Notices

This paper presents the small sample optimum choice of the k(≤ r ₂) order statistics for the best linear unbiased estimate (BLUE) of the parameters μ and σ or σ alone (μ known) when the sample is Type II censored on the right. For n = 2(1)10, k = 1(1)r ₂ and Tr ₂ = ([.504n]+1}(1)n, the optimum ranks, the coefficients of the BLUES have been presented in Table I.     

On the Interval Estimation of Stress–Strength Reliability for Exponentiated Scale Family of Distributions

The stress–strength reliability R=P(X₁<X₂) where X₁ and X₂ represent the stress applied and strength of an equipment, respectively, plays a crucial role in setting warranty periods while launching new brands of a product. The paper addresses the issue of estimating R when X₁ and X₂ belong to the exponentiated scale family, which includes the popular EED that has proven to be an excellent model for lifetime distributions. The cases of known/unknown and equal/unequal scale parameters are handled separately. For known scale parameter, a generalized pivot quantity for the shape parameter and R are developed. The interval estimates of R based on the proposed generalized pivot quantity exhibited uniformly best performance. For an unknown scale parameter, a maximum scale invariant likelihood estimator of the shape and an allied estimator of the scale are introduced. The parametric bootstrap interval estimates of R based on a proposed maximum scale invariant likelihood estimator of the shape parameter exhibited best performance among others. An application in setting warranty periods is illustrated based on two real data sets. Copyright © 2017 John Wiley & Sons, Ltd.     

Minimum Distance Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.     

A short note on the effect of sample size on the estimation error in Cp

Process capability indices such as C_p are used extensively in manufacturing industries to assess processes in order to decide about purchasing. In practice, the parameter for calculating C_p is rarely known and is frequently replaced with estimates from an in-control reference sample. This article explores the optimal sample size required to achieve a desired error of estimation using absolute percentage error of different C_p estimates. Moreover, some practical tools are created to allow practitioners to find sample size in different situations.     

Design of Over-Stress Life-Test Experiments When Failure Times Have the Two-Parameter Weibull Distribution

This paper deals with estimation of reliability parameters when life testing is conducted at stress levels above that which would normally be applied in standard usage. A range of stress for testing is prescribed, and a two-parameter Weibull model for failure times is assumed. The logarithm of the Weibull scale parameter is assumed to be a polynomial function of known degree k of the reciprocal of stress level. The Weibull shape parameter is assumed to be independent of stress level.

The problem considered is that of determining the design for obtaining the least-squares-curve intercept with minimum variance at the nominal testing level. The design obtained specifies the number and location of stress levels in the prescribed range at which the life tests will be conducted and proportion of the total sample of specified size to be randomly allocated to each testing level.     

Choosing the smoothing parameter for unordered multinomial data

We consider the estimation of multinomial probabilities in the non-sparse univariate unordered case. We describe a number of explicit methods (mostly pre-existing) for the choice of the smoothing parameter in this context. In simulations, we compare these methods in terms of mean root mean squared error performance. Our recommendation is for the routine use of the simplest Bayesian estimation formula in which the probability in thek'th cell is estimated by (n _k +1)/(N+K) wheren _k is the count in thek'th cell,N is the sample size andK is the number of cells.     

When is optimal experimental design advantageous for the analysis of Michaelis–Menten kinetics?

A systematic analysis is performed for the estimation of parameters for the Michaelis–Menten kinetic using the dynamic mechanistic model. As quality measure the Cramer–Rao lower bond (CRLB) is used. Depending on three different optimization criteria, two different measurement errors, the process operation mode (batch and fed-batch) as well as the error of the rough estimates of the parameters used for optimal experimental design the CRLB is calculated. These values are used to decide, if optimal experimental design is favourable compared to equidistant measurement points. It will be demonstrated, that optimization criteria A and E are just slightly more favourable than criteria D. The considered measurement errors give different results, but will not change the design and the precision of the parameters significant. The fed-batch process operation gives always a significant higher precision of the parameters compared to the batch mode. If the rough estimated parameter values, with which the optimal experimental design is carried out, are just known with low accuracy equidistant measurement points are in favour to model-based optimal experimental design. Using a batch process the error of the rough value of k₂ (=v_max/E_tot) must be smaller than 17% and of K_m smaller than 40% so an optimal experimental design is worth for the parameter estimation. For fed-batch the errors of the rough estimates can be even bigger (up to 40% for k₂ and 68% for K_m).     

