Maximum Likelihood Estimates Of Parameters Research Articles

Statistical Inference of Weighted Exponential Distribution under Joint Progressive Type-II Censoring

The weighted exponential distribution is a promising skewed distribution in the life-testing experiment. The joint progressive type-II censoring (JPC) scheme is an effective approach to reducing costs. In this paper, we consider the estimates of parameters of the weighted exponential distribution with the JPC data. Two populations, whose scale parameters are the same but the shape parameters of which are different, are chosen to be studied. We first evaluate the parameters with the maximum likelihood method. Because the maximum likelihood estimates of parameters cannot be obtained in closed form, we apply the Newton–Raphson method in this part. Bayesian estimates and the corresponding credible intervals under the squared error loss function are computed by using the Markov Chain Monte Carlo method. After that, we use the bootstrap method to calculate the associated confidence intervals of the unknown parameters. A simulation has been performed to test the feasibility of the above methods and real data analysis is also provided for illustrative purposes.

MLE Evolution Equation for Fractional Diffusions and Berry-Esseen Inequality of Stochastic Gradient Descent Algorithm for American Option

We study recursive parameter estimation in fractional diffusion processes. First, stability and asymptotic properties of the global maximum likelihood estimator (MLE) of the drift parameter are obtained under some regularity conditions. Then we obtain an evolution equation for the MLE of the drift parameter in nonhomogeneous fractional stochastic differential equation (fSDE) driven by fractional Brownian motion. This equation is then modified to yield an algorithm which is consistent, asymptotically efficient and converges to the MLE. The gradient and Newton type algorithm are firstorder approximations. Finally we study the Berry-Esseen inequality for stochastic gradient descent in continuous time (SGDCT) algorithm for American option. We compare it with Longstaff-Schwartz regression based method.

MULTILEVEL NON-LINIER REGRESSION FOR REPEATED MEASURMENT DATA AS STUDY OF PEANUT GROWTH

Peanut is one of the most important legume commodities in Indonesia. In its implementation, a lot of research has been done related to this plant. However, in studies conducted by growth models, it is very rarely studied. Therefore, researchers are interested in modeling the growth of peanuts. One of the models that can be used is a multilevel regression model for the case of repeated measurement data. Multilevel regression was chosen because it is considered to provide more information than other regression models. On the other hand, the nonlinear model was chosen based on the tendency of the initial plot of the data obtained. The research method used is a case study in the study of peanut growth. This study aims to build the best model based on the tested model. The Restricted Estimator Maximum Likelihood (REML) parameter estimation method was chosen because it is considered to have unbiased parameter estimates. The best model is based on the lowest Akaike Information Criterion (AIC) generated from a predetermined model. The results obtained indicate that the multilevel parabolic regression model is the model with the best AIC size. In addition, it was found that there was an Interclass Correlation (ICC) of 81.19% which indicated a difference in variability between levels.

Optimal Test Plan of Step-Stress Model of Alpha Power Weibull Lifetimes under Progressively Type-II Censored Samples

In this study, the estimation of the unknown parameters of an alpha power Weibull (APW) distribution using the concept of an optimal strategy for the step-stress accelerated life testing (SSALT) is investigated from both classical and Bayesian viewpoints. We used progressive type-II censoring and accelerated life testing to reduce testing time and costs, and we used a cumulative exposure model to examine the impact of various stress levels. A log-linear relation between the scale parameter of the APW distribution and the stress model has been proposed. Maximum likelihood estimators for model parameters, as well as approximation and bootstrap confidence intervals (CIs), were calculated. Bayesian estimation of the parameter model was obtained under symmetric and asymmetric loss functions. An optimal test plan was created under typical operating conditions by minimizing the asymptotic variance (AV) of the percentile life. The simulation study is discussed to demonstrate the model’s optimality. In addition, real-world data are evaluated to demonstrate the model’s versatility.

