ML Estimation Research Articles

Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.

Negative Binomial Three Parameter Lindley Distribution and Its Properties

Many researchers have proposed mixed distributions as one of the most important methods for obtaining new probability distributions. Several studies have shown that mixed Negative Binomial distributions fits count data better than Poisson and Negative Binomial distribution itself. In this paper, we introduce a mixed distribution by mixing the distributions of negative binomial and three Parameter Lindley distribution. This new distribution has a thick tail and may be considered as an alternative for fitting count data with over dispersion. The parameters of the new distribution are estimated using MLE method and properties studied. Special cases of the new distribution and also identified. A simulation study carried out shows that the ML estimators give the parameter estimates close to the parameter when the sample is large, that is, the bias and variance of the parameter estimates decrease with increase in sample size showing the consistent nature of the new compound distribution. The study also compares the performance of the new distribution over distributions of Poisson, Negative Binomial, Negative Binomial oneParameter Lindley Distribution, Negative Binomial two Parameter distribution, three parameter Lindley distribution using a real count over dispersed dataset and the results shows that Negative Binomial three parameter Lindley distribution gave the smallest Kolmogorov Smirnov test statistic, AIC and BIC as compared to other distributions, hence the new distribution provided a better fit compared to other distributions under study for fitting over dispersed count data.

Identifiability and estimation of possibly non-invertible SVARMA Models: The normalised canonical WHF parametrisation

This article focuses on the parametrisation, identifiability, and (quasi-) maximum likelihood (QML) estimation of possibly non-invertible structural vector autoregressive moving average (SVARMA) models. SVAR models are routinely adopted due to their well-known implementation strategy. However, for various economic and statistical reasons, multivariate SVARMA settings are often more suitable. These settings introduce complexity in the analysis, primarily due to the presence of the moving average (MA) polynomial. We propose a novel representation of the MA polynomial matrix using the Wiener–Hopf factorization (WHF). A significant advantage of the WHF is its ability to handle possible non-invertibility and thus models with informational asymmetry between economic agents and outside observers. Since solutions of Dynamic Stochastic General Equilibrium (DSGE) models often involve this informational asymmetry, SVARMA models in WHF parametrisation can be considered data-driven alternatives to DSGE models and used for their evaluation. Furthermore, we provide low-level conditions for the asymptotic normality of the (Q)ML estimator and analytic expressions for the score and information matrix. As application, we estimate the Blanchard and Quah model, and compare our results and implied impulse response function with the ones in the SVAR model by Blanchard and Quah and a non-invertible SVARMA model by Gouriéroux and co-authors. Importantly, we have implemented this novel method in a well-documented R-package, making it readily accessible for researchers and practitioners.

Characterizing Existence and Location of the ML Estimate in the Conway–Maxwell–Poisson Model

Characterizing Existence and Location of the ML Estimate in the Conway–Maxwell–Poisson Model

The application of machine learning techniques for smart irrigation systems: A systematic literature review

Smart Irrigation System is a complex concept used to control, monitor and automate the irrigation of yields by integrating artificial intelligence techniques such as Machine Learning strategies. SIS has endorsed various machine learning models. However, there has been no attempt to analyze the empirical evidence on ML models in a systematic way. Moreover, ML based SIS often face problems and raise questions that must be resolved. This article presents a systematic literature review of ML based SIS; an overview of the literature on ML is designed, revealing a premier and unbiased survey of the existing empirical research. 55 selected studies published from 2017 to 2023 and nine broadly used ML models were identified. Furthermore, four analysis aspects namely type of ML technique, estimation accuracy, model comparison, and estimation context have been outlined. The findings of this review prove the performance capability of the ML techniques endorsed within SIS. The ML techniques outperform other conventional approaches. However, the application of ML models in SIS is still limited and more effort is needed to obtain well-formed and generalizable results. To this end, and based on the outcomes obtained in this work, future guidelines have been provided to practitioners and researchers to grasp the major contributions and challenges in the state-of-the-art research.

Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.

