Complete convergence of weighted sums under negative dependence

In this paper, we study the complete convergence of weighted sums. In fact, we extend the result of Amini and Bozorgnia (J Appl Math Stoch Anal 16(2):121–126, 2003) on unweighted average to a weighted average under mild conditions.     

论综合指数和平均数指数

综合指数和平均数指数在由简单形式发展到加权形式的过程中,源于L氏指数和P氏指数中同度量因素（权数）的选择和时间的固定的不同,在其编制和计算中形成了L氏指数法和P氏指数法—由此,加权综合指数和加权平均数指数构成了当代编制与计算总指数中最重要最基本的两种形式。于是在编制质量指标指数和数量指标指数时,分别采用P氏指数法和L氏指数法。加权综合指数和加权平均数指数在二者的变量（或指数化因素或指数化指标）和同度量因素（权数）、分子分母资料的表现形式、对资料的要求和使用等三方面有不同之处,但后者是前者的变形。     

Randomly weighted average with Dirichlet component proportions

This article introduces a new average of n independent continuous random variables X₁, …, X_n weighted by Dirichlet random components. A relation between the Cauchy–Stieltjes transforms of the distribution functions of this weighted average and X₁, …, X_n is established. Several examples illustrate usefulness and applicability of the result.     

Classes of power semicircle laws that are randomly weighted average distributions

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n independent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of size n?1, for each n. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.     

On the strong convergence for weighted sums of random variables

A strong convergence result is obtained for weighted sums of identically distributed negatively associated random variables which have a suitable moment condition. This result improves the result of Cai (Metrika 68:323–331, 2008).     

Estimating coefficients of single-index regression models by minimizing variation

This paper proposes a novel estimation of coefficients in single-index regression models. Unlike the traditional average derivative estimation [Powell JL, Stock JH, Stoker TM. Semiparametric estimation of index coefficients. Econometrica. 1989;57(6):1403–1430; Hardle W, Thomas M. Investigating smooth multiple regression by the method of average derivatives. J Amer Statist Assoc. 1989;84(408):986–995] and semiparametric least squares estimation [Ichimura H. Semiparametric least squares (sls) and weighted sls estimation of single-index models. J Econometrics. 1993;58(1):71–120; Hardle W, Hall P, Ichimura H. Optimal smoothing in single-index models. Ann Statist. 1993;21(1):157–178], the procedure developed in this paper is to estimate the coefficients directly by minimizing the mean variation function and does not involve estimating the link function nonparametrically. As a result, it avoids the selection of the bandwidth or the number of knots, and its implementation is more robust and easier. The resultant estimator is shown to be consistent. Numerical results and real data analysis also show that the proposed procedure is more applicable against model free assumptions.     

An optimal k of kth MA-ARIMA models under a class of ARIMA model

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated ARIMA(p, d, q) model. We run a simulation using the three above examining methods under specific conditions. The main finding is that 5th exponential weighted moving average (5th EWMA) ARIMA model is the best forecasting model among others, which means the optimal k = 5. For Turkish Telecommunications (TTKOM) stock market, real data reveal the similar results of simulation study.     

Using the predictive distribution to determine control limits for the Bayesian MEWMA chart

Bayesian control charts have been proposed for monitoring multivariate processes with the multivariate exponentially weighted moving average (MEWMA) statistic. It has been suggested that we use limits based on the predictive distribution of the MEWMA statistic. This analysis, however is based on the erroneous result that the average run length (ARL) is a function of the exceedance probability, that is, the probability that the first point exceeds the control limit. We show how this result can be corrected and we discuss how the Bayesian MEWMA chart with limits based on the predictive distribution compares with other multivariate control chart procedures.     

A comparison of statistical methods for combining relative bioactivities from parallel line bioassays

This paper compares the ordinary unweighted average, weighted average, and maximum likelihood methods for estimating a common bioactivity from multiple parallel line bioassays. Some of these or similar methods are also used in meta‐analysis. Based on a simulation study, these methods are assessed by comparing coverage probabilities of the true relative bioactivity and the length of the confidence intervals computed for these methods. The ordinary unweighted average method outperforms all statistical methods by consistently giving the best coverage probability but with somewhat wider confidence intervals. The weighted average methods give good coverage and smaller confidence intervals when combining homogeneous bioactivities. For heterogeneous bioactivities, these methods work well when a liberal significance level for testing homogeneity of bioactivities is used. The maximum likelihood methods gave good coverage when homogeneous bioactivities were considered. Overall, the preferred methods are the ordinary unweighted average and two weighted average methods that were specifically developed for bioassays. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximations to the p-values of tests for a change-point under non-standard conditions

Three test statistics for a change-point in a linear model, variants of those considered by Andrews and Ploberger [Optimal tests when a nusiance parameter is present only under the alternative. Econometrica. 1994;62:1383–1414]: the sup-likelihood ratio (LR) statistic; a weighted average of the exponential of LR-statistics and a weighted average of LR-statistics, are studied. Critical values for the statistics with time trend regressors, obtained via simulation, are found to vary considerably, depending on conditions on the error terms. The performance of the bootstrap in approximating p-values of the distributions is assessed in a simulation study. A sample approximation to asymptotic analytical expressions extending those of Kim and Siegmund [The likelihood ratio test for a change-point in simple linear regression. Biometrika. 1989;76:409–423] in the case of the sup-LR test is also assessed. The approximations and bootstrap are applied to the Quandt data [The estimation of a parameter of a linear regression system obeying two separate regimes. J Amer Statist Assoc. 1958;53:873–880] and real data concerning a change-point in oxygen uptake during incremental exercise testing and the bootstrap gives reasonable results.     

