Uniformly Minimum Variance Unbiased Estimator Research Articles

A new procedure for monitoring the range and standard deviation of a quality characteristic

The Shewhart and the Bonferroni-adjustment R and S chart are usually applied to monitor the range and the standard deviation of a quality characteristic. These charts are used to recognize the process variability of a quality characteristic. The control limits of these charts are constructed on the assumption that the population follows approximately the normal distribution with the standard deviation parameter known or unknown. In this article, we establish two new charts based approximately on the normal distribution. The constant values needed to construct the new control limits are dependent on the sample group size (k) and the sample subgroup size (n). Additionally, the unknown standard deviation for the proposed approaches is estimated by a uniformly minimum variance unbiased estimator (UMVUE). This estimator has variance less than that of the estimator used in the Shewhart and Bonferroni approach. The proposed approaches in the case of the unknown standard deviation, give out-of-control average run length slightly less than the Shewhart approach and considerably less than the Bonferroni-adjustment approach.

Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.

STATISTICAL TESTING FOR ASSESSING LIFETIME PERFORMANCE INDEX OF THE RAYLEIGH LIFETIME PRODUCTS

ABSTRACT In the service (or manufacturing) industries, process capability indices (PCIs) are utilized to assess whether product quality meets the required level. And the lifetime performance index (or larger-the-better process capability index) CL is frequently used as a means of measuring product performance, where L is the lower specification limit. Hence, this study constructs an uniformly minimum variance unbiased estimator (UMVUE) of CL based on the type II multiply censored sample from the Rayleigh distribution. The UMVUE of CL is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give an example regarding the ball bearing in endurance test to illustrate the use of the testing procedure under given significance level. Moreover, the purchasers can then employ the testing procedure to determine whether the product performance adheres to the required level. Manufacturers can also utilize this testing procedure to improve the process capability.

Comparing UMVU and ML estimators of the generalized variance for natural exponential families

In this paper we compare the uniformly minimum variance unbiased (UMVU) estimator and maximum likelihood (ML) estimator of the generalized variance in the context of natural exponential families (NEFs) on , d>1. We conjecture that for irreducible NEFs the proportionality holds if and only if the generalized variance has a specific form. In particular, we show that the estimators are proportional in the simple and homogeneous quadratic NEFs and prove that the UMVU estimator is preferable in terms of mean squared error except for the case of multinomial family.

Optimum plan for step-stress model with progressive type-II censoring

This paper considers statistical inference on a simple step-stress accelerated life test with progressively type-II censored exponential data. In this case, the uniformly minimum variance unbiased estimators (UMVUEs) of the mean life and the failure rate at a design stress are derived. It is shown that the UMVUEs have smaller mean squared errors than the maximum likelihood estimators (MLEs). A numerical example is used to illustrate the superiority of UMVUEs to MLEs. The optimum test plans of the models are also discussed based on the variance of UMVUEs. Furthermore, we discuss the generalization of the optimal plan to more than two levels of stress.

Implementing lifetime performance index for the pareto lifetime businesses of the service industries

Process capability analysis is an effective means of measuring process performance and potential capability. In the service industries, process capability indices (PCIs) are utilized to assess whether business quality meets the required level. Hence, the performance index CL is used as a means of measuring business performance, where L is the lower specification limit. In the technology of data transformation, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of CL based on the right type II censored sample from the pareto distribution. The UMVUE of CL is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give one practical example and the Monte Carlo simulation to assess the behavior of this test statistic for testing null hypothesis under given significance level. Moreover, the managers can then employ the new testing procedure to determine whether the business performance adheres to the required level.

Estimation of Reliability for (k + 1) Component Lightly Loaded Standby System with Single Repair

A (k + 1)-component lightly loaded standby system with k basic components and one standby component is considered. If a component fails irrespective of basic or standby, the failed component undergoes repair. During the repair time of the failed component if any of the k basic components fails, then the system fails. The system reliability function is obtained. Using the sample data obtained from the life testing experiment, the maximum likelihood estimators (MLEs) and the uniformly minimum variance unbiased estimators (UMVUEs) of parameters of the distribution of the system are obtained. Using the estimators of these parameters in the expression for R (τ), the estimators of R (τ) are obtained. Monte-Carlo simulation experiments were conducted in order to compare the two estimators.

