Uniformly Minimum Variance Unbiased Estimator Research Articles

Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample

With the high demands on the quality of high-tech products for consumers, assuring the lifetime performance is a very important task for competitive manufacturing industries. The lifetime performance index CL is frequently used to monitor the larger-the-better lifetime performance of products. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Chen lifetime distribution with a bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. We derived the uniformly minimum variance unbiased estimator (UMVUE) for CL, and we used this estimator to develop a hypothesis testing procedure of CL under a lower specification limit based on the progressive type-II censored sample. The Bayesian estimator for CL is also derived, and it is used to develop another hypothesis testing procedure. A simulation study is conducted to compare the average confidence levels for two procedures. Finally, one practical example is given to illustrate the implementation of our proposed non-Bayesian and Bayesian testing procedure.

Multi-component stress–strength parameter estimation of a non-identical-component strengths system under the adaptive hybrid progressive censoring samples

To produce more flexible models in reliability theory field, statistical inference of stress–strength parameter was studied in a multi-component system with the non-identical-component strengths based on bathtub-shaped distribution under adaptive Type-II hybrid progressive censoring samples. The problem is considered in three cases. First, with assuming that common first parameter is unknown for strengths and stress variables, the maximum likelihood estimation (MLE) and two Bayes approximations are obtained. Also, asymptotic, two bootstrap, and highest posterior density (HPD) intervals are derived. Second, with assuming that common first parameter is known, MLE, exact Bayes estimation, uniformly minimum variance unbiased estimator (UMVUE), and different confidence intervals are provided. Finally, in the third case, with assuming that all parameters are different and unknown, MLE and Bayesian estimation are studied. For comparing different estimates, the Monte Carlo simulation is used and one real data set is employed to implement theoretical methods.

Comparison of Some of Estimation methods of Stress-Strength Model: R = P(Y < X < Z)

In this study, the stress-strength model R = P(Y < X < Z) is discussed as an important parts of reliability system by assuming that the random variables follow Invers Rayleigh Distribution. Some traditional estimation methods are used to estimate the parameters namely; Maximum Likelihood, Moment method, and Uniformly Minimum Variance Unbiased estimator and Shrinkage estimator using three types of shrinkage weight factors. As well as, Monte Carlo simulation are used to compare the estimation methods based on mean squared error criteria.

Uniform Minimum Variance Unbiased Estimator of Fractal Dimension

The paper introduced the concept of a fractal distribution using a power-law distribution. It proceeds to determining the relationship between fractal and exponential distribution using a logarithmic transformation of a fractal random variable which turns out to be exponentially distributed. It also considered finding the point estimator of fractional dimension and its statistical characteristics. It was shown that the maximum likelihood estimator of the fractional dimension λ is biased. Another estimator was found and shown to be a uniformly minimum variance unbiased estimator (UMVUE) by Lehmann-Scheffe’s theorem.

On a complete and sufficient statistic for the correlated Bernoulli random graph model

Inference on vertex-aligned graphs is of wide theoretical and practical importance. There are, however, few flexible and tractable statistical models for correlated graphs, and even fewer comprehensive approaches to parametric inference on data arising from such graphs. In this paper, we consider the correlated Bernoulli random graph model (allowing different Bernoulli coefficients and edge correlations for different pairs of vertices), and we introduce a new variance-reducing technique—called balancing—that can refine estimators for model parameters. Specifically, we construct a disagreement statistic and show that it is complete and sufficient; balancing can be interpreted as Rao-Blackwellization with this disagreement statistic. We show that for unbiased estimators of functions of model parameters, balancing generates uniformly minimum variance unbiased estimators (UMVUEs). However, even when unbiased estimators for model parameters do not exist—which, as we prove, is the case with both the heterogeneity correlation and the total correlation parameters—balancing is still useful, and lowers mean squared error. In particular, we demonstrate how balancing can improve the efficiency of the alignment strength estimator for the total correlation, a parameter that plays a critical role in graph matchability and graph matching runtime complexity.

