Behavior Of Maximum Likelihood Estimator Research Articles

Flexible Extension of the Lomax Distribution for Asymmetric Data under Different Failure Rate Profiles: Characteristics with Applications for Failure Modeling and Service Times for Aircraft Windshields

A novel four-parameter lifetime Lomax model is presented and investigated within the scope of this paper. The failure rate of the innovative model can be “monotonically decreasing failure rate,” “monotonically increasing failure rate,” or “constant failure rate,” and the density of the model can be “asymmetric right skewed,” “symmetric,” “asymmetric left skewed,” or “uniform density”. The new density is expressed as a blend of the Lomax densities that have been multiplied by an exponent. New bivariate Lomax types were created for our research. The maximum likelihood technique was utilized. We performed simulated experiments for the purpose of evaluating the finite sample behavior of maximum likelihood estimators, using “biases” and “mean squared errors” as our primary metrics of analysis. The novel distribution was evaluated based on a number of pertinent Lomax models, including Lomax extensions that were generated on the basis of odd log-logistic, Kumaraswamy, beta, gamma, and Topp–Leone families, among others. The newly developed extension demonstrated its relevance by predicting the service and failure times of datasets pertaining to aircraft windshields.

Kth-order equilibrium Weibull distribution: properties, simulation, and its applications

A generalization of some well-known probability distributions (viz., Weibull, exponential, gamma, Maxwell, and Chi-square) is introduced using the concept of weighted probability distributions. The introduced model is named Kth-order equilibrium Weibull distribution (KEWD). Various properties of the new distribution are studied in detail. One of the important properties of KEWD is that its hazard rate shows increasing, decreasing, and constant behavior. It is also shown that the new model belongs to the log exponential family. Various ordering relations of the KEWD are studied in comparison with the baseline model, i.e. Weibull distribution. Parameters are estimated using the concept of the maximum likelihood estimation technique. Using the Anderson–Darling test statistic, a simulation study is carried out to analyze the asymptotic normality behavior of maximum likelihood estimators. The behaviors of bias and mean square error are observed with the increase in sample size. The applications of new distribution are illustrated through its fitting to some incorporated real-life data sets. Finally, a comparison is made between KEWD, its sub-models, and some other introduced extensions of Weibull distribution in terms of fitting using the Akaike information criterion (AIC).

DUS-neutrosophic multivariate inverse Weibull distribution: properties and applications

AbstractThe existing DUS-multivariate inverse Weibull distribution under classical statistics can be applied when all observations in the data are imprecise. In this paper, we introduce DUS-neutrosophic multivariate inverse Weibull distribution that can be used when the observations in the data are imprecise or in intervals. We derive some statistical properties and functions of DUS-neutrosophic multivariate inverse Weibull distribution. We also discuss the maximum likelihood estimation method for estimating the parameters. Monte-Carlo simulation study is performed to study the behavior of maximum likelihood estimates. We compare the efficiency of the proposed DUS-neutrosophic multivariate inverse Weibull distribution with the existing distributions under classical statistics. From the comparison, it is found that the proposed DUS-neutrosophic multivariate inverse Weibull distribution provides smaller values of Akaike’s information criteria and Bayesian information criteria than the existing distributions under classical statistics. The proposed study can be extended for other statistical distributions as future research.

Discrete Logistic Exponential Distribution with Application

In this paper, a new two-parameter discrete logistic exponential distribution is proposed based on the survival discretization approach. Some statistical properties are derived, and it is found that the proposed model can be used to discuss several kinds of failure rates including unimodal, bathtub, and increasing-shaped. Moreover, it can be utilized effectively to model under- and over-dispersed data. The distribution parameters are estimated using the maximum likelihood technique. The behavior of maximum likelihood estimators is assessed utilizing a comprehensive simulation study. In the end, two real data are analyzed to show the usefulness of the new discrete distribution.

