Uniformly Minimum Variance Unbiased Research Articles

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.

Estimation of P(X ≤ Y) for discrete distributions with non-identical support

Abstract The Uniformly Minimum Variance Unbiased (UMVU) and the Maximum Likelihood (ML) estimations of R = P(X ≤ Y) and the associated variance are considered for independent discrete random variables X and Y. Assuming a discrete uniform distribution for X and the distribution of Y as a member of the discrete one parameter exponential family of distributions, theoretical expressions of such quantities are derived. Similar expressions are obtained when X and Y interchange their roles and both variables are from the discrete uniform distribution. A simulation study is carried out to compare the estimators numerically. A real application based on demand-supply system data is provided.

Abundance‐ and incidence‐based estimation of total number of rare species in under‐sampled sites

AbstractQuestionsEcologists collecting field samples of biological data have a keen interest in addressing the following question: how many rare species are there in as‐yet unsurveyed additional samples? Depending on the size of a targeted additional sample, statistical models for estimating the number of rare species have not been systematically established and compared.LocationGlobal.MethodsFor fairly comparing and predicting rare‐species richness at the same sample‐size baseline, we systematically developed and compared four estimators for rarefaction and extrapolation of rare‐species richness with a given specific abundance. These four estimators included a uniformly minimum variance unbiased (UMVUE), Bayesian‐weighted, Chao‐derived unweighted and naïve estimator.ResultsAfter extensive numerical tests, for conducting rarefaction of rare‐species richness (i.e., when additional sample size was not larger than the original one) it is recommended to implement UMVUE, as it has zero bias and coverage percentage closest to 0.95. However, the performance of Bayesian‐weighted and Chao‐derived estimators is also satisfactory. By contrast, for conducting extrapolation of rare‐species richness (i.e., when the additional sample size is larger than the original one), the Bayesian‐weighted estimator is recommended, as it has the best performance among the four estimators (here UMVUE is inapplicable).ConclusionsThere was no absolute winner, as the different estimators have their own merits and are recommended under different settings. When conducting rarefaction of rare‐species richness, UMVUE, which has the highest accuracy, is recommended. By contrast, when conducting extrapolation of rare‐species richness, the Bayesian‐weighted estimator is recommended, as it has the overall best performance. To facilitate the potential application in the comparison and prediction of rare‐species diversity using rarefaction and extrapolation techniques, an R package (fRSE) has been developed; it is freely distributed at the following URL: https://github.com/ecomol/fRSE.

Lifetime performance evaluation model based on quick response thinking

In practice, lifetime performance index CL has been a method commonly applied to the evaluation of quality performance. L is the upper or lower limit of the specification. The product lifetime distribution is mostly abnormal distribution. This study explored that the lifetime of commodities comes from exponential distribution. Complete data collection is the primary goal of analysis. However, the censoring type is one of the most commonly used methods due to considerations of manpower and material cost or the timeliness of product launch. This study adopted Type-II right censoring to find out the uniformly minimum variance unbiased (UMVU) estimator of the lifetime performance index CL and its probability density function. Afterward this study obtained the 100×(1-α)% confidence interval of the lifetime performance index CL as well as created the uniformly most powerful (UMP) test and the power of the test for the product lifetime performance index. Last, this study came up with a numerical example to demonstrate the suggested method as well as the application of the model.

Bayesian shrinkage approaches to unbalanced problems of estimation and prediction on the basis of negative multinomial samples

In this paper, we treat estimation and prediction problems where negative multinomial variables are observed and, in particular, consider unbalanced settings. First, the problem of estimating multiple negative multinomial parameter vectors under the standardized squared error loss is treated and a new empirical Bayes estimator which dominates the uniformly minimum variance unbiased (UMVU) estimator under suitable conditions is derived. Next, we consider estimation of the joint predictive density of several multinomial tables under the Kullback–Leibler divergence and obtain a sufficient condition under which the Bayesian predictive density with respect to a hierarchical shrinkage prior dominates the Bayesian predictive density with respect to the Jeffreys prior. Finally, our proposed Bayesian estimator and predictive density give risk improvements in simulations and our methods are applied to real data resulting from the inverse sampling method.

Classical and Bayesian inference on traffic intensity of multiserver Markovian queuing system

In this paper we consider multi-server single queue system in which inter-arrival and service times are exponentially distributed. When assessing the performance of such queuing model, information regarding the parameter traffic intensity (ρ), also called the utilization factor of the service station, is very essential. The unknown factor ρ is therefore our parameter of interest in the present work. Maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimators of ρ are proposed in the context of classical paradigm. A Bayes estimator of ρ is derived assuming that the prior density of ρ belongs to the family of Beta distributions. The performance of estimators is evaluated in terms of their relative efficiency. The proposed procedures are illustrated through a simulation study.

