Minimum Variance Unbiased Estimator Research Articles

Vibration reduction control by a cable actuation system for reflector antenna

Vibration deformation of a reflector antenna under a moving load will reduce pointing performance. Current vibration suppression technology for reflector antennas, such as the application of mesh reflector antennas, the construction of large radomes, the use of antennas with high stiffness, and the improvement of the shaft servo systems, can reduce the impact of vibrations. However, the disadvantages of high-frequency transmission, high cost, increased weight, and servo bandwidth limitations, are unavoidable. A cable actuation system is proposed as a method for reducing the vibration of reflector antennas while having little influence on the original structure. In this method, the acceleration of the reflector is measured, and an unbiased minimum variance estimator of the system state is introduced to observe the vibration state of the reflector. Then using the LQR approach, drivers on the slide rail pull the cables to suppress the vibration of the reflector. Moreover, the position of the drivers on the slide rail can be adjusted depending on the moving load condition. Finally, the experimental results show that the vibration time of the reflector without control was approximately 7 s, while the vibration time of the reflector with control was reduced to approximately 1 s when the pendulum fell from the same height to excite the reflector.

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.

Reliability estimation in a multicomponent stress-strength model for unit Burr III distribution under progressive censoring

ABSTRACT We study the problem of estimating multicomponent stress-strength reliability under progressive Type II censoring when stress and strength variables follow unit Burr III distributions with a common shape parameter. Point and interval estimates of this parametric function are derived using both classical and Bayesian approaches. Different approximation methods are used to obtain Bayes estimates. In particular, uniformly minimum variance unbiased estimate of the reliability is discussed when the common shape parameter is known. Monte Carlo simulations are performed to compare proposed methods. Finally, we provide two numerical examples for illustration purposes.

Estimation of Stress-Strength Reliability Following Extended Chen Distribution

This paper considers the estimation of the stress-strength parameter [Formula: see text] under progressive type-II censored samples, where [Formula: see text] is the strength and [Formula: see text] is the stress follow extended Chen distribution. It is assumed that [Formula: see text] and [Formula: see text] are independent. Two cases are studied. The uniformly minimum variance unbiased estimator is derived when a common scale parameter and a common shape parameter are known. In both the cases, the maximum likelihood estimates are computed using numerical iteration technique. Sufficient conditions are derived under which the MLEs of [Formula: see text] exist and unique. The Bayes estimates with respect to the squared error loss function are evaluated using Lindley’s and MCMC methods. Interval estimates are also constructed using asymptotic and bootstrap techniques. A simulation study is carried out to compare the proposed methods. Finally, data analysis has been performed on two real-life data sets for the purpose of illustration.

Multicomponent stress-strength reliability based on a right long-tailed distribution

This article deals with the problem of reliability in a multicomponent stress-strength (MSS) model when both stress and strength variables are from inverse Kumaraswamy distribution. The reliability of the system is estimated using classical and Bayesian approaches when the common second shape parameter is known or unknown. The maximum likelihood estimation and its asymptotic confidence interval for the reliability of the system are obtained. Furthermore, two other asymptotic confidence intervals are computed based on Logit and Arcsin transformations. The uniformly minimum variance unbiased estimator for the reliability of the MSS model is obtained when the common second shape parameter is known. The Bayes estimate is obtained exactly when the second shape parameter is known and it is approximated by using the Monte Carlo Markov Chain method when the second shape parameter is unknown. The highest probability density credible interval is established using the Gibbs sampling technique. Monte Carlo simulations are implemented to compare the different proposed methods. Finally, two real data sets are presented in support of the suggested procedures.

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.

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.

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.

A study on the g and h control charts

In this paper, we revisit the g and h control charts that are commonly used for monitoring the number of conforming cases between the two consecutive appearances of nonconformities. It is known that the process parameter of these charts is usually unknown and estimated by using the maximum likelihood estimator and the minimum variance unbiased estimator. However, the minimum variance unbiased estimator in the control charts has been inappropriately used in the quality engineering literature. This observation motivates us to provide the correct minimum variance unbiased estimator and investigate the theoretical and empirical biases of these estimators under consideration. Given that these charts are developed based on the underlying assumption that samples from the process should be balanced, which is often not satisfied in practice, we propose a method for constructing these charts with unbalanced samples. Finally, two real-data applications are provided for illustrative purposes.