Asymptotic behavior of maximum likelihood estimators for Ornstein–Uhlenbeck process with large linear drift

In this paper, we study the asymptotic behavior of maximum likelihood estimators for Ornstein–Uhlenbeck process with large linear drift [Formula: see text], [Formula: see text], where [Formula: see text], and [Formula: see text] is a given standard Brownian motion. The law of iterated logarithm, consistency and asymptotic distributions of the estimators are discussed based on the continuous observation [Formula: see text] as [Formula: see text].

On bivariate modelling using a new statistical distribution with homogeneous marginals

AbstractRecent studies have shown that the univariate exponentiated Teissier distribution is an effective model for analysing rainfall data. A bivariate probability distribution offers greater insight into a process like a flood or drought than a univariate method for analysing the characteristics of environmental occurrences. As a result, in this study we developed a bivariate exponentiated Teissier distribution. The marginals of the proposed distribution model follow the exponentiated Teissier distribution. The cumulative distribution function of this bivariate model consists of continuous and singular components. The proposed model is shown to have several fundamental statistical properties, including joint cumulative distribution, joint density, marginals, and conditional distributions. Additionally, some copula functions are examined. The proposed model's parameter estimations are derived using the maximum likelihood method. Furthermore, the asymptotic confidence intervals are given for these estimates. A Monte Carlo simulation study with various sample sizes was performed to evaluate the behaviour of maximum likelihood estimates. Finally, we used football and flood datasets to demonstrate how the proposed model may be employed.

The Slashed Power Half-Normal Distribution with Applications

In this paper, an extension of the power half-normal (PHN) distribution is introduced. This new model is built on the application of slash methodology for positive random variables. The result is a distribution with greater kurtosis than the PHN; i.e., its right tail is heavier than the PHN distribution. Its probability density, survival and hazard rate function are studied, and moments, skewness and kurtosis coefficientes are obtained, along with relevant properties of interest in reliability. It is also proven that the new model can be expressed as the scale mixture of a PHN and a uniform distribution. Moreover, the new model holds the PHN distribution as a limit case when the new parameter tends to infinity. The parameters in the model are estimated by the method of moments and maximum likelihood. A simulation study is given to illustrate the good behavior of maximum likelihood estimators. Two real applications to survival and fatigue fracture data are included, in which our proposal outperforms other models.

Inverse Kies distribution: Properties and applications

In this paper, a new class of distribution namely “the inverse Kies distribution” is proposed and some of its important aspects are studied by deriving expressions for its percentile measures, raw moments, reliability measures etc. The maximum likelihood estimation of the parameters of the distribution is discussed and the distribution has been fitted to certain real life data sets. The asymptotic behaviour of maximum likelihood estimators of the parameters of the distribution are also studied by using simulated data sets. Keywords: Beta generalised inverse Weibull distribution, Failure rate, Generalised inverse Weibull distribution, Maximum likelihood estimation, Model selection, Percentile measures

Weighted Lindley frailty model: estimation and application to lung cancer data.

In this paper, we propose a novel frailty model for modeling unobserved heterogeneity present in survival data. Our model is derived by using a weighted Lindley distribution as the frailty distribution. The respective frailty distribution has a simple Laplace transform function which is useful to obtain marginal survival and hazard functions. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions. A classical inference procedure based on the maximum likelihood method is presented. Extensive simulation studies are further performed to verify the behavior of maximum likelihood estimators under different proportions of right-censoring and to assess the performance of the likelihood ratio test to detect unobserved heterogeneity in different sample sizes. Finally, to demonstrate the applicability of the proposed model, we use it to analyze a medical dataset from a population-based study of incident cases of lung cancer diagnosed in the state of São Paulo, Brazil.

Properties, Inference and Applications of Inverse Power Two-Parameter Weighted Lindley Distribution

We proposed “a new extension of three-parametric distribution” called the inverse power two-parameter weighted Lindley (IPWL) distribution capable of modeling a upside-down bathtub hazard rate function. This distribution is studied to get basic structural properties such as reliability measures, moments, inverse moments and its related measures. Simulation studies are done to present the performance and behavior of maximum likelihood estimates of the IPWL distribution parameters. Finally, we perform goodness of fit measures and test statistics using a real data set to show the performance of the new distribution.

