Uniformly Minimum Variance Unbiased Estimator Research Articles

Estimation of Bonferroni Curve and Bonferroni Index of the Pareto Distribution

Abstract In this article, we consider classical and Bayesian estimations of some economic measures, specially Bonferroni Curve and Bonferroni Index of the Pareto distribution. We obtain the Maximum Likelihood Estimator and Uniform Minimum Variance Unbiased Estimator in the classical setup, and their properties are studied. Additionally, we conduct Bayesian estimation procedures based on symmetric loss functions and a truncated gamma prior. The precision of the estimators is evaluated under different sample sizes via Monte Carlo simulation. Furthermore, a real dataset is provided to compute all the estimators.

Classical and Bayesian estimations of performance measures in G e o / G e o / 1 queue

. In this article, we consider an early arrival G e o / G e o / 1 queue from the statistical perspective. The estimations of parameters and system performances under Classical and Bayesian frameworks are considered. Based on queue length data, we attempt to study the uniform minimum-variance unbiased estimators (UMVUEs) and the closed Bayesian estimators for various queueing characteristics. We compare their properties using Monte Carlo (MC) simulations. As a comparison and improvement, based on arrival and departure data over a period, we use a Bivariate Beta distribution as joint prior, and solve the performance estimators using the properties of Gaussian hypergeometric function. We derive Bayesian estimators and conduct numerical simulations using Markov Chain Monte Carlo (MCMC) method.

On Point Estimators for Gamma and Beta Distributions

Let X 1 , … , X n be a random sample from the gamma distribution with density f ( x ) = λ α x α − 1 e − λ x / Γ ( α ) , x > 0, where both α > 0 (the shape parameter) and λ > 0 (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, α, exists if and only if n ≥ 4 ; moreover, it has finite variance if and only if n ≥ 6 . More precisely, the form of the UMVUE is given for all parametric functions α, λ, 1 / α , and 1 / λ . Furthermore, a highly efficient estimating procedure for the two-parameter beta distribution is also given. This is based on a Stein-type covariance identity for the beta distribution, followed by an application of the theory of U-statistics and the delta-method.

Asymptotic UMVUE: Asymptotic Moments Matching the UMVUE under the Ewens Sampling Formula

The Ewens sampling formula is a distribution related to the random partition of a positive integer. In this study, we investigate the issue of non-existent solutions in parameter estimation under the distribution. We derive the first and second moments matching estimators to the uniformly minimum variance unbiased estimator (UMVUE) in the asymptotic sense (hereafter referred to as Asymptotic UMUVE) using the Ewens sampling formula. A Monte Carlo simulation study is performed to evaluate the efficiency of the resulting estimators. AMS Subject Classification: 62F12, 62P25

Estimating reciprocals of scale parameters of two exponential populations with common location and ordered scales using censored samples

The problem of estimating reciprocals of scale parameters (hazard rates) from two exponential populations with a common location and ordered scale parameters have been considered under the progressive type-II censoring scheme from a decision-theoretic viewpoint. The loss function is considered as quadratic. The maximum-likelihood estimators (MLEs) and the uniformly minimum variance unbiased estimators (UMVUEs) are derived. Sufficient conditions are derived for improving estimators in affine and scale equivariant classes. Consequently, improved estimators over the MLEs and the UMVUEs are derived. A numerical comparison among all the proposed estimators is made, and conclusions are drawn regarding their performances with respect to the quadratic loss function.

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.

Extending Normality: A Case of Unit Distribution Generated from the Moments of the Standard Normal Distribution

This paper presents an important theorem, which shows that, heading from the moments of the standard normal distribution, one can generate density functions originating a family of models. Additionally, we discussed that different random variable domains are achieved with transformations. For instance, we adopted the moment of order two, from the proposed theorem, and transformed it, which enabled us to exemplify this class as a unit distribution. We named it as Alpha-Unit (AU) distribution, which contains a single positive parameter α (AU(α)∈[0,1]). We presented its properties and demonstrated two estimation methods for the α parameter, the maximum likelihood estimator (MLE) and uniformly minimum-variance unbiased estimator (UMVUE) methods. In order to analyze the statistical consistency of the estimators, a Monte Carlo simulation study was carried out, in which the robustness was demonstrated. As a real-world application, we adopted two sets of unit data, the first regarding the dynamics of Chilean inflation in the post-military period, and the other one regarding the daily maximum relative humidity of the air in the Atacama Desert. In both cases presented, the AU model is competitive, whenever the data present a range greater than 0.4 and extremely heavy asymmetric tail. We compared our model with other commonly used unit models, such as the beta, Kumaraswamy, logit-normal, simplex, unit-half-normal, and unit-Lindley distributions.