Extraction and characterization of novel cellulose fibers from Dracaena draco plant

This study investigates the potential of Dracaena draco fiber (DDF) as a viable option for the development of biodegradable natural composites, with the objective of making a valuable contribution to the field of environmentally sustainable materials. This study presents a comprehensive investigation into the morphological, thermophysical, and mechanical properties of DDF. The findings indicate an average density of 1.2043 g/cm3, a linear density of 18.87 Tex, and moisture recovery and moisture content values of 12.88 % and 11.54 %, respectively. These results represent the first full examination of these parameters for DDF. X-ray diffraction (XRD) and Fourier transform infrared spectroscopy (FTIR) examinations offer valuable information regarding the crystallinity and functional groups present in DDF. These analyses reveal a crystallite size of 3.14 nm and a crystallinity index of 30.42 %. The TGA analysis indicates that the DDF fiber exhibits a consistent temperature of 329.5 °C. The behavior of a material can be effectively demonstrated through the exact modeling of water absorption, as exemplified by the models developed by Sikame et al. The mechanical testing results reveal that DDF has an impressive average yield strength of 553 MPa, a strain at rupture of 2.5 %, and a Young's modulus of 24.9 GPa. In this study, we utilize ML estimation and statistical analysis based on the two-parameter Weibull distribution to conduct a thorough assessment of the mechanical properties of DDF. The comparison of DDF with other natural fibers discussed in the available literature establishes its potential as a suitable reinforcement for lightweight biocomposites. This makes it a feasible option for various applications that involve thermoplastic or thermosetting polymers. The aforementioned results highlight the potential of DDF in the development of sustainable materials for a wide range of industrial applications.

Restricted Liu estimator under stochastic linear restrictions in generalized linear models: theory and applications

It is a fact that the problem of multicollinearity adversely effects the estimation of parameters not only in normal linear models but also in generalized linear models (GLMs). In this study, we propose a new estimator by imposing the stochastic restrictions on the parameters and combining the Liu estimator in GLMs to address the problem of multicollinearity. The new estimator is referred to as the stochastic restricted Liu estimator. The novel estimator’s first order approximated form (FOA) is presented to compare it to its competitors using the asymptotic mean square error criterion. Furthermore, two empirical applications using a response variable from the Poisson and Gamma distributions are studied to evaluate the performance of estimators. In addition, a simulation study is conducted by considering a response variable from Binomial distribution for comparisons. Based on simulation study and empirical applications, it is found that the performance of the proposed estimator is better than ML, Liu, and mixed estimators for the appropriate selected biasing parameter.

Bayesian Inference for the Loss Models via Mixture Priors

Constructing an accurate model for insurance losses is a challenging task. Researchers have developed various methods to model insurance losses, such as composite models. Composite models combine two distributions: one for part of the data with small and high frequencies and the other for large values with low frequencies. The purpose of this article is to consider a mixture of prior distributions for exponential–Pareto and inverse-gamma–Pareto composite models. The general formulas for the posterior distribution and the Bayes estimator of the support parameter θ are derived. It is shown that the posterior distribution is a mixture of individual posterior distributions. Analytic results and Bayesian inference based on the proposed mixture prior distribution approach are provided. Simulation studies reveal that the Bayes estimator with a mixture distribution outperforms the Bayes estimator without a mixture distribution and the ML estimator regarding their accuracies. Based on the proposed method, the insurance losses from natural events, such as floods from 2000 to 2019 in the USA, are considered. As a measure of goodness-of-fit, the Bayes factor is used to choose the best-fitted model.

Gaussian-Shaped Free-Space Optical Beam Intensity Estimation in Detector Arrays

Photon counting detector arrays are commonly used for deep space optical communication receivers operating on the principle of intensity modulation/direct detection (IM/DD). In scenarios where beam parameters can vary at the receiver due to scattering, it is important to estimate beam parameters in order to minimize the probability of error. The use of array of detectors increases the sensitivity of the receiver as compared to single photo-detector of the same size. In this paper, we present the derivation of a maximum likelihood estimator (ML) for peak optical intensity, providing both numerical and closed form expressions for the estimator. Performance of both forms of ML estimator are compared using the mean squared error (MSE) criterion and Cramer–Rao Lower Bound (CRLB) is also derived to assess the proposed estimator’s efficiency. This research contributed to the advancement of estimation techniques and has practical implications for optimizing deep space optical communication systems.

