Common Scale Parameter Research Articles

Classical and Bayesian inferences for two Topp-Leone models under joint progressive Type-II censoring

Recently, the progressive Type-II censoring has been extended to a more general censoring scheme, called joint progressive Type-II censoring, which studies the lifetimes of two or more populations simultaneously. In this article, we consider the joint Type-II progressive censoring scheme for two populations when their lifetimes follow Topp-Leone models with unknown common scale parameter but different shape parameters. Classical and Bayesian inferences are studied. Expectation-Maximization (EM) algorithm is implemented for obtaining the maximum likelihood estimators (MLEs) and the associated asymptotic confidence intervals of the unknown parameters. Bayesian inferences are discussed based on a beta-gamma prior for the shape parameters and an incomplete inverse gamma prior for the scale parameter. Importance sampling method is proposed to approximate the Bayes estimates. The associated Bayesian credible intervals are also established. Monte Carlo simulation study is performed to compare the performance of the proposed methods. Finally, a real data set representing two different algorithms for estimating unit capacity factors is analyzed for illustrative purposes.

Erlang mixture modeling for Poisson process intensities

We develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function which is modeled nonparametrically with a gamma process prior. Such model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting. The prior model structure supports general shapes for the point process intensity function, and it also enables effective handling of the Poisson process likelihood normalizing term resulting in efficient posterior simulation. The Erlang mixture modeling approach is further elaborated to develop an inference method for spatial Poisson processes. The methodology is examined relative to existing Bayesian nonparametric modeling approaches, including empirical comparison with Gaussian process prior based models, and is illustrated with synthetic and real data examples.

A control chart for time truncated life tests using type-ii generalized log logistic distribution

In this paper, an attribute control chart is aimed when the lifetime of the item follows Type-II generalized log-logistic distribution (TGLLD) under a time truncated life test assuming that the common scale parameter is known. Average run length (ARL) is used to assess the performance of the aimed control chart. Simulation technique is developed to present the performance of the control charts at a specified average run length (ARL), shift constant and for different parametric values of shape and scale parameters, sample size. The results are illustrated with live data example.

Inference for constant-stress Weibull competing risks model under generalized progressive hybrid censoring

In this paper, accelerated life test is considered when the latent failure times follow Weibull competing risks models. When both scale and shape parameters are nonconstant and affected by operating stress, inference is discussed for a constant-stress model under a generalized progressive hybrid censoring. Maximum likelihood estimates together with existence and uniqueness are established, and the approximate confidence intervals for unknown parameters are proposed based on the asymptotic theory. Further, when the distributions of failure causes have common shape or scale parameters, associated point and interval estimates are also proposed. In addition, to compare the equivalence of the parameters of different Weibull competing risks, likelihood ratio tests for interested hypotheses are presented for complementary. Finally, extensive numerical studies and a real-life example are provided for illustrations.

Bayesian predictive density estimation for a Chi-squared model using information from a normal observation with unknown mean and variance

In this paper, we consider the problem of estimating the density function of a Chi-squared variable on the basis of observations of another Chi-squared variable and a normal variable under the Kullback–Leibler divergence. We assume that these variables have a common unknown scale parameter and that the mean of the normal variable is also unknown. We compare the risk functions of two Bayesian predictive densities: one with respect to a hierarchical shrinkage prior and the other based on a noninformative prior. The hierarchical Bayesian predictive density depends on the normal variable while the Bayesian predictive density based on the noninformative prior does not. Sufficient conditions for the former to dominate the latter are obtained. These predictive densities are compared by simulation.

Inference on P[Y < X] for Geometric Extreme Exponential Distribution

Geometric Extreme Exponential Distribution (GEE) is one of the statistical models that can be useful in fitting and describing lifetime data. In this paper, the problem of estimation of the reliability R = P(Y < X) when X and Y are independent GEE random variables with common scale parameter but different shape parameters has been considered. The probability R = P(Y < X) is also known as stress-strength reliability parameter and demonstrates the case where a component has stress X and is subjected to strength Y. The reliability R = P(Y < X) has applications in engineering, finance and biomedical sciences. We present the maximum likelihood estimator of R and study its asymptotic behavior. We first study the asymptotic distribution of the maximum likelihood estimators of the GEE parameters. We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution. A bootstrap confidence interval for R is also presented. Monte Carlo simulations are performed to assess he performance of the proposed estimation method and validity of the confidence interval. We found that the performance of the maximum likelihood estimator and also the bootstrap confidence interval is satisfactory even for small sample sizes. Analysis of a dataset has been given for illustrative purposes.

Stress-Strength Reliability for P(T < X < Z) using Dagum Distribution

In this paper, the reliability formula of the stress-strength model is derived for probability P(T<X<Z) of a component having strength X falling between two stresses T and Z, that these stresses and strength variables have Dagum distribution with unknown shape parameters and known common scale parameters. Three methods for estimating the Dagum parameters are discussed which are the Maximum Likelihood, Regression method and Method of Moment, and the comparison between these estimation method based on a simulation study by the mean square error criteria for each of the small, medium and large samples. The most important conclusion is that this comparison confirms that the performance of the maximum likelihood estimator works better for most of the experiments studied.