Type-II progressive censoring with GLM-based random removal mechanism dependent on the experimental conditions

This article presents a novel stochastic removal mechanism under Type-II progressive random censoring in which removal probabilities are allowed to be dependent on the lifetime conditions through Generalized Linear Models (GLM). These conditions potentially include failure distances (the time required to observe the next failure) or other covariate information available in the experiment. The proposed GLM-based random removal mechanism includes a set of tuning parameters that are determined by the researcher according to the possible failure distance category. These parameters allow flexible determination of the removal probabilities leading to necessary experimental cost and time reductions. To establish the proposed mechanism, the Proportional Hazard Rate (PHR) family of distributions is considered. Also, the maximum likelihood estimators of parameters and their asymptotic variances are derived for the Weibull distributed lifetime data. A simple simulation algorithm for generating Type-II progressive censoring samples with GLM-based dependent removal probabilities is also presented. The expected experiment time required to complete the life test under this censoring scheme is also investigated using the Monte Carlo integration method. Several simulation studies are conducted to evaluate and compare the performance of the proposed mechanism. A sensitivity analysis is also considered to study the effect of misspecification of removal mechanism coefficients. Finally, two real data sets are analyzed for illustrative purposes.

Statistical Inference of Spatial Econometric Models Based on Likelihood Functions

This paper is devoted to studying the probabilistic and statistical properties of maximum likelihood estimation of parameters in spatially mixed autoregressive models. When the response variable in the mixed autoregressive model obeys a continuous distribution, this paper verifies the monotonicity of the likelihood function of the mixed autoregressive model with respect to the autoregressive parameter P, which proves the existence and uniqueness of the maximum likelihood estimation of the autoregressive parameter. The results show that when the condition n > rank(x+1) is satisfied, the quasi-likelihood function of the mixed autoregressive model has a unique maximum value with probability 1 in the parameter space; when n > rank(x+1) and the regression coefficient When the matrix column is full rank, the maximum likelihood estimates of all parameters in the spatial mixed autoregressive model exist with probability 1 and are unique in the parameter space. In order to detect the strong influence points and abnormal points in the spatial mixed autoregressive model, this paper uses the first derivative method in the local influence analysis to obtain the statistics of the strong influence points and abnormal points in the spatial mixed autoregressive model in the form of variance perturbation. Simulation studies have shown that the first derivative of the maximum likelihood estimate of variance in the spatial mixed autoregressive model should be chosen to obtain the test statistic, which can effectively avoid the mearing and masking effects that often occur and are difficult to handle in local influence analysis. . As application and verification, this paper analyzes a real data to show that the conclusions obtained in this paper are reliable and practical.

Bayesian Estimation of Land Deformation Combining Persistent and Distributed Scatterers

Persistent Scatterer Interferometry (PSI) has been widely used for monitoring land deformation in urban areas with millimeter accuracy. In natural terrain, combining persistent scatterers (PSs) and distributed scatterers (DSs) to jointly estimate deformation, such as SqueeSAR, can enhance PSI results for denser and better coverage. However, the phase quality of a large number of DSs is far inferior to that of PSs, which deteriorates the deformation measurement accuracy. To solve the contradiction between measurement accuracy and coverage, a Bayesian estimation method of land deformation combining PSs and DSs is proposed in this paper. First, a two-level network is introduced into the traditional PSI to deal with PSs and DSs. In the first-level network, the Maximum Likelihood Estimation (MLE) of deformation parameters at PSs and high-quality DSs is obtained accurately. In the secondary-level network, the remaining DSs are connected to the nearest PSs or high-quality DSs, and the deformation parameters are estimated by Maximum A Posteriori (MAP) based on Bayesian theory. Due to the poor phase quality of the remaining DSs, MAP can achieve better estimation results than the MLE based on the spatial correlation of the deformation field. Simulation and Sentinel-1A satellite data results verified the feasibility and reliability of the proposed method. Regularized by the spatial deformation field derived from the high-quality PSs and DSs, the proposed method is expected to achieve robust results even in low-coherence areas, such as rural areas, vegetation coverage areas, or deserts.