A hypothesis testing procedure on assessing process performance

Process incapability index has been introduced to provide quantitative measures on process performance. Investigations of the estimated incapability index based on multiple samples have been proposed and arrested substantial research attention. In this paper, investigation based on multiple samples is considered for normally distributed processes. A reliable inferential procedure based on the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the incapability index is proposed. Useful critical values required to ensure the process reaching a certain desirable level of the time are also tabulated. A practical example is provided to demonstrate how the proposed procedure may be applied for judging whether the process runs under the desirable quality requirement.

Evaluating the performance of the official document

The rapidity and quality of official documents process in a government unit is one of the most important characteristics relating to its quality and efficiency of administrative management and performance. We always use the speed and quality of document-process to assess a government unit if it is capable and efficient. In this paper, the document-process means the whole procedure: receiving, dealing, transmitting, and filing a document. We propose an efficiency index to evaluate the efficiency assessment of the document-process. We study the characteristics of the proposed efficiency index and derive the uniformly minimum variance unbiased estimator (UMVUE) of the efficiency index. We show that the UMVUE of the efficiency index is distributed as a non-central t -distribution random variable multiplied by a constant when sampling from the normal distribution. In addition, based on the theory of statistical testing hypothesis we develop a step-by-step procedure using the UMVUE of the efficiency index for practitioners to use in judging whether a document-process meets the efficiency requirement.

The Structure of the UMVUEs from Categorical Data

Let an observation X take finitely many values with probabilities $p_1(\theta),\ldots,p_N(\theta)$ depending on an abstract parameter $\theta\in \Theta$. It is proved that a statistic is a uniformly minimum variance unbiased estimator (UMVUE) if and only if it is measurable with respect to a subalgebra of the finite algebra generated by X. In general, this subalgebra is smaller than the minimal sufficient subalgebra for θ and is explicitly described. It is related to a special partition of a finite set of elements of an abstract linear space.

Estimating Average Worth of the Selected Subset from Two-Parameter Exponential Populations

ABSTRACT Suppose independent random samples are available from k(k ≥ 2) exponential populations ∏1,…,∏ k with a common location θ and scale parameters σ1,…,σ k , respectively. Let X i and Y i denote the minimum and the mean, respectively, of the ith sample, and further let X = min{X 1,…, X k } and T i  = Y i  − X; i = 1,…, k. For selecting a nonempty subset of {∏1,…,∏ k } containing the best population (the one associated with max{σ1,…,σ k }), we use the decision rule which selects ∏ i if T i  ≥ c max{T 1,…,T k }, i = 1,…, k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE).

Statistical aspects of linkage analysis in interlaboratory studies

This paper investigates statistical issues that arise in interlaboratory studies known as Key Comparisons when one has to link several comparisons to or through existing studies. An approach to the analysis of such a data is proposed using Gaussian distributions with heterogeneous variances. We develop conditions for the set of sufficient statistics to be complete and for the uniqueness of uniformly minimum variance unbiased estimators (UMVUE) of the contrast parametric functions. New procedures are derived for estimating these functions with estimates of their uncertainty. These estimates lead to associated confidence intervals for the laboratories (or studies) contrasts. Several examples demonstrate statistical inference for contrasts based on linkage through the pilot laboratories. Monte Carlo simulation results on performance of approximate confidence intervals are also reported.

A precise estimator for the log-normal mean

The log-normal distribution is frequently encountered in applications. The uniformly minimum variance unbiased (UMVU) estimator for the log-normal mean is given explicitly by a formula found by Finney in 1941. In contrast to this the most commonly used estimator for a log-normal mean is the sample mean. This is possibly due to the complexity of the formula given by Finney. A modified maximum likelihood estimator which approximates the UMVU estimator is derived here. It is sufficiently simple to be implemented in elementary spreadsheet applications. An elementary approximate formula for the root-mean-square error of the suggested estimator and the UMVU estimator is presented. The suggested estimator is compared with the sample mean, the maximum likelihood, and the UMVU estimators by Monte Carlo simulation in terms of root-mean-square error.