Unbiasedness and efficiency of non-parametric and UMVUE estimators of the probabilistic index and related statistics.

In reliability theory, diagnostic accuracy, and clinical trials, the quantity , also known as the Probabilistic Index (PI), is a common treatment effect measure when comparing two groups of observations. The quantity , a linear transformation of PI known as the net benefit, has also been advocated as an intuitively appealing treatment effect measure. Parametric estimation of PI has received a lot of attention in the past 40 years, with the formulation of the Uniformly Minimum-Variance Unbiased Estimator (UMVUE) for many distributions. However, the non-parametric Mann-Whitney estimator of the PI is also known to be UMVUE in some situations. To understand this seeming contradiction, in this paper a systematic comparison is performed between the non-parametric estimator for the PI and parametric UMVUE estimators in various settings. We show that the Mann-Whitney estimator is always an unbiased estimator of the PI with univariate, completely observed data, while the parametric UMVUE is not when the distribution is misspecified. Additionally, the Mann-Whitney estimator is the UMVUE when observations belong to an unrestricted family. When observations come from a more restrictive family of distributions, the loss in efficiency for the non-parametric estimator is limited in realistic clinical scenarios. In conclusion, the Mann-Whitney estimator is simple to use and is a reliable estimator for the PI and net benefit in realistic clinical scenarios.

Inclusion of non-inferiority analysis in superiority-based clinical trials with single-arm, two-stage Simon's design

IntroductionNon-inferiority (NI) analysis is not usually considered in the early phases of clinical development. In some negative phase II trials, a post-hoc NI analysis justified additional phase III trials that were successful. However, the risk of false positive achievements was not controlled in these early phase analyses. We propose to preplan NI analyses in superiority-based Simon's two-stage designs to control type I and II error rates. MethodsSimulations have been proposed to assess the control of type I and II errors rates with this method. A total of 12,768 two-stage Simon's design trials were constructed based on different assumptions of rejection response probability, desired response probability, type I and II errors, and NI margins. P-value and type II error were calculated with stochastic ordering using Uniformly Minimum Variance Unbiased Estimator. Type I and II errors were simulated using the Monte Carlo method. The agreement between calculated and simulated values was analyzed with Bland-Altman plots. ResultsWe observed the same level of agreement between calculated and simulated type I and II errors from both two-stage Simon's superiority designs and designs in which NI analysis was allowed. Different examples has been proposed to explain the utility of this method. ConclusionInclusion of NI analysis in superiority-based single-arm clinical trials may be useful for weighing additional factors such as safety, pharmacokinetics, pharmacodynamic, and biomarker data while assessing early efficacy. Implementation of this strategy can be achieved through simple adaptations to existing designs for one-arm phase II clinical trials.

Bayes Estimation of the Reliability Function of Pareto Distribution Under Three Different Loss Functions

In this paper, we have proposed Bayes estimators of shape parameter of Pareto distribution as well as reliability function under SELF, QLF and APLF loss functions. For better understanding of Bayesian approach, we consider Jeffrey’s prior as non-informative prior, exponential and gamma priors as informative priors. The proposed estimators have been compared with Maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE). Moreover, the current study also derives the expressions for risk function under these three loss functions. The results obtained have been illustrated with the real as well as simulated data set.

Bayes Estimation of the Reliability Function of Pareto Distribution Under Three Different Loss Functions

In this paper, we have proposed Bayes estimators of shape parameter of Pareto distribution as well as reliability function under SELF, QLF and APLF loss functions. For better understanding of Bayesian approach, we consider Jeffrey’s prior as non-informative prior, exponential and gamma priors as informative priors. The proposed estimators have been compared with Maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE). Moreover, the current study also derives the expressions for risk function under these three loss functions. The results obtained have been illustrated with the real as well as simulated data set.