Statistical inference of reliability in multicomponent stress strength model for pareto distribution based on upper record values

ABSTRACT In this article, inferences about the multicomponent stress strength reliability are drawn under the assumption that strength and stress follow independent Pareto distribution with different shapes and common scale parameter under the setup of upper record values. The maximum likelihood estimator, Bayes estimator under-squared error and Linear exponential loss function, of multicomponent stress-strength reliability are constructed with corresponding highest posterior density interval for unknown For known uniformly minimum variance unbiased estimator and asymptotic distribution of multicomponent stress-strength reliability with asymptotic confidence interval is discussed. Also, various Bootstrap confidence intervals are constructed. A simulation study is conducted to numerically compare the performances of various estimators of multicomponent stress-strength reliability. Finally, a real life example is presented to show the applications of derived results in real life scenarios. Several researchers have attempted such problems under different distributions e.g., [1] considered the Weibull distribution for estimation of multicomponent stress-strength reliability, [2] considered generalized Rayleigh distribution for the reliability estimation of multicomponent stress-strength setup.

Estimating the Reliability of a Component between Two Stresses from Gompertz-Frechet Model

In this paper, the reliability of the stress-strength model is derived for probability P( &lt;X&lt; ) of a component strength X between two stresses , , when X and , are independent Gompertz Fréchet distribution with unknown and known shape parameters and common known scale parameters. Different methods used to estimate R and Gompertz Fréchet distribution parameters which are [Maximum Likelihood, Least square, Weighted Least square, Regression and Ranked set sampling methods], and the comparison between these estimations by simulation study based on mean square error criteria. The comparison confirms that the performance of the maximum likelihood estimator works better than the other estimators.

Reliability inference for stress-strength model based on inverted exponential Rayleigh distribution under progressive Type-II censored data

In this paper, stress-strength model is studied for an inverted exponential Rayleigh distribution (IERD) when the latent failure times are progressively Type-II censored. When both strength and stress random variables follow common IERD scale parameters, the maximum likelihood estimate of stress-strength reliability (SSR) is established and the associated approximate confidence interval is also constructed using the asymptotic distribution theory and delta method. By constructing pivotal quantities, another alternative generalized estimates for SSR are also proposed for comparison. Moreover, when there are arbitrary strength and stress parameters, likelihood and generalized pivotal based estimates are also presented. In addition, testing problem is gave for comparing the equality of different strength and stress parameters. Finally, simulation study and a real data example are provided for illustration.

On a progressively censored competing risks data from Gompertz distribution

We study a competing risks model using Gompertz distribution under progressive Type-II censoring when probability distributions of failure causes are identically distributed with common scale and different shape parameters. Maximum likelihood estimates (MLEs) of these parameters are obtained and their uniqueness and existence behavior are also discussed. The asymptotic intervals are derived from the observed Fisher information matrix. We compare the performance of all the estimators numerically using simulations. Analysis of a real data set is presented as well. We further determine optimal censoring scheme using expected Fisher information matrix. The design parameters are selected based on suitable measures like cost-based and variance-based criteria functions. Finally, we discuss single and multi-objective optimization approaches to find optimal censoring schemes.

Jaya algorithm in estimation of P[X &amp;gt; Y] for two parameter Weibull distribution

&lt;abstract&gt; &lt;p&gt;Jaya algorithm is a highly effective recent metaheuristic technique. This article presents a simple, precise, and faster method to estimate stress strength reliability for a two-parameter, Weibull distribution with common scale parameters but different shape parameters. The three most widely used estimation methods, namely the maximum likelihood estimation, least squares, and weighted least squares have been used, and their comparative analysis in estimating reliability has been presented. The simulation studies are carried out with different parameters and sample sizes to validate the proposed methodology. The technique is also applied to real-life data to demonstrate its implementation. The results show that the proposed methodology's reliability estimates are close to the actual values and proceeds closer as the sample size increases for all estimation methods. Jaya algorithm with maximum likelihood estimation outperforms the other methods regarding the bias and mean squared error.&lt;/p&gt; &lt;/abstract&gt;

Estimating Common Scale Parameter of Two Logistic Populations: A Bayesian Study

Estimation under equality restrictions is an age old problem and has been considered by several researchers in the past due to practical applications and theoretical challenges involved in it. Particularly, the problem has been extensively studied from classical as well as decision theoretic point of view when the underlying distribution is normal. In this paper, we consider the problem when the underlying distribution is non-normal, say, logistic. Specifically, estimation of the common scale parameter of two logistic populations has been considered when the location parameters are unknown. It is observed that closed forms of the maximum likelihood estimators (MLEs) for the associated parameters do not exist. Using certain numerical techniques the MLEs have been derived. The asymptotic confidence intervals have been derived numerically too, as these also depend on the MLEs. Approximate Bayes estimators are proposed using non-informative as well as conjugate priors with respect to the squared error (SE) and the LINEX loss functions. A simulation study has been conducted to evaluate the proposed estimators and compare their performances through mean squared error (MSE) and bias. Finally, two real life examples have been considered in order to show the potential applications of the proposed model and illustrate the method of estimation.

