Max control chart with adaptive sample sizes for jointly monitoring process mean and standard deviation

This paper develops the two-state and three-state adaptive sample size control schemes based on the Max chart to simultaneously monitor the process mean and standard deviation. Since the Max chart is a single variables control chart where only one plotting statistic is needed, the design and operation of adaptive sample size schemes for this chart will be simpler than those for the joint X? and S charts. Three types of processes including on-target initial, off-target initial and steady-state conditions are considered to evaluate the chart performance. The results of this study show that both two-state and three-state schemes are more efficient than the conventional non-adaptive joint X? and S charts. The three-state procedure is only slightly better than the two-state scheme, and the most dramatic improvement occurs when the two-state scheme is compared with the non-adaptive joint X? and S charts. Moreover, with the ease of implementation, the two-state scheme is likely adequate in most practical applications.     

Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

Revisiting the PERT mean and variance

Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (1959) to suggest in the Program Evaluation and Review Technique (PERT) their by now classical expressions for the mean and variance for activity completion for practical applications. In this note, we shall provide an alternative for the PERT variance expression addressing a concern raised by Hahn (2008) regarding the constant PERT variance assumption given the range for an activity’s duration, while retaining the original PERT mean expression. Moreover, our approach ensures that an activity’s elicited most likely value aligns with the beta distribution’s mode. While this was the original intent of Malcolm et al. (1959), their method of selecting beta parameters via the PERT mean and variance is not consistent in this manner.     

Portfolio selection and duality under mean variance preferences

This paper uses duality to analyze an investor’s behavior in a n-asset portfolio selection problem when the investor has mean variance preferences. The indirect utility and wealth requirement functions are used to derive Roy’s identity, Shephard’s lemma and the Slutsky equation. In our simple Slutsky equation the income effect is characterized by decreasing absolute risk aversion (DARA) and the substitution effect is always positive [negative] with respect to an asset’s holding if the asset’s mean return [risk] increases. Substitution effect and income effect work in the same direction presupposed mean variance preferences display DARA.     

Evaluation of mean and variance integrals without integration

The mean and variance of some continuous distributions, in particular the exponentially decreasing probability distribution and the normal distribution, are considered. Since they involve integration by parts, many students do not feel comfortable. In this note, a technique is demonstrated for deriving mean and variance through differential calculus. The general nature of the technique has potential for wider applications.     

Optimization designs and performance comparison of two CUSUM schemes for monitoring process shifts in mean and variance

In statistical process control (SPC), when dealing with a quality characteristic x that is a variable, it is usually necessary to monitor both the mean value and variability. This article proposes an optimization algorithm (called the holistic algorithm) to design the CUSUM charts for this purpose. It facilitates the determination of the charting parameters of the CUSUM charts and considerably or significantly increases their overall detection effectiveness. A single CUSUM chart (called the ABS CUSUM chart) has been developed by the holistic algorithm and fully investigated. This chart is able to detect two-sided mean shifts and increasing variance shifts by inspecting the absolute value of sample mean shift. The results of performance studies show that the overall performance of the ABS CUSUM chart is nearly as good as an optimal 3-CUSUM scheme (a scheme incorporating three individual CUSUM charts). However, since the ABS CUSUM chart is easier for implementation and design, it may be more suitable for many SPC applications in which both mean and variance of a variable have to be monitored.     

Individuals control schemes for monitoring the mean and variance of processes subject to drifts

This paper investigates control chart schemes for detecting drifts in the process mean μ and/or process standard deviation σ when individual observations are sampled. Drifts may be due to causes such as gradual deterioration of equipment, catalyst aging, waste accumulation, or human causes, such as operator fatigue or close supervision. The standard Shewhart X chart and moving range (MR) chart are evaluated, as well as several types of exponentially weighted moving average (EWMA) charts and combinations of charts involving these EWMA charts. We show that the combinations of the EWMA charts detect slow-rate and moderate-rate drifts much faster than the combined X and MR charts. We also show that varying the sampling interval adaptively as a function of the process data results in notable reductions in the detection delay of drifts in μ and/or σ.     

Asymptotic risk behavior of mean vector and variance estimators and the problem of positive normal mean

Asymptotic risk behavior of estimators of the unknow variance and of the unknown mean vector in a multivariate normal distribution is considered for a general loss. It is shown that in both problems this characteristic is related to the risk in an estimation problem of a positive normal mean under quadratic loss function. A curious property of the Brewster-Zidek variance estimator of the normal variance is also noticed.Research supported by NSF Grant DMS 9000999 and by Alexander von Humboldt Foundation Senior Distinguished Scientist Award.University of Münster     

Semiparametric bounds of mean and variance for exotic options

Finding semiparametric bounds for option prices is a widely studied pricing technique. We obtain closed-form semiparametric bounds of the mean and variance for the pay-off of two exotic (Collar and Gap) call options given mean and variance information on the underlying asset price. Mathematically, we extended domination technique by quadratic functions to bound mean and variances. This work was supported by National Science Foundation of the United States (Grant Nos. DMS-0720977 and DMS-0805929)     

