An Economic Design of the Bounded Reset Chart Using the Integral Controller for Batch-Oriented Processes

In batch-oriented processes, the quality of the process tends to deteriorate over time. A bounded reset chart (BRC) using the integral controller is proposed to adjust such processes effectively. The effectiveness of the BRC procedure is evaluated in terms of the average cost per unit interval (ACU). Optimal values for the reset limit, considering various process and cost parameters, are studied to minimize the ACU. The BRC procedure is shown to be effective for adjusted processes with deterioration over time compared to the traditional adjustment procedure where resetting is not considered.     

OPTIMAL CONTROL OF A DETERIORATING PRODUCTION PROCESS

We study the optimal control of a production process subject to a deterministicdrift and to random shocks. The process mean is observable at discrete points of time after producing a batch and, at each such point, a decision is made whether to reset the process mean to some initial value or to continue with the production. The objective is to find the initial setting of the process mean and the resetting time that minimizes the expected average cost per unit time. It is shown that the optimal control policy is of a control limit type. An algorithm for finding the optimal control parameters is presented.     

On an adjustment of degrees of freedom in the minimim mean squared error ertimator

In this paper, we consider an adjustment of degrees of freedom in the minimum mean squared error (MMSE) estimator, We derive the exact MSE of the adjusted MMSE (AMMSE) estimator, and compare the MSE of the AMMSE estimator with those of the Stein-(SR), positive-part Stein-rule (PSR) and MMSE estimators by numerical evaluations. It is shown that the adjustment of degrees of freedom is effective when the noncentrality parameter is close to zero, and the MSE performance of the MMSE estimator can be improved in the wide region of the noncentrality parameter by the adjustment, ft is also shown that the AMMSE estimator can have the smaller MSE than the PSR estimator in the wide region of the noncentrality parameter     

Some results on maximum likelihood estimators of parameters of exponential distribution under type I progressive censoring with changing failure rates

In this paper the study of relative bias (RB), exact variance and mean square error (MSE) of the maximum likelihood estimators of the exponential distribution under type I progressive censoring with changing failure rates is considered. A minimum mean square error (MMSE) estimator for the parameter at each stage is proposed. The numerical evalution of their relative performance is made for selected values of n and p. Further results concerning group-censoring, total expected waiting time and optimal spacings of the times of censoring are derived and results obtained by Kendell and Anderson (1971) are deduced as special cases.     

Setting Policy for On-Line Process Control with Dynamic Characteristics

In this article, we investigate a control policy for the choice of sampling interval and control limit by minimizing the expected quality cost. The study is based on the environment in which (i) the stochastic disturbances are assumed to follow an IMA(1, 1) process, (ii) there is process dynamics between the input series and the output series, (iii) a feedback control scheme is imposed, and (iv) the expected quality cost contains off-target cost, adjustment cost, and inspection cost. Modeling and forecasting for (i), (ii), and (iii) are performed according to the transfer function plus noise model. Minimizing the expected quality cost for (iv) is carried out by a modified pattern search procedure. An example is given to demonstrate the advantage of using the pattern search method over the usual 3-sigma control scheme. The penalty of ignoring the process dynamics and for the case of choosing incorrect value of θ of an IMA(1, 1) disturbance is discussed. The pattern search method is also compared favorably with the modified Taguchi's method in quality cost for the cases considered therein.     

Bias Adjustment Minimizing the Asymptotic Mean Square Error

A method of bias adjustment which minimizes the asymptotic mean square error is presented for an estimator typically given by maximum likelihood. Generally, this adjustment includes unknown population values. However, in some examples, the adjustment can be done without population values. In the case of a logit, a reasonable fixed value for the adjustment is found, which gives the asymptotic mean square error smaller than those of the asymptotically unbiased estimator and the maximum likelihood estimator. The weighted-score method, which yields directly the estimator with the minimized asymptotic mean square error, is also given.     

Economical process adjustment with sampling interval

Srivastava and Wu and Box and Kramer considered an integrated moving average process of order one with sampling interval for process adjustment. However, the results were obtained by asymptotic methods and simulations respectively. In this paper, these results are obtained analytically. It is assumed that there is a sampling cost and an adjustment cost. The cost of deviating from the target-value is assumed to be proportional to the square of the deviations. The long-run average cost is evaluated exactly in terms of moments of the randomly stopped random walk. Two approximations are given and shown by simulation to be close to the exact value One of these approximations is used to obtain an explicit expression for the optimum value of the inspection interval and the control limit where an adjustment is to be made.     