Prevalence and associated factors of neonatal mortality in Ethiopia

Neonatal mortality is the death of a live-born baby within the first 28 days of birth. For the selected households, neonatal mortality was collected from children aged 0–28 days and women aged 15–49. The neonatal period is a significant 4-week period in human life because it carries a greater mortality risk. To identify the determinant factors of neonatal mortality in Ethiopia based on EDHS 2016 data with the application of count regression models. In this study, all neonates in Ethiopia were born within the 5 years preceding EDHS 2016 of the source population in the selected EAs from September to December 2015. Count regression models were used to analyze the data. A total of 10,641 live-born neonates within the previous 5 years of EDHS 2016 had neonatal mortality of women aged 15–49, which was considered in the study to be 7193. The data were found to have excess zeros (96.6%), and the variance (0.052) was higher than its mean (0.04). The count regression model (ZINB) was best fitted to the data with maximum likelihood parameter estimation methods. The average neonatal mortality difference in multiple births was increased by IRR = 8.53 times compared with a single birth. The average number of neonatal deaths experienced during breastfeeding was lower (IRR = 0.38) than that experienced by mothers who did not experience breastfeeding their child. The average neonatal mortality difference in rural residences was increased by IRR = 3.99 times compared to urban mothers' residences. In this study, the prevalence of Neonatal mortality in Ethiopia was higher. For selected ZINB count regression models of explanatory variables, such as multiple birth types, having rural residence factors of neonatal mortality increased the risk of death. However, having early breastfeeding, a female household head, and antenatal visits (1–4) and (5–10) during pregnancy decrease the risk of neonatal death.

Comparison of Statistical Methods for Claims Reserve Estimation Using R Language

Stochastic methods of reserves estimation serve to assess the technical provisions of outstanding claims and forecast cash payments of claims in the coming years. The chain ladder model developed by Mack is the more prevalent model. The main deficiency in the chain-ladder model is that the chain-ladder model depends on the last observation on the diagonal. If this last observation is an outlier, this outlier will be projected to the ultimate claim. One of the possibilities to smooth outliers on the last observed diagonal is to robustify such observations, making use of the maximum likelihood estimation along with the common Loss Development Factor (LDF) curve fitting and Cape Cod (CC) techniques. This paper aims to highlight the advantages of using these methods for the best estimate of claims reserves in the Domestic Motor Third Party Liability portfolio. The maximum–likelihood parameter estimation and Chi-square test, are used to specify the probability distribution that best fits the data. Using the Standard Chain Ladder method, LDF, and CC method the claims reserve is calculated based on the run-off triangles of paid claims or the run-off triangles of the incurred claims. Many times, the projections based on the paid claims are different than the projections based on the incurred claims. The solution for this problem is the Munich Chain Ladder method.

Single-cell generalized trend model (scGTM): a flexible and interpretable model of gene expression trend along cell pseudotime.

Modeling single-cell gene expression trends along cell pseudotime is a crucial analysis for exploring biological processes. Most existing methods rely on nonparametric regression models for their flexibility; however, nonparametric models often provide trends too complex to interpret. Other existing methods use interpretable but restrictive models. Since model interpretability and flexibility are both indispensable for understanding biological processes, the single-cell field needs a model that improves the interpretability and largely maintains the flexibility of nonparametric regression models. Here, we propose the single-cell generalized trend model (scGTM) for capturing a gene's expression trend, which may be monotone, hill-shaped or valley-shaped, along cell pseudotime. The scGTM has three advantages: (i) it can capture non-monotonic trends that are easy to interpret, (ii) its parameters are biologically interpretable and trend informative, and (iii) it can flexibly accommodate common distributions for modeling gene expression counts. To tackle the complex optimization problems, we use the particle swarm optimization algorithm to find the constrained maximum likelihood estimates for the scGTM parameters. As an application, we analyze several single-cell gene expression datasets using the scGTM and show that scGTM can capture interpretable gene expression trends along cell pseudotime and reveal molecular insights underlying biological processes. The Python package scGTM is open-access and available at https://github.com/ElvisCuiHan/scGTM. Supplementary data are available at Bioinformatics online.

Model-based clustering via skewed matrix-variate cluster-weighted models

Cluster-weighted models (CWMs) extend finite mixtures of regressions (FMRs) in order to allow the distribution of covariates to contribute to the clustering process. In this article, we introduce 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates in each cluster to be modelled by one of four existing skewed matrix-variate distributions or the matrix-variate normal distribution. Endowed with greater flexibility, our matrix-variate CWMs are able to handle this kind of data in a more suitable manner. As a by-product, the four skewed matrix-variate FMRs are also introduced. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Lastly, our matrix-variate CWMs, along with the matrix-variate normal CWM and matrix-variate FMRs, are applied to two real datasets for illustrative purposes.