Robust strong tracking unscented Kalman filter for non‐linear systems with unknown inputs

This paper proposes a state estimation approach ‘robust strong tracking unscented Kalman filter with unknown inputs’ that can be applied to non-linear systems with unknown inputs. Specifically, the non-linear state and measurement equations are linearised by statistical linearisation. Then, the estimation equation of the unknown input is derived based on the weighted least squares method. The multiple suboptimal fading factor is introduced into a priori error covariance matrix to improve the tracking ability for the inaccuracy of the system model and the abrupt change of state variables caused by unknown inputs. Finally, based on the unbiased minimum variance estimation, the unbiased state estimation and the error covariance matrix are derived. Singular value decomposition is performed on the error covariance matrix to improve the stability of the algorithm. Simulated results validate the effectiveness of the proposed method.

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.

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.

Weak derivative-based expansion of functions: ANOVA and some inequalities

Functional analysis of variance (FANOVA) is widely used in statistical modeling and sensitivity analysis. Knowing the important role of derivatives in models’ analysis and/or development such as variational formulation of phenomena, we develop new expansions of functions using their weak derivatives, the density and distribution functions of input variables. We also derive the expressions of ANOVA components based on weak cross-partial derivatives. We investigate its application in uncertainty quantification by proposing the derivative-based multivariate sensitivity analysis. Our approach allows for better assessing the main, interaction and total effects of inputs using weak derivatives, and we provide minimum variance unbiased estimators of the cross-covariances of ANOVA functionals for computing such effects in presence of multivariate and/or functional response models.

On estimating of the stress-strength model with more than two states under Gompertz distribution

The Gompertz distribution, one of statistical distributions, is considered in survival analysis because it has an increasing hazard rate function. It has many applications in various fields such as medical and reliability. The problem of estimation of stress-strength model with more than two states is considered by other authors under exponential and Weibull distribution. The main aim of this paper is to consider the same problem under Gompertz distribution based on progressive type-II censoring data. Three estimation methods are considered to estimate the reliability parameter, maximum likelihood (MLE), uniformly minimum variance unbiased estimator (UMVE) and Bayes estimator. To compare and study the performance of three estimators, Monte Carlo simulation study is performed.

Determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator of a DC Signal in AWGN Using Laplace Transform

This paper presents an alternative approach for the determination of Cramer-Rao Lower Bound (CRLB) and Minimum Variance Unbiased Estimator (MVUE) using Laplace transformation. In this work, a DC signal in Additive White Gaussian Noise (AWGN) was considered. During the investigation, a number of experiments were conducted to analyze different possible outputs under different conditions, and then the patterns of the outcomes were studied. Finally closed-form expressions for the CRLB and MVUE were deduced employing the Laplace transformation. The resulting expressions showed that the proposed method has almost the same number of steps as the existing method. However, the latter requires only the knowledge of algebra to arrive at the CRLB expressions contrary to the existing approach where a strong mathematical background is required and hence making it superior over the existing method, in that sense.
 Keywords: Additive White Gaussian Noise, Cramer-Rao Lower Bound, DC-Value, Laplace transform, and Minimum Variance Unbiased Estimator (MVUE).

USP: an independence test that improves on Pearson's chi-squared and the G-test.

We present the -statistic permutation (USP) test of independence in the context of discrete data displayed in a contingency table. Either Pearson’s -test of independence, or the -test, are typically used for this task, but we argue that these tests have serious deficiencies, both in terms of their inability to control the size of the test, and their power properties. By contrast, the USP test is guaranteed to control the size of the test at the nominal level for all sample sizes, has no issues with small (or zero) cell counts, and is able to detect distributions that violate independence in only a minimal way. The test statistic is derived from a -statistic estimator of a natural population measure of dependence, and we prove that this is the unique minimum variance unbiased estimator of this population quantity. The practical utility of the USP test is demonstrated on both simulated data, where its power can be dramatically greater than those of Pearson’s test, the -test and Fisher’s exact test, and on real data. The USP test is implemented in the R package USP.

Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution

Abstract The exponentiated Burr Type XII (EBXII) distribution has wide applications in reliability and economic studies. In this article, the estimation of the probability density function and the cumulative distribution function of EBXII distribution is considered. We examine the maximum likelihood estimator, the uniformly minimum variance unbiased estimator, the least squares estimator, the weighted least squares estimator, the maximum product spacing estimator, the Cramér–von-Mises estimator, and the Anderson–Darling estimator. We derive analytical forms for the bias and mean square error. A simulation study is performed to investigate the consistency of the suggested methods of estimation. Data relating to the wind speed and service times of aircraft windshields are used with the studied methods. The simulation studies and real data applications have revealed that the maximum likelihood estimator performs more efficiently than its remaining counterparts.

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.