Development of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of Kumaraswamy-G Distributions Based on Progressive Type-II Censoring

Classical estimators and preliminary test estimators (PTEs) of the powers of model parameter and the two measures of reliability, namely $$R\left( t \right) = P(X > t)$$ and $$P = P(X > Y)$$ of Kumaraswamy-G distribution, are developed under progressive type-II censoring. The preliminary test confidence intervals (PTCIs) are also developed based on uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). Merits of PTEs and PTCIs are established. Simulation results are presented, and real-life data set is also analysed.

On a Generalization of the Positive Exponential Family of Distributions and the Estimation of Reliability Characteristics

A generalization of positive exponential family of distributions developed by Liang (2008) is taken into consideration. Its properties are studied. Two measures of reliability are discussed. Uniformly minimum variance unbiased estimators (UMVUES), maximum likelihood estimators (MLES) and method of moment estimators (MMES) are developed for the reliability functions. The performances of three types of estimators are compared through Monte Carlo simulation. Real life data sets are also analyzed.

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.

Classical and Bayes estimation in the queueing system

By considering an queueing system, the maximum likelihood and consistent estimators of traffic intensity are derived by observing the number of entity arrivals during the service time of an entity. Uniform minimum variance unbiased estimators for the expected waiting times per entity in the system and queue are obtained. Further, Bayes estimators of traffic intensity, measures of system performance, minimum posterior risk, and minimum Bayes risk associated with these estimators are derived. Also, the Bayes estimator of traffic intensity and its risk function are derived under LINEX loss function.

Bayesian Estimation of Parameters of Weibull Distribution Using Linex Error Loss Function

This paper develops a Bayesian analysis of the scale parameter in the Weibull distribution with a scale parameter  θ  and shape parameter  β (known). For the prior distribution of the parameter involved, inverted Gamma distribution has been examined. Bayes estimates of the scale parameter, θ  , relative to LINEX loss function are obtained. Comparisons in terms of risk functions of those under LINEX loss and squared error loss functions with their respective alternate estimators, viz: Uniformly Minimum Variance Unbiased Estimator (U.M.V.U.E) and Bayes estimators relative to squared error loss function are made. It is found that Bayes estimators relative to squared error loss function dominate the alternative estimators in terms of risk function.

On Estimation of Stress-Strength Reliability Using Lower Record Values from Proportional Reversed Hazard Family

In this article, a study on the stress-strength parameter based on lower record values from one parameter proportional reversed hazard family (PRHF) has been conducted. The classical and Bayesian results of Khan and Arshad (UMVU Estimation of Reliability Function and Stress-Strength Reliability from Proportional Reversed Hazard Family Based on Lower Records. American Journal of Mathematical and Management Sciences, 35(2), 171–181) and Condino et al. (Likelihood and Bayesian estimation of P(Y < X) using lower record values from a general class of distributions. Statistical Papers), when the strength and stress variables belong to different family of distributions from PRHF have been generalized. Uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE) and Bayes estimator (BE) are obtained for the powers of the parameter and reliability functions and The estimators of three parametric functions, namely, powers of parameter, and are interrelated, whereas, in the literature, researchers have handled the three estimation problems separately. Moreover, it is has been shown that the expressions for and are not required to estimate them. In this article, the technique of obtaining estimators of and is simpler as it does not require Rao-Blackwellization. Simulation studies have been performed for analyzing the behavior of the proposed estimators. An example using real data has also been considered as an illustration.

On estimating the location parameter of the selected exponential population under the LINEX loss function

Suppose that $\pi_{1},\pi_{2},\ldots ,\pi_{k}$ be $k(\geq2)$ independent exponential populations having unknown location parameters $\mu_{1},\mu_{2},\ldots,\mu_{k}$ and known scale parameters $\sigma_{1},\ldots,\sigma_{k}$. Let $\mu_{[k]}=\max \{\mu_{1},\ldots,\mu_{k}\}$. For selecting the population associated with $\mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $\mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $\delta_{N,1},\delta_{N,2}$ and $\delta_{N,3}$ of $\mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $\mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $\mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $\mu_{S}$ is derived. Using this result, estimator better than the natural estimator $\delta_{N,1}$ is obtained. We also shown that the estimator $\delta_{N,1}$ is dominated by the natural estimator $\delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $\mu_{S}$.