Fitting multivariate Erlang mixtures to data: A roughness penalty approach

The class of multivariate Erlang mixtures with common scale parameter has many desirable properties and has widely been used in insurance loss modeling. The parameters of a multivariate Erlang mixture are normally estimated using an expectation–maximization (EM) algorithm as shown in Lee and Lin (2012) and Verbelen et al. (2016). However, when fitting the mixture to data of high dimension, the fitted density surface is often not smooth (with deep peaks and valleys) and the tail fitting may also be rather unsatisfactory. In this paper, we propose a generalized expectation conditional maximization (GECM) algorithm that maximizes a penalized likelihood with a proposed roughness penalty. The roughness penalty is based on integrated squared second derivative of the density function of aggregate data, which is used in functional data analysis. We illustrate the performance of the proposed method through some numerical experiments and real data applications.

Multicomponent stress-strength system reliability estimation for generalized exponential-poison distribution

This study deals with (s -out of- k) Multicomponent Stress (Y) and Strength (X) System Reliability Estimation. Both stress and strength assumed to have Generalized Exponential-Poisson Distribution with common and known scale parameters (θ and λ). The aim here is to estimate the unknown shape parameters (α and β) for X and Y respectively using two methods of estimation ML and Bayes analysis by one prior with five loss functions. Then estimate Reliability using the same methods and compared the results by Mean square error criteria from simulation study to find the best performance of the estimators. the results show that the best estimator for R(s,k) is Bayes estimator under Quadratic loss function using Gamma prior function, followed by GD, GW, GP, MLE and GS estimators, respectively.

Estimation of reliability in a multicomponent stress-strength model for inverted exponentiated Rayleigh distribution under progressive censoring

We consider estimation of the multicomponent stress-strength reliability for inverted exponentiated Rayleigh distributions under progressive Type II censoring. It is assumed that stress and strength variables follow inverted exponentiated Rayleigh distributions with a common scale parameter. Point and interval estimates of the reliability are obtained using maximum likelihood and Bayesian approaches when common parameter is unknown. Bayes estimates are derived using Lindley approximation and Markov chain Monte Carlo methods. The case of known common parameter is also considered. Then uniformly minimum variance unbiased estimator of the reliability is derived. We have also computed the exact Bayes estimates under the squared error loss function. The asymptotic and HPD intervals of the reliability are constructed under this case also. Proposed methods are compared numerically using simulations and comments are obtained. Finally, a real data set is analyzed for illustration purposes.

Multicomponent Inverse Lomax Stress-Strength Reliability

In this article we derive two reliability mathematical expressions of two kinds of s-out of -k stress-strength model systems; and . Both stress and strength are assumed to have an Inverse Lomax distribution with unknown shape parameters and a common known scale parameter. The increase and decrease in the real values of the two reliabilities are studied according to the increase and decrease in the distribution parameters. Two estimation methods are used to estimate the distribution parameters and the reliabilities, which are Maximum Likelihood and Regression. A comparison is made between the estimators based on a simulation study by the mean squared error criteria, which revealed that the maximum likelihood estimator works the best.

Minimum Volume Confidence Regions for Parameters of Exponential Distributions from Different Samples

Independent samples from different exponential distributions are available in many statistical applications, for example, in queuing or reliability models, where the random variables describe waiting times and lifetimes, respectively. When two samples are observed, inference is then carried out for two location and two scale parameters. In multivariate setups including common location and common scale parameter assumptions, we provide confidence regions for the parameters of interest, which have minimum Lebesgue measure among all those based on the usual pivotal quantities and with the same or higher confidence level; in particular, they improve in terms of area (volume) upon the standard ‘trapezoidal’ confidence regions being constructed by combining independent univariate pivot statistics. The proposed confidence regions do not require any factorization of the overall confidence level, and their calculations need simple Monte Carlo simulations, only. Although focusing on two complete samples, generalizations of the results to more than two and doubly type-II censored samples are possible.

Measuring Uncertainty Under Prior Information

There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: $(i)$ i.i.d. sampling, $(ii)$ record values, $(iii)$ Type-II censoring and $(iv)$ progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.

Optimal confidence interval for the largest exponential location parameter

A single sampling procedure is proposed for obtaining an optimal confidence interval for the largest location parameter of several k independent exponential populations in the cases of known and unknown common scale parameter, where “optimal” is defined in the sense of minimal expected interval width and best allocation at a fixed confidence. The optimal result can be applied to type II censored samples. Tables of percentage points are provided for practitioners.