maximum likelihood estimation

In this paper‎, ‎based on progressively Type-II censored samples‎, ‎the maximum likelihood and Bayes estimators are derived for some lifetime parameters‎. ‎ ‎In the Bayesian framework‎, ‎the point estimations of unknown parameters under both symmetric and asymmetric loss functions are discussed‎. ‎The Bayesian estimations have been obtained using the conjugate prior and discrete priors for the shape and scale parameters‎, ‎respectively‎. ‎We also provide Bayes prediction intervals for the times to failure of units censored in multiple stages in a progressively censored sample‎. ‎Finally‎, ‎two numerical examples are presented to illustrate the results.

This paper examines a novel extension of the geometric distribution characterized by two parameters, that is not created based on discretizing existing continuous models. This model, due to its analytical form of the cumulative distribution function and simple structure, can be of interest from mathematical perspectives, particularly in cases where the analysis of stochastic orders is desired. In addition, it is a suitable candidate for analyzing monotone hazard rate discrete data, in view of the fact that its hazard rate function exhibits monotonicity in both increasing and decreasing directions. Additionally, the behavior of the survival function of residual lifetime is briefly addressed. The parameters of the distribution are estimated using the maximum likelihood method, and a real-world data set is scrutinized to assess the distribution's adequacy in providing satisfactory fits.

Circular data are measured in angles or directions. In many cases of sampling, instead of a random sample, we deal with a weighted model. In such sampling, observations are provided throughout with a positive function, weight function. This article deals with weight distributions in circular data. According to von Mises distrinution is the most widely used distribution for modeling circular data, maximum likelihood estimation of parameters in weighted von Mises distributions is investigated. In a simulation study, different weights are compared in the Van Mises circular distribution.

The two-parameter discrete Weibull distribution is an important model especially in reliability studies when the data are reported on a discrete scale‎. ‎The hazard rate function of a discrete Weibull distribution is monotonically increasing and decreasing‎. ‎The present paper provides a family of parametric discrete distributions which is an infinite mixture of exponentiated discrete Weibull distributions‎, ‎and versatile in fitting increasing‎, ‎decreasing‎, ‎and bathtub-shaped failure rate models to different discrete life-test data‎. ‎Some important distributional properties of the model such as the moments‎, ‎order statistics‎, ‎and infinite divisibility are investigated and the parameters of the distribution are estimated by the maximum likelihood method‎. ‎In addition‎, ‎a real data set is analyzed to show the effectiveness of the model‎. ‎Finally we conclude the paper.

‎ In this paper, we introduce a new extension of the XLindley distribution, called the exponentiated new XLindley ‎distribution.‎ The new model has an increasing or bathtub-shaped hazard rate function, making it suitable for modeling real-life phenomena. We study important properties of the new model, such as ‎the ‎moments, moment generating function, incomplete moments, mean deviations from the mean and ‎the ‎median, Bonferroni and Lorenz curves, mean residual life function, Rényi entropy, order statistics, and ‎‎‎k‎‎-record values. We also address the estimation of parameters using ‎the ‎maximum likelihood and bootstrap methods. A Monte Carlo simulation study is conducted to evaluate the estimators discussed in the paper. Additionally, we analyze two real data applications, including rainfall and COVID-19 data sets, to demonstrate the applicability and flexibility of the new distribution. Our results show that the new model fits the data sets better than ‎several ‎other recognized or recently introduced distributions, based on ‎some ‎well-known ‎goodness-‎of-‎fit ‎criteria.

The novel corona virus (covid-19) spread quickly from person to another and one of the basic aspects of the country management has been to prevent the spread of this disease. So the prediction of its expansion is very important. In such matters, the estimation of new cases and deaths in covid-19 has been considered by researchers. we propose an estimation of the statistical model for predicting the new cases and the new deaths by using the vector autoregressive (VAR) model with the multivariate skew normal (MSN) distribution for the asymmetric shocks and predict the samples data. The maximum likelihood (ML) method is applied to estimation of this model for the weekly data of the new cases and the new deaths of covid-19. Data are taken from World Health Organization (WHO) from March 2020 until March 2023 Iran country. The performance of the model is evaluated with the Akaike and the Bayesian information criterions and the mean absolute prediction error (MAPE) is interpreted.

In this paper‎, ‎we consider the estimation of the stress-strength reliability of a coherent system‎. ‎The distributions of stress and strength random variables are the members of a general class of distributions‎. ‎For a series-parallel system‎, ‎the reliability of the stress-strength model is estimated using the maximum likelihood estimation‎, ‎asymptotic confidence interval‎, ‎uniformly minimum variance unbiased estimation‎, ‎and Bayes estimation‎. ‎Also‎, ‎simulation studies are performed‎, ‎and two real data sets are analyzed.