A review and extensions of sample size determination for estimating process precision and loss with a designated accuracy ratio

Numerous process capability indices, including Cp, Cpk, Cpm, and Cpmk, have been proposed in the manufacturing industry providing numerical measures of process capability based on various criteria. The index Cp provides a measure of process precision, which reflects the consistency of product quality. The index Cpm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information regarding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower confidence bounds, but investigation on the sample sizes required for a specified accuracy ratio of the estimation has been comparatively neglected. The sample size determination is important, as it is directly related to the cost of the data collection plan. In this paper, we review some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different approach, with efficient MATLAB programs, to obtain the sample sizes using the UMVUE (uniformly minimum variance unbiased estimator) of Cp, and the MLE (maximum likelihood estimator) of Cpm. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications.

Estimating parameters of a selected Pareto population

Let Π 1 , … , Π k be k populations with Π i being Pareto with unknown scale parameter α i and known shape parameter β i ; i = 1 , … , k . Suppose independent random samples ( X i 1 , … , X i n ) , i = 1 , … , k of equal size are drawn from each of k populations and let X i denote the smallest observation of the i th sample. The population corresponding to the largest X i is selected. We consider the problem of estimating the scale parameter of the selected population and obtain the uniformly minimum variance unbiased estimator (UMVUE) when the shape parameters are assumed to be equal. An admissible class of linear estimators is derived. Further, a general inadmissibility result for the scale equivariant estimators is proved.

Assessing process performance with incapability index based on multiple samples

The process incapability index, which provides an uncontaminated separation between information concerning the process accuracy and the process precision, has been introduced to provide quantitative measures on process performance. Contributions based on a single sample of the estimated incapability index have been proposed extensively. In this paper, estimation of the incapability index based on multiple samples is considered for normally distributed processes. The uniformly minimum variance unbiased estimator (UMVUE) is derived. Upper confidence limits based on the UMVUE are constructed. A useful procedure for judging whether the process satisfies the preset quality reliability requirement is also proposed for practitioners to use.

Best unbiased estimation of fixed effects in mixed ANOVA models

In a linear model with an arbitrary variance–covariance matrix, Zyskind (Ann. Math. Statist. 38 (1967) 1092) provided necessary and sufficient conditions for when a given linear function of the fixed-effect parameters has a best linear unbiased estimator (BLUE). If these conditions hold uniformly for all possible variance–covariance parameters (i.e., there is a UBLUE) and if the data are assumed to be normally distributed, these conditions are also necessary and sufficient for the parametric function to have a uniformly minimum variance unbiased estimator (UMVUE). For mixed-effects ANOVA models, we show how these conditions can be translated in terms of the incidence array, which facilitates verification of the UBLUE and UMVUE properties and facilitates construction of designs having such properties.

UMVU estimation of the reliability function of the generalized life distributions

The problems of estimating the reliability function and P=PrX > Y are considered for the generalized life distributions. Uniformly minimum variance unbiased estimators (UMVUES) of the powers of the parameter involved in the probabilistic model and the probability density function (pdf) at a specified point are derived. The UMVUE of the pdf is utilized to obtain the UMVUE of the reliability function and ‘P’. Our method of obtaining these estimators is quite simple than the traditional approaches. A theoretical method of studying the behaviour of the hazard-rate is provided.

A construction of the UMVU estimator for simple quadratic natural exponential families

This paper presents a construction of the uniformly minimum variance unbiased (UMVU) estimator of real-valued functions for the simple quadratic natural exponential families on R d . A polynomial expansion of the estimator is derived and a condition for its existence is given. The exact variance of the UMVU estimator is calculated. It is also shown that the series of the multidimensional Bhattacharyya bounds converges to this variance. These results are extensions of Morris (Ann. Statist. 11 (1983) 515) and Blight and Rao (Biometrika 61 (1974) 137). Some illustrations are indicated. Non-i.i.d. and biased cases are also discussed.

A reliable procedure for testing linear regulators with one-sided specification limits based on multiple samples

Process capability indices C PU and C PL are developed in the manufacturing industry to provide quantitative measures on process potential and performance, for processes with one-sided specification limits. Statistical properties of the estimators of C PU and C PL have been investigated extensively, but only restricted to cases with single samples. In this paper, we consider the estimation and capability testing of C PU and C PL based on multiple samples. We show that the proposed estimator of C PU and C PL are indeed the uniformly minimum variance unbiased estimators (UMVUEs). A simple procedure based on a hypothesis testing using the UMVUE is developed for the practitioners to use for judging whether a stable process meets the preset capability requirement.