On Identifying the Population of Origin of Each Observation in a Mixture of Observations from Two Gamma Populations

The gamma distribution, known also as the Erlangian distribution, and its special case the exponential distribution arise in many technological applications of statistics. The present note is on the problem of identifying the population of origin of each observation in a sample thought to be the result of mixing a random sample of size N ₁, from a gamma distribution with scale parameter σ₁ and an independent random sample of size N ₂ from another gamma distribution with scale parameter σ₂. We shall also be interested in the estimation of σ₁ and σ₂. The method of moments and the maximum likelihood method are applied to the solution of these problems.     

Identification of parameters with different orders of magnitude in chaotic systems

The synchronization and parameter identification of six unknown parameters in a chaotic neuron model, which one parameter (about 0.006) is 3 orders of magnitude smaller than the others (about 1–5), is investigated by using Lyapunov stability theory and adaptive synchronization in detail. Two gain coefficients (δ₁,?δ₂) are introduced into the Lyapunov function to obtain certain optimized controllers and parameter observers. A selectable amplification factor k ₀ is presented using scale conversion and it is used to improve the accuracy of parameter estimation with the smallest order. The parameter space for gain coefficient (δ) versus amplification factor k ₀, and the parameter space δ₁ versus δ₂ at certain fixed amplification factor k ₀ are calculated numerically. It is found that the selection values of optimized gain coefficients and amplification factor are critical to estimate the six unknown parameters, particularly for the smallest unknown parameters with an order 0.001. The extensive numerical results show that it is more effective to estimate the smallest unknown parameter r when the two gain coefficients δ₁ and δ₂ are given the same value and a higher amplification factor k ₀ is used. It could be useful to estimate the unknown parameters with large deviation of order magnitude, such as a single chaotic Josephson junction coupled to a Resonant tank and other chaotic systems with potential application [Z.Y. Wang, H.Y. Liao, and S.P. Zhou, Study of the DC biased Josephson junction coupled to a Resonant tank, Acta. Phys. Sin. 50(10) (2001), pp. 1996–2000 (in Chinese)].     

Unbiased estimates of the Weibull parameters by the linear regression method

For sample sizes from 5 to 100, the bias of the scale parameter was investigated for probability estimators, P = (i − a)/(n + b), which yield unbiased estimates of the shape parameter. A class of unbiased estimators for both the shape and scale parameters was developed for each sample size. In addition, the percentage points of the distribution of unbiased estimate of the shape parameter were determined for all sample sizes. The distribution of the scale parameter was found to be normal by using the Anderson-Darling goodness-of-fit test. How the results can be used to establish confidence intervals on both the shape and scale parameters are demonstrated in the paper.     

Bayesian approach to parameter estimation of the generalized pareto distribution

Several methods have been used for estimating the parameters of the generalized Pareto distribution (GPD), namely maximum likelihood (ML), the method of moments (MOM) and the probability-weighted moments (PWM). It is known that for these estimators to exist, certain constraints have to be imposed on the range of the shape parameter,k, of the GPD. For instance, PWM and ML estimators only exist fork>−0.5 andk≤1, respectively. Moreover, and particularly for small sample sizes, the most efficient method to apply in any practical situation highly depends on a previous knowledge of the most likely values ofk. This clearly suggests the use of Bayesian techniques as a way of using prior information onk. In the present work, we address the issue of estimating the parameters of the GPD from a Bayesian point of view. The proposed approach is compared via a simulation study with ML, PWM and also with the elemental percentile method (EPM) which was developed by Castillo and Hadi (1997). The estimation procedure is then applied to two real data sets.     

Testing the Mean and Standard Deviation of a Normal Distribution Using Quantiles

Assuming a normally distributed population and a large sample size, three tests of hypotheses are presented using sample quantiles. First, we test whether μ = μ₁ and σ = σ₁ or μ = μ₂ and σ = σ₂, using one optimum quantile. Then test statistics and acceptance regions are given for testing the mean and standard deviation independently. The efficiency of each test is given relative to the best test using all the sample values.     

On the Estimation of Safe Life When the Underlying Life Distribution Is Weibull

The problem of estimating Y ₍₁₎, the smallest of a future sample of k observations from the Weibull distribution, based on an observed sample from the same distribution, is considered. A conditional confidence interval solution is proposed for the estimation of Y ₍₁₎. The results have direct application in reliability theory, where the time until the first failure in a group of k items in service provides a measure of assurance regarding the operation of the items.     