Point estimation following a two-stage group sequential trial.

Repeated testing in a group sequential trial can result in bias in the maximum likelihood estimate of the unknown parameter of interest. Many authors have therefore proposed adjusted point estimation procedures, which attempt to reduce such bias. Here, we describe nine possible point estimators within a common general framework for a two-stage group sequential trial. We then contrast their performance in five example trial settings, examining their conditional and marginal biases and residual mean square error. By focusing on the case of a trial with a single interim analysis, additional new results aiding the determination of the estimators are given. Our findings demonstrate that the uniform minimum variance unbiased estimator, whilst being marginally unbiased, often has large conditional bias and residual mean square error. If one is concerned solely about inference on progression to the second trial stage, the conditional uniform minimum variance unbiased estimator may be preferred. Two estimators, termed mean adjusted estimators, which attempt to reduce the marginal bias, arguably perform best in terms of the marginal residual mean square error. In all, one should choose an estimator accounting for its conditional and marginal biases and residual mean square error; the most suitable estimator will depend on relative desires to minimise each of these factors. If one cares solely about the conditional and marginal biases, the conditional maximum likelihood estimate may be preferred provided lower and upper stopping boundaries are included. If the conditional and marginal residual mean square error are also of concern, two mean adjusted estimators perform well.

On comparing design-based estimation versus model-based prediction to assess the abundance of biological populations

Assessing the abundance of biological populations is a central technical challenge in ecology, whether fundamental or applied. Although the question faced is simple, it is actually a complicated topic that has produced a vast array of methods. The first step is always sampling the geographical space, ideally by means of a randomized selection process. In that case, with a frequentist interpretation of probability, abundance can be estimated by taking into account randomization, or predicted conditionally on the sample at hand, by specifying a statistical model. This leads to a choice between so-called design-based estimation and model-based prediction. The goal of this methodological article is to contribute to the understanding of fundamental notions regarding these two statistical frameworks by the targeted audience, mainly quantitative ecologists. For this purpose, we illustrate the comparison between design-based estimation in the case of simple random sampling without replacement (SRSWOR) and model-based prediction. As an example, we model count data with a delta-lognormal distribution and rely on uniformly minimum-variance unbiased estimators (UMVUEs) for the prediction of abundance. We investigate the robustness of the predictor by contaminating the delta-lognormal distribution using actual count data. Data from a survey concerning wintering populations in France of two wader species, namely, northern lapwing (Vanellus vanellus) and European golden plover (Pluvialis apricaria) serve as illustrative examples. By means of Monte Carlo simulations, we highlight the lack of robustness of the predictor based on the delta-lognormal distributional model, in terms of both actual bias and precision. We organize the discussion around the illustrative examples in the context of the sampling design, the model and the data.

Ridge Regression Method and Bayesian Estimators under Composite LINEX Loss Function to Estimate the Shape Parameter in Lomax Distribution.

In this paper, the Ridge Regression method is employed to estimate the shape parameter of the Lomax distribution (LD). In addition to that, the approaches of both classical and Bayesian are considered with several loss functions as a squared error (SELF), Linear Exponential (LLF), and Composite Linear Exponential (CLLF). As far as Bayesian estimators are concerned, informative and noninformative priors are used to estimate the shape parameter. To examine the performance of the Ridge Regression method, we compared it with classical estimators which included Maximum Likelihood, Ordinary Least Squares, Uniformly Minimum Variance Unbiased Estimator, and Median Method as well as Bayesian estimators. Monte Carlo simulation compares these estimators with respect to the Mean Square Error criteria (MSE's). The result of the simulation mentioned that the Ridge Regression method is promising and can be used in a real environment. where it revealed better performance the than Ordinary Least Squares method for estimating shape parameter.