Lifetime performance index of electronic products

Product lifetime is one of the most important issues associated with circular and sharing economies. In the face of global warming, increasing product quality and product lifetime can enhance product value and industry competitiveness. Many electronic products use designs with spare electronic components to enhance their reliability. To prevent the loss of generality, we assumed that an electronic product has m electronic components, that is, one primary electronic component and m-1 spare electronic components. When the primary electronic component fails, then the product will immediately switch to a spare. Under the assumption that the warranty period is L units of time, we proposed a lifetime performance index for electronic products. With an identical mean time between failure (MTBF) for each electronic component, we investigated the relationships between the number of electronic components and the lifetime performance index, failure rate, and reliability of the product. We next developed uniformly minimum variance unbiased (UMVU) estimator for the lifetime performance index of electronic products and its probability density function. Then we established the uniformly most powerful (UMP) test and the power of the test for the lifetime performance index for electronic products. Finally, we presented a numerical example to demonstrate the applicability of the proposed method and model.

Evaluation for estimating of the PDF and the CDF of Generalized Inverted Exponential Distribution with Application in Industry

The generalized inverted exponential distribution is introduced as a lifetime model with good statistical properties. This paper, the estimation of the probability density function and the cumulative distribution function of with five different estimation methods: uniformly minimum variance unbiased(UMVU), maximum likelihood(ML), least squares(LS), weighted least squares (WLS) and percentile(PC) estimators are considered. The performance of these estimation procedures, based on the mean squared error (MSE) by numerical simulations are compared. Simulation studies express that the UMVU estimator performs better than others and when the sample size is large enough the ML and UMVU estimators are almost equivalent and efficient than LS, WLS and PC. Finally, the result using a real data set are analyzed.

Minimum Variance Unbiased Estimation of Reliability Function for a Class of Generalizations of Lindley Distribution

Uniformly minimum variance unbiased (UMVU) estimators of reliability function (both stress–strength and mission time) and the associated variance are derived for a class of generalization of Lindley distribution. The application of the results is further validated by a real data study.AMS 2020 subject classification: 62N05 62F10

A study on methods for estimating the PDF and the CDF in the exponentiated gamma distribution

The exponentiated gamma distribution is a two parameters lifetime distribution with monotone and non-monotone failure rates. In this article, some estimation methods of the probability density function and the cumulative distribution function of the exponentiated gamma distribution such as uniformly minimum variance unbiased (UMVU), maximum likelihood (ML), least squares, weighted least squares and Minimum distance estimators are studied and their performances through numerical simulations are compared. By the mean integrated squared error (MISE), the UMVU and ML are approximately equivalent and more efficient than the others based on simulation studies when the sample size is more than 40. A real data set is analyzed for illustrative purposes.

Estimation and testing procedures of the reliability functions of generalized inverted scale family of distributions

ABSTRACTWe consider here the generalized inverted scale family of distributions proposed by Potdar and Shirke [Inference for the parameters of generalized inverted family of distributions. ProbStat Forum. 2013;06:18–28] by zeroing in on two measures of reliability, and , based on Type II censoring and the sampling scheme of Bartholomew [The sampling distribution of an estimate arising in life testing. Technometrics. 1963;5:361–374]. Thereafter, we develop point estimators of powers of α, and ; and develop uniformly minimum variance unbiased (UMVU) estimators and maximum likelihood (ML) estimators. A new technique of obtaining these estimators is introduced. Exact confidence intervals for the parameter α, and are also constructed based on Type II censoring. Then we derive testing procedures for the hypotheses related to different parametric functions. Finally, we compare these results using the Monte Carlo simulation method.

A study of methods for estimating in the exponentiated Gumbel distribution

The exponentiated Gumbel model has been shown to be useful in climate modeling including global warming problem, flood frequency analysis, offshore modeling, rainfall modeling and wind speed modeling. Here, we consider estimation of the PDF and the CDF of the exponentiated Gumbel distribution. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.

ML and UMVU estimation in the M/D/1 queuing system

ABSTRACTIn the imbedded Markov chain (IMC) analysis of M/G/1 queuing system, X1, X2, …, Xn, … form a sequence of i.i.d random variables. where Xn denotes the number of customer arrivals during the service time of customer. In the M/D/1 queue, the distribution of common random variable X is the Poisson distribution with mean ρ, the traffic intensity. This fact is utilized for maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimation of traffic intensity, performance measures, transition probabilities of IMC, and correlation functions of departure process, based on a sample of fixed size n from P(ρ) distribution. Also, consistent asymptotic normality (CAN) property of ML estimators (MLEs) is established. The MLEs and UMVUEs are compared.