Analysis of Covid-19 data using discrete Marshall–Olkinin Length Biased Exponential: Bayesian and frequentist approach

The paper presents a novel statistical approach for analyzing the daily coronavirus case and fatality statistics. The survival discretization method was used to generate a two-parameter discrete distribution. The resulting distribution is referred to as the "Discrete Marshall–Olkin Length Biased Exponential (DMOLBE) distribution". Because of the varied forms of its probability mass and failure rate functions, the DMOLBE distribution is adaptable. We calculated the mean and variance, skewness, kurtosis, dispersion index, hazard and survival functions, and second failure rate function for the suggested distribution. The DI index demonstrates that the proposed model can represent both over-dispersed and under-dispersed data sets. We estimated the parameters of the DMOLBE distribution. The behavior of ML estimates is checked via a comprehensive simulation study. The behavior of Bayesian estimates is checked by generating 10,000 iterations of Markov chain Monte Carlo techniques, plotting the trace, and checking the proposed distribution. From simulation studies, it was observed that the bias and mean square error decreased with an increase in sample size. To show the importance and flexibility of DMOLBE distribution using two data sets about deaths due to coronavirus in China and Pakistan are analyzed. The DMOLBE distribution provides a better fit than some important discrete models namely the discrete Burr-XII, discrete Bilal, discrete Burr-Hatke, discrete Rayleigh distribution, and Poisson distributions. We conclude that the new proposed distribution works well in analyzing these data sets. The data sets used in the paper was collected from 2020 year.

Estimating a new stress–strength index for several exponential populations with a common location

In design and development of products in various industries, a key characteristic is stress–strength reliability. In this article, we consider estimation of a new stress–strength index for several exponential populations with a common location. We derive various estimators such as the maximum likelihood, the uniformly minimum variance unbiased (UMVU), and Bayes estimators. We additionally apply Brewster–Zidek technique for improving upon estimators based on UMVU or best affine equivariant estimators of scale parameters. We derive the asymptotic distribution of the ML estimator and prove that the Bayes estimators' limit under a suitable prior distribution is a generalized Bayes estimator. We then evaluate the risk performance of the obtained estimators in an extensive simulation study. Two applications are given on real data sets to illustrate the new methods. One example relates to the duration analysis and the other to a problem of comparing strengths of different fibres in jute industry.

Effects of Organic Fertilizer Substitution on the Technical Efficiency among Farmers: Evidence from Bohai Rim Region in China

Using organic fertilizer to replace chemical fertilizer is an effective way to promote sustainable agricultural development. Based on survey data of 514 farmers from five provinces and cities in the Bohai Rim region of China, collected in 2019, organic fertilizer substitution’s effects on farmers’ technical efficiency were analyzed. The PSM method is used to solve the problem of farmers’ self-selection. Considering the causal relationship between the substitution behavior of organic fertilizer and farmers’ technical efficiency, the IV-Tobit model, 2SLS, and ML estimation methods are used to solve the endogenous problem. The study found that: (1) quality and safety awareness, part-time employment level, participation in insurance, and training significantly promote using organic fertilizers to replace chemical fertilizers; (2) the technical efficiency of farmers who use organic fertilizer to substitute the chemical fertilizers is 0.625, 9.46% higher than their counterparts; and (3) the promotion effect of organic fertilizer substitution on technical efficiency has heterogeneity, which has a significant role among small-scale farmer households with fewer and low-educated family laborers. Regarding the policy implications, it is necessary to strengthen agricultural subsidies by improving the accuracy of policy formulation and implementation, fully considering the heterogeneity of farmers for implementation.

Statistical analysis of Markov switching vector autoregression models with endogenous explanatory variables

We consider multivariate Markov switching first-order autoregression models with endogenous explanatory variables, propose a joint estimation algorithm of type EM, written at vector–matrix level, to account for endogeneity, and derive matrix formulas for the ML estimators of model parameters. Then we prove the consistency of such estimators, provide matrix expressions for their asymptotic covariances, and present some tests for endogeneity. Further, a simulation study is proposed to illustrate the theoretical results and provide evidence on the usefulness of the considered model.