A Note on Superiority Comparisons of Homogeneous Linear Estimators

This note extends some results on homogeneous linear estimators to the general, even nonlinear case.A Sufficient condition for the difference of mean square error matrices of minimum conditional mean square error estimator and minimum average risk linear estimator to be postive definite is derived.     

Randomized and random run order experiments

Randomization in industrial and scientific experiments on equipment has meant randomizing the order of application of levels of treatments to units. This definition is inadequate because it does not render independent error terms. Randomization also requires independent resettings of treatment levels when the levels for the preceding run are the same. We review how the literature incorrectly explains how randomization is to be carried out. The need to reset levels of a treatment from one run to the next is never emphasized. Using a simple example we show why statistical tests are biased for all treatments even when levels for just one treatment are not independently reset. Even if the expected mean squares recognize the restrictions on randomization, the usual F test will not give predictable results because its numerator and denominator are correlated.Experimental design on equipment includes experiments from the chemical, automobile, pharmaceutical, and aeronautical industries. The statistical interpretation of data from such experiments will be misleading. Books on experimental design must emphasize the independent resetting of levels just as carefully as they emphasize the random assignment of treatment levels.     

Reinterpreting a Temporally Aggregated Consumption CAP Model

Recent literature has found that the coefficient of relative risk aversion estimated through consumption-based asset-pricing models is implausibly high, even when the phenomenon of temporal aggregation is taken into account. This article suggests that an IMA (1,1) process be assumed as the generating mechanism of consumption, instead of the standard random-walk process. In this case, if the coefficient of the moving average component is negative, the implied value of the coefficient of relative risk aversion can be reduced to plausible levels. Some empirical and theoretical support for the IMA (1,1) hypothesis is also presented and discussed.     

On moment stability properties for a class of state-dependent stochastic networks

We consider a class of stochastic networks with state-dependent arrival and service rates. The state dependency is described via multi-dimensional birth/death processes, where the birth/death rates are dependent upon the current population size in the system. Under the uniform (in state) stability condition, we establish several moment stability properties of the system:

• (i) 

  the existence of a moment generating function in a neighborhood of zero, with respect to the unique invariant measure of the state process;

• (ii) 

  the convergence of the expected value of unbounded functionals of the state process to the expectation under the invariant measure, at an exponential rate;

• (iii) 

  uniform (in time and initial condition) estimates on exponential moments of the process;

• (iv) 

  growth estimates of polynomial moments of the process as a function of the initial conditions.

Our approach provides elementary proofs of these stability properties without resorting to the convergence of the scaled process to a stable fluid limit model.     

Generalized value at risk forecasting

Abstract

In this paper, using estimating function approach, a new optimal volatility estimator is introduced and based on the recursive form of the estimator a data-driven generalized EWMA model for value at risk (VaR) forecast is proposed. An appropriate data-driven model for volatility is identified by the relationship between absolute deviation and standard deviation for symmetric distributions with finite variance. It is shown that the asymptotic variance of the proposed volatility estimator is smaller than that of conventional estimators and is more appropriate for financial data with larger kurtosis. For IBM, Microsoft, Apple stocks and SP 500 index the proposed method is used to identify the model, estimate the volatility, and obtain minimum mean square error(MMSE) forecasts of VaR.     

Optimal estimation of a finite population mean under generalized random permutation models

Employing certain generalized random permutation models and a general class of linear estimators of a finite population mean, it is shown that many of the conventional estimators are “optimal” in the sense of minimum average mean square error. Simple proofs are provided by using a well-known theorem on UMV estimation. The results also cover certain simple response error situations.     

MSE performance of the weighted average estimators consisting of shrinkage estimators

In this paper, we consider a regression model and propose estimators which are the weighted averages of two estimators among three estimators; the Stein-rule (SR), the minimum mean squared error (MMSE), and the adjusted minimum mean-squared error (AMMSE) estimators. It is shown that one of the proposed estimators has smaller mean-squared error (MSE) than the positive-part Stein-rule (PSR) estimator over a moderate region of parameter space when the number of the regression coefficients is small (i.e., 3), and its MSE performance is comparable to the PSR estimator even when the number of the regression coefficients is not so small.     