The Design of the Multivariate Synthetic Exponentially Weighted Moving Average Control Chart

A multivariate synthetic exponentially weighted moving average (MSEWMA) control chart is presented in this study. The MSEWMA control chart consists of a multivariate exponentially weighted moving average (MEWMA) control chart and a conforming run length control chart. The average run length of the MSEWMA control chart is obtained using a Markov chain approach. From the numerical comparisons, it is shown that the MSEWMA control chart is more efficient than the multivariate synthetic T ² control chart and the MEWMA control chart for detecting shifts in the process mean vector.     

An optimal k of kth MA-ARIMA models under MA(q) models

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated MA(q) model. We run a simulation using the three above examining methods under specific conditions. The main finding is that, 5th Exponential Weighted Moving Average (5-th EWMA) Autoregressive Integrated Moving Average (ARIMA) model is the best forecasting model among others, which means the optimal k = 5. For Turkish Telecommunications (TTKOM), stock market real data reveals the similar results of the simulation study.     

Maximum Chi-Square Generally Weighted Moving Average Control Chart for Monitoring Process Mean and Variability

In this article, we propose a new control chart called the maximum chi-square generally weighted moving average (MCSGWMA) control chart. This control chart can effectively combine two generally weighted moving average (GWMA) control charts into a single one and can detect both increases as well as decreases in the process mean and/or variability simultaneously. The average run length (ARL) characteristics of the MCSGWMA and maximum exponentially weighted moving average (MaxEWMA) charts are evaluated by performing computer simulations. The comparison of the ARLs shows that the MCSGWMA control chart performs better than the MaxEWMA control chart.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

A Double Moving Average Control Chart

The double exponentially weighted moving average (DEWMA) technique has been investigated in recent years for detecting shifts in the process mean and has been shown to be more efficient than the corresponding exponentially weighted moving average (EWMA) technique. In this article, we extend the DEWMA technique of performing exponential smoothing twice to the double moving average (DMA) technique by computing the moving average twice. Using simulation, we show that our proposed DMA chart improves upon the ARL performance of the moving average (MA) chart in detecting mean shifts of small to moderate magnitudes. It is also shown through simulation that, generally, the DMA charts with spans, w = 10 and 15 provide comparable average run length (ARL) performances to the EWMA and cumulative sum (CUSUM) charts, designed for detecting small shifts.     

Average kappa coefficient: a new measure to assess a binary test considering the losses associated with an erroneous classification

The weighted kappa coefficient of a binary diagnostic test (BDT) is a measure of performance of a BDT, and is a function of the sensitivity and the specificity of the diagnostic test, of the disease prevalence and the weighting index. Weighting index represents the relative loss between the false positives and the false negatives. In this study, we propose a new measure of performance of a BDT: the average kappa coefficient. This parameter is the average function of the weighted kappa coefficients and does not depend on the weighting index. We have studied three asymptotic confidence intervals (CIs) for the average kappa coefficient, Wald, logit and bias-corrected bootstrap, and we carried out some simulation experiments to study the asymptotic coverage of each of the three CIs. We have written a program in R, called ‘akcbdt’, to estimate the average kappa coefficient of a BDT. This program is available as supplementary material. The results were applied to two examples.     

最优加权几何平均组合预测方法研究

Based on theory of weighted arithmetic average prediction method, the paper introduces the optimal weighted geometric combined prediction method, arrives at some important conclusions about judging existence of optimal weighted geometric combined model and finally the author make an error analysis and comparison between optimal weighted average arithmetic combined prediction and optimal weighted geometric average combined prediction.     

Strong uniform consistency results of the weighted average of conditional artificial data points

In this paper, we study strong uniform consistency of a weighted average of artificial data points. This is especially useful when information is incomplete (censored data, missing data …). In this case, reconstruction of the information is often achieved nonparametrically by using a local preservation of mean criterion for which the corresponding mean is estimated by a weighted average of new data points. The present approach enlarges the possible scope for applications beyond just the incomplete data context and can also be useful to treat the estimation of the conditional mean of specific functions of complete data points. As a consequence, we establish the strong uniform consistency of the Nadaraya–Watson [Nadaraya, E.A., 1964. On estimating regression. Theory Probab. Appl. 9, 141–142; Watson, G.S., 1964. Smooth regression analysis. Sankhyā Ser. A 26, 359–372] estimator for general transformations of the data points. This result generalizes the one of Härdle et al. [Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16, 1428–1449]. In addition, the strong uniform consistency of a modulus of continuity will be obtained for this estimator. Applications of those two results are detailed for some popular estimators.