On the existence of solutions to the quadratic mixed-integer mean–variance portfolio selection problem

In the standard mean–variance portfolio selection approach, several operative features are not taken into account. Among these neglected aspects, one of particular interest is the finite divisibility of the (stock) assets, i.e. the obligation to buy/sell only integer quantities of asset lots whose number is pre-established. In order to consider such a feature, we deal with a suitably defined quadratic mixed-integer programming problem. In particular, we formulate this problem in terms of quantities of asset lots (instead of, as usual, in terms of capital per cent quotas). Secondly, we provide necessary and sufficient conditions for the existence of a non-empty mixed-integer feasible set of the considered programming problem. Thirdly, we present some rounding procedures for finding, in a finite number of steps, a feasible mixed-integer solution which is better than the one detected by the necessary and sufficient conditions in terms of the value assumed by the portfolio variance. Finally, we perform an extensive computational experiment by means of which we verify the goodness of our approach.     

On the possibilistic mean value and variance of multiplication of fuzzy numbers

In this paper, we introduce the definitions of the possibilistic mean, variance and covariance of multiplication of fuzzy numbers, and show some properties of these definitions. Then, we apply these definitions to build the possibilistic models of portfolio selection under the situations involving uncertainty over the time horizon, by considering the portfolio selection problem from the point of view of possibilistic analysis. Moreover, numerical experiments with real market data indicate that our approach results in better portfolio performance.     

Semiparametric estimation of mean and variance functions for non-Gaussian data

Flexible modelling of the response variance in regression is interesting for understanding the causes of variability in the responses, and is crucial for efficient estimation and correct inference for mean parameters. In this paper we describe methods for mean and variance estimation where the responses are modelled using the double exponential family of distributions and mean and dispersion parameters are described as an additive function of predictors. The additive terms in the model are represented by penalized splines. A simple and unified computational methodology is presented for carrying out the calculations required for Bayesian inference in this class of models based on an adaptive Metropolis algorithm. Application of the adaptive Metropolis algorithm is fully automatic and does not require any kind of pretuning runs. The methodology presented provides flexible methods for modelling heterogeneous Gaussian data, as well as overdispersed and underdispersed count data. Performance is considered in a variety of examples involving real and simulated data sets.     

The robustness of mean and variance approximations in risk analysis

This paper examines further the problem of estimating the mean and variance of a continuous random variable from estimates of three points within the distribution, typically the median or mode and two extreme fractiles. The problem arises most commonly in PERT and risk analysis where it can usually be assumed that the distribution in question is bell-shaped and positively skewed, often typified by a Beta distribution. Over the years, a number of alternative approximations have been proposed, usually as modifications to the original PERT formulae. The accuracy of a number of these approximations is investigated based not only on a Beta distribution, but also for three other commonly used bell-shaped, positively skewed distributions, namely the Gamma, Lognormal and F distributions. It is shown that a balanced weighted average of the median and the 4% fractiles provides a consistent estimator of the distribution mean across all four distributions. Furthermore, reasonably accurate estimates of the variance can also be obtained by treating the three fractiles as defining an equivalent discrete distribution with the same probability weight as in the formula for the mean.     

Reliability bounds in DFRA class with known mean and variance

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Bayesian model selection approach to the one way analysis of variance under homoscedasticity

An objective Bayesian model selection procedure is proposed for the one way analysis of variance under homoscedasticity. Bayes factors for the usual default prior distributions are not well defined and thus Bayes factors for intrinsic priors are used instead. The intrinsic priors depend on a training sample which is typically a unique random vector. However, for the homoscedastic ANOVA it is not the case. Nevertheless, we are able to illustrate that the Bayes factors for the intrinsic priors are not sensitive to the minimal training sample chosen; furthermore, we propose an alternative pooled prior that yields similar Bayes factors. To compute these Bayes factors Bayesian computing methods are required when the sample sizes of the involved populations are large. Finally, a one to one relationship—which we call the calibration curve—between the posterior probability of the null hypothesis and the classical $p$ value is found, thus allowing comparisons between these two measures of evidence. The behavior of the calibration curve as a function of the sample size is studied and conclusions relating both procedures are stated.     

The variance function of the Erlang process

Summary A relatively simple exact expression of closed form is obtained for the varianceσ ²(t) of the asynchronous counting distribution for a counting period of lengtht,t>0, in an Erlang process. Useful bounds are placed upon the error of the linear approximation toσ ²(t). Implications of these results are examined. In particular, a new exact expression and related bounds are obtained for the mean function of the synchronous counts (also known as the renewal function of the process). All bounds given are sharp in asymptotic order of magnitude as the length of the counting period is allowed to increase. Prepared under Contract No. FH-11-6890, Bureau of Public Roads, Federal Highway Administration, U.S. Department of Transportation. The opinions, findings and conclusions expressed herein are those of the author and not necessarily those of the Bureau of Public Roads.     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.