Recursive inversion models for permutations

We develop a new exponential family model for permutations that can capture hierarchical structure in preferences, and that has the well known Mallows models as a subclass. The Recursive Inversions Model (RIM), unlike most distributions over permutations of n items, has a flexible structure, represented by a binary tree. We describe how to compute marginals in the RIM, including the partition function, in closed form. Further we introduce methods for the Maximum Likelihood estimation of parameters and structure search for this model. We demonstrate that this added flexibility both improves predictive performance and enables a deeper understanding of collections of permutations.

Inference of a competing risks model with partially observed failure causes under improved adaptive type-II progressive censoring

In this paper, a competing risks model is analyzed based on improved adaptive type-II progressive censored sample (IAT-II PCS). Two independent competing causes of failures are considered. It is assumed that lifetimes of the competing causes of failure follow exponential distributions with different means. Maximum likelihood estimators (MLEs) for the unknown model parameters are obtained. Using asymptotic normality property of MLE, the asymptotic confidence intervals are constructed. Existence and uniqueness properties of the MLEs are studied. Further, bootstrap confidence intervals are computed. The Bayes estimators are obtained under symmetric and asymmetric loss functions with non-informative and informative priors. For informative priors, independent gamma distributions are considered. Highest posterior density (HPD) credible intervals are obtained. A Monte Carlo simulation study is carried out to compare performance of the established estimates. Furthermore, three different optimality criteria are proposed to obtain the optimal censoring plan. Finally, a real-life data set is considered for illustrative purposes.

Probabilistic study of snow loads on flat roofs considering the effects of wind at representative sites in China

Based on meteorological data, this paper simulates the evolution of ground and roof snow loads over time in winter. The snowmelt model is used to simulate the ground snow load, and the snow transport model is employed in cooperation with the snowmelt model to simulate the flat roof snow load. Through the simulations, samples of annual maximum ground snow loads, annual maximum roof snow loads, and ground-to-roof conversion factors for at least 60 years at 50 representative snowy sites in China are obtained. First, the statistical characteristics of the above samples are analysed. Then, the probabilistic models with different parameter estimation methods are used to fit the samples, and each model's goodness of fit is evaluated by using K-S test, Akaike information criterion (AIC), and Q-Q plot. The results show that the Generalized Extreme Value (GEV) distribution based on Maximum Likelihood Estimation (MLE) of parameters is the preferred probabilistic model for snow loads both on the ground and roofs. However, none of the models selected in this paper can accurately describe the ground-to-roof conversion factor. Therefore, this paper recommends using the R-year ground and roof snow loads obtained by the probabilistic model for calculating the ground-to-roof conversion factor. Finally, the ground-to-roof conversion factor is found to be quite scattered for the most of the studied sites, thus it could be more reasonable to use site-specific values estimated by the method proposed in this paper rather than a uniform value in consideration of safety and economy of the structure.

An extended type I half-logistic family of distributions: Properties, applications and different method of estimations

Abstract We propose a new family of continuous distributions with two shape parameters called the Extended type I(ET1HL-G). We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics. Then we study half-logistic distribution as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two data sets are illustration the purposes.

Multi‐Tracer Groundwater Dating in Southern Oman Using Bayesian Modeling

AbstractIn the scope of assessing aquifer systems in areas where freshwater is scarce, estimation of transit times is a vital step to quantify the effect of groundwater abstraction. Transit time distributions of different shapes, mean residence times, and contributions are used to represent the hydrogeological conditions in aquifer systems and are typically inferred from measured tracer concentrations by inverse modeling. In this study, a multi‐tracer sampling campaign was conducted in the Salalah Plain in Southern Oman including CFCs, SF6, 39Ar, 14C, and 4He. Based on the data of three tracers, a two‐component Dispersion Model (DMmix) and a nonparametric model with six age bins were assumed and evaluated using Bayesian statistics. In a Markov Chain Monte Carlo approach, the maximum likelihood parameter estimates and their uncertainties were determined. Model performance was assessed using Bayes factor and leave‐one‐out cross‐validation. Both models suggest that the groundwater in the Salalah Plain is composed of a very young component below 30 yr and a very old component beyond 1,000 yr, with the nonparametric model performing slightly better than the DMmix model. All wells except one exhibit reasonable goodness of fit. Our results support the relevance of Bayesian modeling in hydrology and the potential of nonparametric models for an adequate representation of aquifer dynamics.

