogeneity and atypical observations is discussed.

Inference for Parameters of Exponential Distribution under Combined Type II Progressive Hybrid Censoring Scheme

In recent years, various forms of progressive hybrid censoring schemes (PHCS) have gained significant traction in survival and reliability analysis studies due to their versatility. However, these PHCS variants are often characterized by complexity stemming from the multitude of parameters involved in their specification. Consequently, the primary objective of this paper is to propose a unified approach termed combined type II progressive hybrid censoring scheme (ComT2PHCS) capable of encompassing several existing PHCS variations. Our analysis focuses specifically on the exponential distribution (ExDist). Bayesian inference techniques are employed to estimate the parameters of the ExDist under the ComT2PHCS. Additionally, we conduct fundamental distributional analyses and likelihood inference procedures. We derive the conditional moment-generating function (CondMGF) of maximum likelihood estimator (MLE) for parameters of the ExDist under ComT2PHCS. Further, we use CondMGF for the distribution of MLE for parameters of ExDist under ComT2PHCS. Finally, we provide an illustrative example to elucidate the inference methods derived in this paper.

Replica analysis of overfitting in regression models for time to event data: the impact of censoring

We use statistical mechanics techniques, viz. the replica method, to model the effect of censoring on overfitting in Cox’s proportional hazards model, the dominant regression method for time-to-event data. In the overfitting regime, Maximum Likelihood (ML) parameter estimators are known to be biased already for small values of the ratio of the number of covariates over the number of samples. The inclusion of censoring was avoided in previous overfitting analyses for mathematical convenience, but is vital to make any theory applicable to real-world medical data, where censoring is ubiquitous. Upon constructing efficient algorithms for solving the new (and more complex) Replica Symmetric (RS) equations and comparing the solutions with numerical simulation data, we find excellent agreement, even for large censoring rates. We then address the practical problem of using the theory to correct the biased ML estimators without knowledge of the data-generating distribution. This is achieved via a novel numerical algorithm that self-consistently approximates all relevant parameters of the data generating distribution while simultaneously solving the RS equations. We investigate numerically the statistics of the corrected estimators, and show that the proposed new algorithm indeed succeeds in removing the bias of the ML estimators, for both the association parameters and for the cumulative hazard.

Two-Stage Limited-Information Estimation for Structural Equation Models of Round-Robin Variables

We propose and demonstrate a new two-stage maximum likelihood estimator for parameters of a social relations structural equation model (SR-SEM) using estimated summary statistics (Σ^) as data, as well as uncertainty about Σ^ to obtain robust inferential statistics. The SR-SEM is a generalization of a traditional SEM for round-robin data, which have a dyadic network structure (i.e., each group member responds to or interacts with each other member). Our two-stage estimator is developed using similar logic as previous two-stage estimators for SEM, developed for application to multilevel data and multiple imputations of missing data. We demonstrate out estimator on a publicly available data set from a 2018 publication about social mimicry. We employ Markov chain Monte Carlo estimation of Σ^ in Stage 1, implemented using the R package rstan. In Stage 2, the posterior mean estimates of Σ^ are used as input data to estimate SEM parameters with the R package lavaan. The posterior covariance matrix of estimated Σ^ is also calculated so that lavaan can use it to calculate robust standard errors and test statistics. Results are compared to full-information maximum likelihood (FIML) estimation of SR-SEM parameters using the R package srm. We discuss how differences between estimators highlight the need for future research to establish best practices under realistic conditions (e.g., how to specify empirical Bayes priors in Stage 1), as well as extensions that would make 2-stage estimation particularly advantageous over single-stage FIML.

Inference on order restricted means of inverse Gaussian populations under heteroscedasticity

The hypothesis testing problem of homogeneity of k(≥2) inverse Gaussian means against ordered alternatives is studied when nuisance or scale-like parameters are unknown and unequal. The maximum likelihood estimators (MLEs) of means and scale-like parameters are obtained when means satisfy some simple order restrictions and scale-like parameters are unknown and unequal. An iterative algorithm is proposed for finding these estimators. It has been proved that under a specific condition, the proposed algorithm converges to the true MLEs uniquely. A likelihood ratio test and two simultaneous tests are proposed. Further, an algorithm for finding the MLEs of parameters is given when means are equal but unknown. Using the estimators, the likelihood ratio test is developed for testing against ordered alternative means. Using the asymptotic distribution, the asymptotic likelihood ratio test is proposed. However, for small samples, it does not perform well. Hence, a parametric bootstrap likelihood ratio test (PB LRT) is proposed. Therefore, the asymptotic validity of the bootstrap procedure has been shown. Using the Box-type approximation method, test statistics are developed for the two-sample problem of equality of means when scale-like parameters are heterogeneous. Using these, two PB-based heuristic tests are proposed. Asymptotic null distributions are derived and PB accuracy is also developed. Two asymptotic tests are also proposed using the asymptotic null distributions. To get the critical points and test statistics of the three PB tests and two asymptotic tests, an ‘R’ package is developed and shared on GitHub. Applications of the tests are illustrated using real data.

Parameter estimation with reluctant quantum walks: a maximum likelihood approach

The parametric maximum likelihood estimation problem is addressed in the context of quantum walk theory for quantum walks on the lattice of integers. A coin action is presented, with the real parameter θ to be estimated identified with the angular argument of an orthogonal reshuffling matrix. We provide analytic results for the probability distribution for a quantum walker to be displaced by d units from its initial position after k steps. For k large, we show that the likelihood is sharply peaked at a displacement determined by the ratio d/k which is correlated with the reshuffling parameter θ. We suggest that this ‘reluctant walker’ behaviour provides the framework for maximum likelihood estimation analysis, allowing for robust parameter estimation of θ via return probabilities of closed evolution loops and quantum measurements of the position of quantum walker with ‘reluctance index’ r = d/k.

Interval estimation of relative risks for combined unilateral and bilateral correlated data

ABSTRACT Measurements are generally collected as unilateral or bilateral data in clinical trials, epidemiology, or observational studies. For example, in ophthalmology studies, the primary outcome is often obtained from one eye or both eyes of an individual. In medical studies, the relative risk is usually the parameter of interest and is commonly used. In this article, we develop three confidence intervals for the relative risk for combined unilateral and bilateral correlated data under the equal dependence assumption. The proposed confidence intervals are based on maximum likelihood estimates of parameters derived using the Fisher scoring method. Simulation studies are conducted to evaluate the performance of proposed confidence intervals with respect to the empirical coverage probability, the mean interval width, and the ratio of mesial non-coverage probability to the distal non-coverage probability. We also compare the proposed methods with the confidence interval based on the method of variance estimates recovery and the confidence interval obtained from the modified Poisson regression model with correlated binary data. We recommend the score confidence interval for general applications because it best controls converge probabilities at the 95% level with reasonable mean interval width. We illustrate the methods with a real-world example.

Stochastic DDM with regime–switching process

This paper introduces a stochastic dividend discount model with regime–switching and conditional heteroscedasticity. We obtain formulas, corresponding to a default probability and $ \ell $-th moment of a random stock price of a company. Next, we connect option pricing, hedging, and equity–linked life insurance products to the stochastic dividend discount model. Also, the paper provides maximum likelihood estimators of parameters of the stochastic dividend discount model. Finally, we present numerical results for the equity–linked life insurance products in the case of Walmart.