Multicomponent Stress-strength Reliability with Exponentiated Teissier Distribution

This article deals with the problem of reliability in a multicomponent stress-strength (MSS) model when both stress and strength variables are from exponentiated Teissier (ET) distributions. The reliability of the system is determined using both classical and Bayesian methods, based on two scenarios where the common scale parameter is unknown or known. In the first scenario, where the common scale parameter is unknown, the maximum likelihood estimation (MLE) and the approximate Bayes estimation are derived. In the second scenario, where the scale parameter is known, the MLE, the uniformly minimum variance unbiased estimator (UMVUE) and the exact Bayes estimation are obtained. In the both scenarios, the asymptotic confidence interval and the highest probability density credible interval are established. Furthermore, two other asymptotic confidence intervals are computed based on the Logit and Arcsin transformations. Monte Carlo simulations are implemented to compare the different proposed methods. Finally, one real example is presented in support of suggested procedures.

Estimating a parametric function involving several exponential populations

This article provides some optimal estimators for a parametric function θR , which arises in the study of reliability analysis involving several exponential populations. Let be independent populations, where the population πi follows an exponential distribution with unknown guarantee time and a known failure rate. These populations may represent the lifetimes of k systems. Let be the reliability function of the ith system, and let denote the largest value of ’s at a fixed t. We call the system associated with the best system. For selecting the best system, a class of natural selection rules is used. The goal is to estimate the parametric function θR , which is a function of parameters and the random variables. The uniformly minimum variance unbiased estimator (UMVUE) and the generalized Bayes estimator of θR are derived. Two natural estimators and of θR are also considered. A general result for improving an equivariant estimator of θR is derived. Further, we show that the natural estimator dominates the UMVUE under the squared error loss function. Finally, the risk functions of the various competing estimators of θR are compared via a simulation study.

Estimation procedures for reliability functions of Kumaraswamy-G Distributions based on Type II Censoring and the sampling scheme of Bartholomew

Abstract In this paper, we consider Kumaraswamy-G distributions and derive a Uniformly Minimum Variance Unbiased Estimator (UMVUE) and a Maximum Likelihood Estimator (MLE) of the two measures of reliability, namely R(t) = P(X > t) and P = P(X > Y) under Type II censoring scheme and sampling scheme of Bartholomew (1963). We also develop interval estimates of the reliability measures. A comparative study of the different methods of point estimation has been conducted on the basis of simulation studies. An analysis of a real data set has been presented for illustration purposes.

Optimised point estimators for multi-stage single-arm phase II oncology trials

ABSTRACT The uniform minimum variance unbiased estimator (UMVUE) is, by definition, a solution to removing bias in estimation following a multi-stage single-arm trial with a primary dichotomous outcome. However, the UMVUE is known to have large residual mean squared error (RMSE). Therefore, we develop an optimisation approach to finding estimators with reduced RMSE for many response rates, which attain low bias. We demonstrate that careful choice of the optimisation parameters can lead to an estimator with often substantially reduced RMSE, without the introduction of appreciable bias.

Estimation and expected sample size in Simon's two-stage designs that stop as early as possible.

In early phase clinical studies in oncology, Simon's two-stage designs are widely used. The trial design could be made more efficient by stopping early in the second stage when the required number of responses is reached, or when it has become clear that this target can no longer be met (a form of non-stochastic curtailment). Early stopping, however, will affect proper estimation of the response rate. We propose a uniformly minimum-variance unbiased estimator (UMVUE) for the response rate in this setting. The estimator is proven to be UMVUE using the Rao-Blackwell theorem. We evaluate the estimator's properties in terms of bias and mean squared error, both analytically and via simulations. We derive confidence intervals based on sample space orderings, and assess the coverage. For various design options, we evaluate the reduction in expected sample size as a function of the true response rate. Our method provides a solution for estimating response rates in case of a non-stochastic curtailment Simon's two-stage design.