Preliminary Test Estimators and Confidence Intervals for the Parametric Functions of the Moore and Bilikam Family of Lifetime Distributions Based on Records

We consider two measures of reliability functions namely R(t)=P(X&gt;t) and P=P(X&gt;Y) for the Moore and Bilikam (1978) family of lifetime distributions which covers fourteen distributions as specific cases. For record data from this family of distributions, preliminary test estimators (PTEs) and preliminary test confidence interval (PTCI) based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), empirical Bayes estimator (EBE) are obtained for the parameter. The bias and mean square error (MSE) (exact and asymptotic) of the proposed estimators are derived to study their relative efficiency and through simulation studies we establish that PTEs perform better than ordinary UMVUE, MLE and EBE. We also obtain the coverage probability (CP) and the expected length of the PTCI of the parameter and establish that the confidence intervals based on MLE are more precise. An application of the ordinary preliminary test estimator is also considered. To the best of the knowledge of the authors, no PTEs have been derived for R(t) and P based on records and thus we define improved PTEs based on MLE and UMVUE of R(t) and P. A comparative study of different methods of estimation done through simulations establishes that PTEs perform better than ordinary UMVUE and MLE.

Establishment of Preliminary Test Estimators and Preliminary Test Confidence Intervals for Measures of Reliability of an Exponentiated Distribution Based on Type-II Censoring

The present paper has developed the preliminary test estimators (PTEs) of the model parameter raised to certain power, σ p , and the two measures of reliability, namely, the reliability function, R(t ) and the reliability of an item or a system, P of an exponentiated distribution, under Type- II censoring, based on their uniformly minimum variance unbiased estimators (UMVUEs) and maximum likelihood estimators (MLEs). The preliminary test confidence intervals (PTCIs) are also developed for σ, R(t ) and P based on their UMVUEs and MLEs. Further, the paper has derived expression for coverage probability of the PTCI of the model parameter, σ. Merits of the proposed PTEs are also established through analysis of simulated numerical data.

Performance Assessment for Sunlight Radiation Resistant Coating Manufacturers

It is of great significance to select suitable manufacturers to improve the ability to respond to the final customer demand. Establishing a strategic partnership with manufacturers is an effective way to increase profits and improve the process efficiency for better connectivity across the supply chain. Therefore, manufacturer performance evaluation and selection is one of the most important decision-making problems in enterprises. This study proposes a manufacturer capability index MCI and gives the uniformly minimum-variance unbiased estimator (UMVUE) of the variance and probability density function (PDF) for index MCI. Finally, an example of performance assessment for sunlight radiation resistant coating (SRRC) manufacturer is given to illustrate the application of the proposed index MCI.

Estimation of Common Scale Parameter of Several Heterogeneous Pareto Populations Based on Records

This article deals with the problem of estimation of common scale parameter $$\theta$$ of k Pareto populations with unknown and unequal shape parameters $$\alpha _1,\alpha _2,\ldots,\alpha _k$$ based on record data. We derive the maximum likelihood estimator (MLE), the modified MLE and the uniformly minimum variance unbiased estimator (UMVUE) of $$\theta$$ . Under the scaled squared error loss function, a class of improved estimators over the MLE of $$\theta$$ is derived. This class of improved estimators contains the modified MLE as well as the UMVUE of $$\theta$$ . A simulation study is carried out for comparing the performances of the estimators of $$\theta$$ . Further, a real data example is provided to illustrate the computation of various estimators of $$\theta$$ .