Bootstrap Methods for Claims Reserving: R Language Approach

Bootstrap methods have been used by actuaries for a long time to predict future claims cash flows and their variability. This work aims to illustrate the use of bootstrap methods in practice, taking as an example the claims development data of the personal accident portfolio from the largest insurance company in Albania, over a period of 10 years. It is not the objective of this work to provide a theoretical analysis of the bootstrap methods, rather, this work focuses on highlighting the benefits of using bootstrap methods to predict the distribution of future claims development, and estimate the standard error, for a better risk assessment of liabilities within insurance companies. This work is divided into two well-differentiated phases: the first is to select the theoretical probability distribution that best fits the available claims dataset. Comparison of distributions is facilitated by the possibilities offered by the R programming languages. Both, the maximum likelihood parameter estimation method and the chi-square goddess goodness of fit test, are used to specify the probability distribution that best fits the data, among a family of predefined distributions. The results show that the Gamma distribution better describes the claim development data. The next phase is to use bootstrap methods, based on the selected distribution, to estimate the ultimate value of claims, the claims reserve, and their standard error.

On Odd Perks-G Class of Distributions: Properties, Regression Model, Discretization, Bayesian and Non-Bayesian Estimation, and Applications

In this paper, we present a new univariate flexible generator of distributions, namely, the odd Perks-G class. Some special models in this class are introduced. The quantile function (QFUN), ordinary and incomplete moments (MOMs), generating function (GFUN), moments of residual and reversed residual lifetimes (RLT), and four different types of entropy are all structural aspects of the proposed family that hold for any baseline model. Maximum likelihood (ML) and maximum product spacing (MPS) estimates of the model parameters are given. Bayesian estimates of the model parameters are obtained. We also present a novel log-location-scale regression model based on the odd Perks–Weibull distribution. Due to the significance of the odd Perks-G family and the survival discretization method, both are used to introduce the discrete odd Perks-G family, a novel discrete distribution class. Real-world data sets are used to emphasize the importance and applicability of the proposed models.

Prediction of electricity prices for non‐regulated markets based on a power transformed mean reverting process

AbstractThe electricity price time series for non‐regulated markets presents two basic properties: (a) a mean reversion trend around a constant or deterministic function and (b) a price process with high volatility and marginal, conditional, and asymptotic distributions skewed to the right. We propose a microeconomic‐based model for the dynamics of electricity prices that provides a satisfactory explanation of the stylized features observed in non‐regulated electricity markets. The suggested model is based on a power transformation of the Ornstein–Uhlenbeck process, which accounts for the unobserved demand process. This means that the model requires only the spot price of the electricity price to be available. Parameter estimates were obtained using the maximum likelihood method. This approach was implemented for data from the Alberta electricity market in Canada, with satisfactory results in terms of residual analysis and forecasting. The success rate of predicting the price jump sign (upward and downward) was approximately . Moreover, the statistical properties for the maximum likelihood parameter estimators are shown via a Monte Carlo simulation of the electricity price time series.

Power Series Generalised Power Weibull Class of Distributions

The power series generalised power Weibull class of distributions were developed in this study by compounding the power series family of distributions and the generalised power Weibull distribution. The statistical properties of this new class were derived. Maximum likelihood parameter estimators were derived for the parameters of the power series generalised power Weibull class of distributions. Four sub-families of distributions were developed from the power series generalised power Weibull class of distribution; the generalised power Weibull geometric distribution, generalised power Weibull Poisson distribution, generalised power Weibull binomial distribution and the generalised power Weibull logarithmic distribution. The hazard rate and probability density function plots of the four sub families of distributions showed that, they can model both monotonic and non-monotonic lifetime data. Monte Carlo simulations performed on these sub-distributions showed that, their estimators were consistent estimators. Application of these sub-distributions to failure data from air conditioning system of an aircraft showed that, the generalised power Weibull geometric distribution provides a better fit to the data. Also, the generalised power Weibull Poisson distribution provides a better fit to the data on service times of 63 aircraft.