Consequences of Model Misspecification for Maximum Likelihood Estimation with Missing Data

Researchers are often faced with the challenge of developing statistical models with incomplete data. Exacerbating this situation is the possibility that either the researcher’s complete-data model or the model of the missing-data mechanism is misspecified. In this article, we create a formal theoretical framework for developing statistical models and detecting model misspecification in the presence of incomplete data where maximum likelihood estimates are obtained by maximizing the observable-data likelihood function when the missing-data mechanism is assumed ignorable. First, we provide sufficient regularity conditions on the researcher’s complete-data model to characterize the asymptotic behavior of maximum likelihood estimates in the simultaneous presence of both missing data and model misspecification. These results are then used to derive robust hypothesis testing methods for possibly misspecified models in the presence of Missing at Random (MAR) or Missing Not at Random (MNAR) missing data. Second, we introduce a method for the detection of model misspecification in missing data problems using recently developed Generalized Information Matrix Tests (GIMT). Third, we identify regularity conditions for the Missing Information Principle (MIP) to hold in the presence of model misspecification so as to provide useful computational covariance matrix estimation formulas. Fourth, we provide regularity conditions that ensure the observable-data expected negative log-likelihood function is convex in the presence of partially observable data when the amount of missingness is sufficiently small and the complete-data likelihood is convex. Fifth, we show that when the researcher has correctly specified a complete-data model with a convex negative likelihood function and an ignorable missing-data mechanism, then its strict local minimizer is the true parameter value for the complete-data model when the amount of missingness is sufficiently small. Our results thus provide new robust estimation, inference, and specification analysis methods for developing statistical models with incomplete data.

The counterintuitive impact of responses and response times on parameter estimates in the drift diffusion model

Given a drift diffusion model with unknown drift and boundary parameters, we analyse the behaviour of maximum likelihood estimates with respect to changes of responses and response times. It is shown analytically that a single fast response time can dominate the estimation in that no matter how many correct answers a test taker provides, the estimate of the drift (ability) parameter decreases to zero. In addition, it is shown that although higher drift rates imply shorter response times, the reverse implication does not hold for the estimates: shorter response times can decrease the drift rate estimate. In the light of these analytical results, we illustrate the actual impact of the findings in a small simulation for a mental rotation test. The method of analysis outlined is applicable to a broader range of models, and we emphasize the need to further check currently used reaction time models within this framework.

Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on (−1,∞). We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.

Maximum Likelihood Estimation for a Smooth Gaussian Random Field Model

Gaussian processes are commonly used for modeling the output of deterministic computer models. We consider the behavior of maximum likelihood estimators (MLEs) of parameters of the commonly used squared exponential covariance function when the computer model has some simple deterministic form. We prove that for regularly spaced observations on the line, the MLE of the scale parameter converges to zero if the computer model is a constant function and diverges to infinity for linear functions. When observing successive derivatives of a $p$th order monomial at zero, we find the asymptotic orders of the MLE of the scale parameter for all $p\ge 0$. For some commonly used test functions, we compare the MLE with cross validation in a prediction problem and explore the joint estimation of range and scale parameters. The correlation matrix is nearly numerically singular even when the sample size is moderate. To overcome numerical difficulties, we perform exact computation by making use of exact results for the correlation matrix and restricting ourselves to parameter values and test functions that yield rational correlations and function values at the observation locations. We also consider the common approach of including a nugget effect to deal with the numerical difficulties, and explore its consequences on model fitting and prediction.

On some properties of Kies distribution

In this paper we study some important aspects of the Kies distribution by deriving expressions for its percentile measures, raw moments, reliability measures etc. The maximum likelihood estimation of the parameters of the distribution have been discussed and the distribution has been fitted to certain real life data sets. The asymptotic behaviour of maximum likelihood estimators are also studied by using simulated data sets.