Comparing the Stress–Strength Reliability of Multicomponent Parallel Systems with Heterogeneous Exponentiated Half Logistic-F Components

In this paper, the stress–strength reliability of two multicomponent parallel systems with heterogeneous exponentiated half logistic-[Formula: see text] components under the same and the different baseline distribution functions are compared. The comparisons are carried out concerning the usual stochastic order and reversed hazard rate order with majorized shape parameters of the distributions. Also, the maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) of the stress–strength reliability of the aforementioned multicomponent parallel system are obtained and then compared via Monte Carlo simulations.

A repetitive group sampling plan based on the lifetime performance index under gamma distribution

AbstractAs technology and living standards improve, people are becoming more concerned with the importance of quality. Quality, thus, becomes the key to company competitiveness, and process capability indices (PCIs) are effective means of measuring process performance. Most PCIs have been developed under the assumption that the quality characteristics are normally distributed; however, the lifetime of products generally possesses a non‐normal distribution. A lifetime performance index (LPI) CL has been proposed and used for measuring product lifetime performance. In this high‐quality era, the use of traditional testing methods incurs very high costs, so acceptance sampling plans have been widely used by industry to perform the inspection. This study considers the product lifetime with a gamma distribution and uses the uniformly minimum‐variance unbiased estimator (UMVUE) of the LPI to develop a repetitive group sampling plan (RGSP). For the practical purpose, this study presents a step‐by‐step procedure and tabulates the plan parameters for various combinations of acceptable and rejectable quality levels, producer's and consumer's risks, and shape parameter. Thus, practitioners can easily know the sample size and the corresponding critical values for decision making. This study also uses the results of the proposed plan to compare with the single sampling plan. Lastly, a practical example is used to illustrate how to apply this acceptance sampling plan.

On evaluating the efficiency of the delta-lognormal mean estimator and predictor

A variable taking positive values from a lognormal distribution and null values with a given probability is distributed according to the so-called delta-lognormal distribution. Two situations arise depending on whether the data are regarded as a random sample from an infinite population (superpopulation) or from a finite population, itself considered as a random sample from a superpopulation. In the case of an infinite population, estimating the mean can be accomplished using a uniformly minimum-variance unbiased estimator (UMVUE). Likewise, the prediction of the mean in the case of a finite population may be based on the UMVUE. In both cases, one expects a gain in precision when taking into account the shape of the distribution by relying on the UMVUE rather than on the sample mean, which is a nonparametric estimator (or predictor).1.For the infinite population case, the relative efficiency results presented in this article are more complete and more accurate than those published so far.2.The article fills a gap regarding the question of relative efficiency in the case of a finite population.3.Calculations were performed using the exact expression for the variance of the UMVUE of the mean, expressed in terms of the confluent hypergeometric limit function.

Uniformly Minimum-Variance Unbiased Estimator (UMVUE) for the Gamma Cumulative Distribution Function with Known and Integer Scale Parameter

Uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameter. This paper applies Rao-Blackwell and Lehmann-Scheffeé Theorems to deduce the uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameters. The paper closes with an example comparing the empirical distribution function with the UMVUE estimates.

Multi-component stress-strength model for Weibull distribution in progressively censored samples

Abstract One of the important issues is risk assessment and calculation in complex and multi-component systems. In this paper, the estimation of multi-component stress-strength reliability for the Weibull distribution under the progressive Type-II censored samples is studied. We assume that both stress and strength are two independent Weibull distributions with different parameters. First, assuming the same shape parameter, the maximum likelihood estimation (MLE), different approximations of Bayes estimators (Lindley’s approximation and Markov chain Monte Carlo method) and different confidence intervals (asymptotic and highest posterior density) are obtained. In the case when the shape parameter is known, the MLE, uniformly minimum variance unbiased estimator (UMVUE), exact Bayes estimator and different confidence intervals (asymptotic and highest posterior density) are considered. Finally, in the general case, the statistical inferences on multi-component stress-strength reliability are derived. To compare the performances of different methods, Monte Carlo simulations are performed. Moreover, one data set for illustrative purposes is analyzed.