A Cautionary Note Regarding Multilevel Factor Score Estimates from Lavaan

To compute factor score estimates, lavaan version 0.6–12 offers the function lavPredict( ) that can not only be applied in single-level modeling but also in multilevel modeling, where characteristics of higher-level units such as working environments or team leaders are often assessed by ratings of employees. Surprisingly, the function provides results that deviate from the expected ones. Specifically, whereas the function yields correct EAP estimates of higher-level factors, the ML estimates are counterintuitive and possibly incorrect. Moreover, the function does not provide the expected standard errors. I illustrate these issues using an example from organizational research where team leaders are evaluated by their employees, and I discuss these issues from a measurement perspective.

A Principal-Component Approach to Antenna Impedance Estimation at MISO Receivers

Impedance matching between receive antenna and front-end significantly impacts channel capacity in wireless channels. To implement the optimal matching, the receiver must know the antenna impedance. This impedance varies with the antenna’s loading conditions. Techniques are derived to estimate antenna impedance, but the optimality of these techniques remains unclear. In this letter, we study antenna impedance estimation at MISO receivers over Rayleigh fading channels and derive the optimal ML estimator in closed-form. Numerical results suggest a computationally efficient, principal-components approach that estimates antenna impedance in real-time and shows sizable improvement against a reference estimator at low SNR.

On some properties of ML and REML estimators in mixed normal models with two variance components

On some properties of ML and REML estimators in mixed normal models with two variance components

A New Cure Rate Model Based on Flory–Schulz Distribution: Application to the Cancer Data

In this article a new flexible survival cure rate model is introduced by assuming that the number of competing causes of the event of interest follows the Flory–Schulz distribution and the competing causes follow the generalized truncated Nadarajah–Haghighi distribution. Parameter estimation for the proposed model is derived based on the maximum likelihood estimation method. A simulation study is performed to show the performance of the ML estimators. We discuss three real data applications related to real cancer data sets to assess the usefulness of the proposed model compared with some existing cure rate models for the sake of comparison.

Bias-Corrected Maximum Likelihood Estimation and Bayesian Inference for the Process Performance Index Using Inverse Gaussian Distribution

In this study, the estimation methods of bias-corrected maximum likelihood (BCML), bootstrap BCML (B-BCML) and Bayesian using Jeffrey’s prior distribution were proposed for the inverse Gaussian distribution with small sample cases to obtain the ML and Bayes estimators of the model parameters and the process performance index based on the lower specification process performance index. Moreover, an approximate confidence interval and the highest posterior density interval of the process performance index were established via the delta and Bayesian inference methods, respectively. To overcome the computational difficulty of sampling from the posterior distribution in Bayesian inference, the Markov chain Monte Carlo approach was used to implement the proposed Bayesian inference procedures. Monte Carlo simulations were conducted to evaluate the performance of the proposed BCML, B-BCML and Bayesian estimation methods. An example of the active repair times for an airborne communication transceiver is used for illustration.

Nonparametric REML-like Estimation in Linear Mixed Models with Uncorrelated Homoscedastic Errors

Restricted Maximum Likelihood (REML) is the most recommended approach for fitting a Linear Mixed Model (LMM) nowadays. Yet, as ML, REML suffers the drawback that it performs such a fitting by assuming normality for both the random effects and the residual errors, a dubious assumption for many real data sets. Now, there have been several attempts at trying to justify the use of the REML likelihood equations outside of the Gaussian world, with varying degrees of success. Recently, a new fitting methodology, code named 3S, was presented for LMMs with only added assumption (to the basic ones) that the residual errors are uncorrelated and homoscedastic. Specifically, the 3S-A1 variant was designed and then shown, for Gaussian LMMs, to differ only slightly from ML estimation. In this article, using the same 3S framework, we develop another iterative nonparametric estimation methodology, code named 3S-A1.RE, for the kind of LMMs just mentioned. However, we show that if the LMM is, indeed, Gaussian with i.i.d. residual errors, then the set of estimating equations defining any 3S-A1.RE iterative procedure is equivalent to the set of REML equations, but while including the nonnegativity constraints on all variance estimates, as well as positive semi-definiteness on all covariance matrices. In numerical tests on some simulated and real world clustered and longitudinal data sets, our new methods proved to be highly competitive when compared to the traditional REML in the R statistical software.