Efficient Estimation of the PDF and the CDF of the Weibull Extension Model

The Weibull extension model is a useful extension of the Weibull distribution, allowing for bathtub shaped hazard rates among other things. Here, we consider estimation of the PDF and the CDF of the Weibull extension model. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator, and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.

Empirical comparison of ML and UMVU estimators of the generalized variance for some normal stable Tweedie models: a simulation study

This paper discuss a comparison of the maximum likelihood (ML) estimator and the uniformly minimum variance unbiased (UMVU) es- timator of generalized variance for some normal stable Tweedie models through simulation study. We describe the estimation of some particular cases of multivariate NST models, i.e. normal gamma, normal Poisson and normal invers-Gaussian. The result shows that UMVU method pro- duces better estimations than ML method on small samples and they both produce similar estimations on large samples.

Statistical inference of P(X Y ) for the Burr Type XII distribution based on records

In this paper, the maximum likelihood and Bayesian approaches have been used to obtain the estimates of the stress-strength reliability R = P (X < Y ) based on upper record values for the two-parameter Burr Type XII distribution. A necessary and sucient condition is studied for the existence and uniqueness of the maximum likelihood estimates of the parameters. When the rst shape parameter of X andY is common and unknown, the maximum likelihood (ML) estimate and asymptotic condence interval of R are obtained. In this case, the Bayes estimate of R has been developed by using Lindley’s approximation and the Markov Chain Monte Carlo (MCMC) method due to lack of explicit forms under the squared error (SE) and linear-exponential (LINEX) loss functions for informative prior. The MCMC method has been also used to construct the highest posterior density (HPD) credible interval. When the rst shape parameter of X and Y is common and known, the ML, uniformly minimum variance unbiased (UMVU) and Bayes estimates, Bayesian and HPD credible as well as exact and approximate intervals of R are obtained. The comparison of the derived estimates is carried out by using Monte Carlo simulations. Two real life data sets are analysed for the illustration purposes.

Efficient Estimation of the PDF and the CDF of the Exponentiated Gumbel Distribution

The exponentiated Gumbel model has been shown to be useful in climate modeling including global warming problem, flood frequency analysis, offshore modeling, rainfall modeling, and wind speed modeling. Here, we consider estimation of the probability density function (PDF) and the cumulative distribution function (CDF) of the exponentiated Gumbel distribution. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least-square (LS) estimator, and weighted least-square (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.

On Point Estimations of some Uniform Distributions

For a population according to a uniform distribution where is known, Thomas Yageen (1996) derived an efficient estimator for . Here for the case when both and 'are unknown, we obtain uniformly minimum variance unbiased (UMVU) estimators for them. Moreover, for a sample drawn without replacement from an integer-valued population which is uniformly distributed on consecutive integers with both and unknown, we obtain a UMVU estimator for and apply it in a number serial analysis problem. Finally, under the criterion of mean squared error, one developed estimator is provided.

On the Construction of Preliminary Test Estimator Based on Record Values for the Burr XII Model

In this paper some different sorts of estimators are proposed based on record breaking observations in the Burr type XII model. We define Bayes as well as empirical Bayes preliminary test estimators in the same fashion as in the ordinary preliminary test estimator using relevant combinations of uniformly minimum variance unbiased (UMVU) and Bayes estimators. Exact and asymptotic bias and mean square error (MSE) expressions for the proposed estimators are derived under two different conditions of knowing the shape parameters. We compare the MSEs and obtain the confidence interval for the parameter of interest in which the preliminary test type estimators outperform the UMVU, Bayes and empirical Bayes estimators. An application of the ordinary preliminary test estimator is also considered. We conclude this approach by a useful discussion for practical purposes and a summary.

Efficient Estimation of the Density and Cumulative Distribution Function of the Generalized Rayleigh Distribution

The uniformly minimum variance unbiased (UMVU), maximum likelihood, percentile (PC), least squares (LS) and weighted least squares (WLS) estimators of the probability density function (pdf) and cumulative distribution function are derived for the generalized Rayleigh distribution. This model can be used quite effectively in modelling strength data and also modelling general lifetime data. It has been shown that MLE is better than UMVUE and UMVUE is better than the others. An application to waiting times (min) of 100 bank customers.