On the outstanding elements and record values in the exponential and gamma populations

Summary Outstanding elements and record values are discussed in this paper as related to exponential and gamma populations. First, the problem of prediction is considered when there are available,k sets of independent observations from a general-type exponential distribution. In such a case, prediction of then _k -th record value in thek-th set is made in terms ofn _i -th (i=1, …,k−1) record values from other (k−1) sets. For this purpose a predictive distribution is obtained. Secondly, the distribution of the sum of record values as well as that of a linear combination of record values are obtained for the exponential case. Probability integrals of the sum of record values and the probability integral of the sum of outstanding, elements are suggested for all values. Then, the distribution of then-th record values in a gamma population is put in a closed form. Further, the distribution of the linear combination of the spacings of outstanding elements as well as that of the linear combination of outstanding elements themselves are obtained. Finally the distribution of a ratio of two record values is obtained.     

On Jonckheere's k-Sample Test Against Ordered Alternatives

In this paper a k-sample non-parametric test for trend is considered. Given a sample of size n_i , i = 1, …, k respectively from each of k populations, the test rejects the hypothesis that the k populations are identical if S = Σ ^k _i=2 S_i ≥ S_i . Here Si is the Mann-Whitney statistic computed when each observation in the i-th sample is compared with the combined observations from the first (i – 1) populations. A recurrence formula is derived for computing the exact distribution of S. Tables of exact probabilities and critical values are given for nominal values of α = 0.5, 0.2, 0.1, 0.05, 0.025, 0.01, and 0.005 for k = 3 and all possible sample sizes from 2 to 8, and for equal sample sizes for values of n = 2(1)6, k = 4(1)6.     

Scattering of Scalar Waves by an Impenetrable Rough Sphere

Scattering of scalar waves by an impenetrable, slightly rough sphere is considered. The incoherent differential cross-section is calculated to the lowest order in hk (where k is the wavenumber and h ² the mean-square deviation from a smooth sphere), but for any value of the size parameter α = kR₀ (where R ₀ is the radius of the sphere). Particular attention is paid to the back-scattering cross-section σ_B for large α, under the assumption that the correlation length of the roughness structure is not larger than the incident wavelength. The resulting limit of σ_B is well known: the same cross-section is obtained by approximating the sphere locally by a tangential plane and using results for slightly rough planar areas.     

A Problem of Ranking and Estimation With Poisson Processes

There are given k Poisson processes with parameters (rates of occurrence) λ₁, …, λ _k . Let λ₍₁₎ ≤ λ₍₂₎ ≤ … ≤ λ_(k) denote the ordered set of values of the parameters. A procedure is given for selecting the process corresponding to λ_(k) and estimating its parameter (λ_(k)). The given procedure controls the joint risk of improper selection and of large error in the estimate. Let θ > 1 and 0 < α, β < 1 be given numbers, and let δ denote the estimate of λ_(k). The joint probability that a correct selection is made and that |(δ/λ_(k)) ? 1| ≤ α is at least as large as β, for (λ_(k)/λ_(k?1)) ≥ θ. Two cases are considered, that is, when the processes are observed continuously in time, and when they are observed at successive intervals of time. Both the cases lead to the same theoretical results.     

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p₀) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p₀ is known; therefore, when p₀ is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p₀ to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p₀ values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p₀. A methodology to identify a continuous variable to monitor is highly recommended when the value of p₀ is low and the required phase I sample size is impractically large. Copyright © 2012 John Wiley & Sons, Ltd.     

Revisiting reweighted robust standard deviation estimators for univariate Shewhart S‐charts

Phase I outliers, unless screened during process parameter estimation, are known to deteriorate Phase II performance of process control charts. Reweighting estimators, ie, trimming outlier subgroups and individual observations, were suggested in the literature to improve both the robustness and efficiency of the resulting parameter estimates. In the current study, effects of various reweighted estimators at different trimming levels on the Phase II performance of S‐charts are elucidated using computer simulations including isolated and mixtures of contamination models. Outlier magnitudes in the simulations are held at a moderately low level to mimic industrial practice. Subtleties, such as varying Type I error rate among different trimming levels with respect to quantiles of dispersion estimates, prevent a single method to be revealed as the best performing one under all circumstances, and choice of estimators and trimming levels should depend on the number of subgroups in Phase I and the specifics of the process. Nevertheless, S‐chart using scale M‐estimator with logistic ρ and location M‐estimator at 2% trimming generally stands out in terms of Phase II performance, and high trimming levels are particularly recommended for high number of Phase I subgroups.     

Handbook of Statistical Tables

The Student's test of H ₀: μ = μ₀ against H ₁: μ ≠ μ₀ of the location parameter μ of the normal, logistic and Cauchy distributions when the scale parameter σ is unknown based on a few selected sample percentiles is considered. Tables facilitating the computation of the testing statistic are provided and the effects of the biases of the linear estimates involved on the test are investigated.