Tracking interval for selecting between non-nested models: An investigation for Type II right censored data

In this paper, we consider the setting where the observed data is incomplete. For the general situation where the number of gaps as well as the number of unobserved values in some gaps go to infinity, the asymptotic behavior of maximum likelihood estimator is not clear. We derive and investigate the asymptotic properties of maximum likelihood estimator under censorship and drive a statistic for testing the null hypothesis that the proposed non-nested models are equally close to the true model against the alternative hypothesis that one model is closer when we are faced with a life-time situation. Furthermore rewrite a normalization of a difference of Akaike criterion for estimating the difference of expected Kullback–Leibler risk between the distributions in two different models.

Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model

Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced, and it is shown that, under each member of this class, the hypothesis H0: λ = 0 is invariant, where λ is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini’s centred parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum likelihood estimators is given under H0, and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.

A Finite‐State Continuous‐Time Approach for Inferring Regional Migration and Mortality Rates from Archival Tagging and Conventional Tag‐Recovery Experiments

Spatially structured population dynamics models are important management tools for harvested, highly mobile species and although conventional tag recovery experiments remain useful for estimation of various demographic parameters of these models, archival tagging experiments are becoming an important data source for analyzing migratory behavior of mobile marine species. We provide a likelihood-based approach for estimating the regional migration and mortality rate parameters intrinsic to these models that may use information obtained from conventional tag recovery and archival tagging experiments. Specifically, we assume that the regional location and survival of animals through time is a finite-state continuous-time stochastic process. The stochastic process is the basis of probability models for observations provided by the different types of tags. Results from application to simulated tagging experiments for western Atlantic bluefin tuna show that maximum likelihood estimators based on archival tagging observations and corresponding confidence intervals perform similar to conventional tagging observations for a given number of tag releases and releasing tags in each region can improve the behavior of maximum likelihood estimators regardless of tag type. We provide an example application with Atlantic bluefin tuna released with conventional tags in 1990-1992.

Comparison of Laparoscopic and Open Partial Nephrectomy for Tumor in a Solitary Kidney

We compared the postoperative and renal functional outcomes of patients undergoing open or laparoscopic partial nephrectomy for tumor in a solitary functioning kidney. Between 1999 and 2006, 169 open and 30 laparoscopic partial nephrectomies were performed for 7 cm or smaller tumors in a solitary functioning kidney. Data were collected in an institutional review board approved registry and median followup was 2.0 years. Preoperative and postoperative glomerular filtration rates were estimated with the abbreviated Modification of Diet in Renal Disease equation. By 3 months after open or laparoscopic partial nephrectomy, the glomerular filtration rate decreased by 21% or 28%, respectively (p = 0.24). Postoperative dialysis was required acutely after 1 open partial nephrectomy (0.6%) and 3 laparoscopic partial nephrectomies (10%, p = 0.01), and dialysis dependent end stage renal failure within 1 year occurred after 1 open partial nephrectomy (0.6%) and 2 laparoscopic partial nephrectomies (6.6%, p = 0.06). In multivariate analysis warm ischemia time was 9 minutes longer (p <0.0001) and the chance of postoperative complications was 2.54-fold higher (p <0.05) with laparoscopic partial nephrectomy. Longer warm ischemia time (more than 20 minutes) and preoperative glomerular filtration rate were associated with poorer postoperative glomerular filtration rate in multivariate analysis. Notwithstanding the association with warm ischemia time, the surgical approach itself was not an independent predictor of postoperative glomerular filtration rate (p = 0.77). While laparoscopic partial nephrectomy is technically feasible for tumor in a solitary kidney, warm ischemia time was longer and complication rates higher compared with open partial nephrectomy. In addition, although average loss of renal function at 3 months is equivalent (after accounting for warm ischemia time), a greater proportion of patients required dialysis temporarily or permanently after laparoscopic partial nephrectomy in this initial series. Therefore, open partial nephrectomy may be the preferred nephron sparing approach at this time for these patients at high risk for chronic kidney disease.

Likelihood based inference for monotone response models

The behavior of maximum likelihood estimates (MLEs) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate $n^{1/3}$ (slower than the usual $\sqrt{n}$ rate) with a non-Gaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLEs and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer $\chi^2$ but can be explicitly characterized in terms of a functional of Brownian motion. Applications of the main results are presented and potential extensions discussed.