We consider the problem of model selection in vector autoregressive model with Normal innovation. Tests such as Vuong's and Cox's tests are provided for order and model selection, i.e. for selecting the order and a suitable subset of regressors, in vector autoregressive model. We propose a test as a modified log-likelihood ratio test for selecting subsets of regressors. The Europe oil prices, Brent, and the real gross domestic product, GDP, data are considered as real data. Since the Brent data does Granger-cause the GDP data, so we suggest the vector autoregressive model and select optimal model based on the model selection test. The analysis provides analytic results show that the Vuong's, Cox's and proposed test are the appropriate test for order and model selection for vector autoregressive models with Normal innovation. In simulation study, the power of proposed test at least is as good as the power of Vuong's test.

A new generalized family of distributions called the Weibull Odd Burr III-G is introduced using the T-X transformation technique. Some of useful mathematical and statistical properties such as the hazard function, quantile function, moments, probability weighted moments, Rényi entropy, order statistics and stochastic orders are derived. The method of maximum likelihood estimation is used to estimate the model parameters. The usefulness of these family of distributions is demonstrated via simulated experiments and its special cases are applied to real life data sets to illustrate flexibility.

A new generalized distribution called the gamma odd power generalized Weibull-G family of distributions is developed and studied. Some special models of the new family of distribution are explored. Statistical properties of the new family of distributions including the quantile function, ordinary and incomplete moments, probability weighted moments, stochastic ordering, distribution of the order statistics, and Rényi entropy are presented. The maximum likelihood method is used for estimating model parameters, and Monte Carlo simulation is conducted to examine the performance of the model. The flexibility of the new family of distributions is demonstrated by means of two applications to real data sets.

In this article, the autoregressive model of order one with exponential innovations is considered. The maximum likelihood and Bayes estimators of the autoregression parameter, under squared error loss function with non-informative prior are examined. A simulation study is conducted to compare the behavior of the estimators via their relative bias and risks. Moreover, a real data example is presented.

Lifetime data has several applications in different fields such as Biology and Engineering. Failures for this type of data may occur due to several causes. In real world, causes of failures are interacting together which violates the independency assumption. Once dependency between failures is satisfied, bivariate families should be used to analyze the data. In literature, the majority of studies handle the case when data come from one source. However, in real life, data could come from different sources. One way to analyze data from different sources together and reduce the time and cost of the experiment is joint type-II censoring. To the best of our knowledge, joint type-II censoring was not yet derived using bivariate lifetime distributions. In this paper, we derive the likelihood function of joint type-II censoring using bivariate family in the presence of dependent competing risks. A simulation study is performed and two real datasets are analyzed.

Finding new families of distributions has become a popular tool in statistical research. In this article, we introduce a new flexible four-parameter discrete model based on the Marshall-Olkin approach, namely, the discrete Kumaraswamy Marshall-Olkin exponential distribution. The proposed distribution can be viewed as another generalization of the geometric distribution and enfolds some important distributions as special cases. Some properties of the new distribution are derived. The model parameters are estimated by the maximum likelihood method, with validation through a complete simulation study. The usefulness of the new model is illustrated via count-type real data sets.

We introduce a flexible lifetime distribution called Burr III-Inverse Weibull (BIII-IW). The new proposed distribution has well-known sub-models. The BIII-IW density function includes exponential, left-skewed, right-skewed and symmetrical shapes. The BIII-IW model’s failure rate can be monotone and non-monotone depending on the parameter values. To show the importance of the BIII-IW distribution, we establish various mathematical properties such as random number generator, ordinary moments, conditional moments, residual life functions, reliability measures and characterizations. We address the maximum likelihood estimates (MLE) for the BIII-IW parameters and estimate the precision of the maximum likelihood estimators via a simulation study. We consider applications to two COVID-19 data sets to illustrate the potential of the BIII-IW model.

In their conventional formulation as linear mixed models (LMMs) and generalized LMMs (GLMMs), a commonly indispensable assumption in settings involving longitudinal non-Gaussian data is that the longitudinal observations from subjects are conditionally independent, given subject-specific random effects. Although conventional Gaussian LMMs are able to incorporate conditional dependence of longitudinal observations, they require that the data are, or some transformation of them is, Gaussian, a serious limitation in a wide variety of practical applications. Here, we introduce the class of Gaussian copula conditional regression models (GCCRMs) as flexible alternatives to conventional LMMs and GLMMs. One advantage of GCCRMs is that they extend conventional LMMs and GLMMs in a way that reduces to conventional LMMs, when the data are Gaussian, and to conventional GLMMs, when conditional independence is assumed. We implement likelihood analysis of GCCRMs using existing software and statistical packages and evaluate the finite-sample performance of maximum likelihood estimates for GCCRM empirically via simulations vis-a-vis the `naivechr('39') likelihood analys is that incorrectly assumes conditionally independent longitudinal data. Our results show that the `naivechr('39') analysis yields estimates with possibly severe bias and incorrect standard errors, leading to misleading inferences. We use bolus count data on patientschr('39') controlled analgesia comparing dosing regimes and data on serum creatinine from a renal graft study to illustrate the applications of GCCRMs.