Computing optimal adjustment schemes for the general tool-wear problem

An important problem in process adjustment using feedback is how often to sample the process and when and by how much to apply an adjustment. Minimum cost feedback schemes based on simple, but practically interesting, models for disturbances and dynamics have been discussed in several particular cases. The more general situation in which there may be measurement and adjustment errors, deterministic process drift, and costs of taking an observation, of making an adjustment, and of being off target, is considered in this article. Assuming all these costs to be known, a numerical method to minimize the overall expected cost is presented. This numerical method provides the optimal sampling interval, action limits, and amount of adjustment; and the resulting average adjustment interval, mean squared deviation from target, and minimum overall expected cost. When the costs of taking an observation, of making an adjustment, and of being off target are not known, the method can be used to choose a particular scheme by judging the advantages and disadvantages of alternative options considering the mean squared deviation they produce, the frequency with which they require observations to be made, and the resulting overall length of time between adjustments. Computer codes that perform the required computations are provided in the appendices and applied to find optimal adjustment schemes in three real examples of application.     

Small Sample Performance of Some Statistical Setup Adjustment Methods

Abstract

The setup adjustment problem occurs when a machine experiences an upset at setup that needs to be compensated for. In this article, feedback methods for the setup adjustment problem are studied from a small-sample point of view, relevant in modern manufacturing. Sequential adjustment rules due to Grubbs (Grubbs, F. E. (1954 Grubbs, F. E. 1954. An optimum procedure for setting machines or adjusting processes. Industrial Quality Control July,  [Google Scholar]). An optimum procedure for setting machines or adjusting processes. Industrial Quality Control 07) and an integral controller are considered. The performance criteria is the quadratic off-target cost incurred over a small number of parts produced. Analytical formulae are presented and numerically illustrated. Two cases are considered, the first one where the setup error is a constant but unknown offset and the second one where the setup error is a random variable with unknown first two moments. These cases are studied under the assumption that no further shifts occur after setup. It is shown how Grubbs' harmonic rule and a simple integral controller provide a robust adjustment strategy in a variety of circumstances. As a by-product, the formulae presented in this article allow to compute the expected off-target quadratic cost when a sudden shift occurs during production (not necessarily at setup) and the adjustment scheme compensates immediately after its occurrence.     

Reliability estimation for the exponential distribution

This paper is concerned with classical statistical estimation of the reliability function for the exponential density with unknown mean failure time θ, and with a known and fixed mission time τ. The minimum variance unbiased (MVU) estimator and the maximum likelihood (ML) estimator are reviewed and their mean square errors compared for different sample sizes. These comparisons serve also to extend previous work, and reinforce further the nonexistence of a uniformly best estimator. A class of shrunken estimators is then defined, and it produces a shrunken quasi-estimator and a shrunken estimator. The mean square errors for both these estimators are compared to the mean square errors of the MVU and ML estimators, and the new estimators are found to perform very well. Unfortunately, these estimators are difficult to compute for practical applications. A second class of estimators, which is easy to compute is also developed. Its mean square error properties are compared to the other estimators, and it outperforms all the contending estimators over the high and low reliability parameter space. Since, for all the estimators, analytical mean square error comparisons are not tractable, extensive numerical analyses are done in obtaining both the exact small sample and large sample results.     

Designing an economic rectifying sampling plan in the presence of two markets

In this paper, an optimization model is developed for the economic design of a rectifying inspection sampling plan in the presence of two markets. A product with a normally distributed quality characteristic with unknown mean and variance is produced in the process. The quality characteristic has a lower specification limit. The aim of this paper is to maximize the profit, which consists the Taguchi loss function, under the constraints of satisfying the producer's and consumer's risk in two different markets simultaneously. Giveaway cost per unit of sold excess material is considered in the proposed model. A case study is presented to illustrate the application of proposed methodology. In addition, sensitivity analysis is performed to study the effect of model parameters on the expected profit and optimal solution. Optimal process adjustment problem and acceptance sampling plan is combined in the economical optimization model. Also, process mean and standard deviation are assumed to be unknown value, and their impact is analyzed. Finally, inspection error is considered, and its impact is investigated and analyzed.     

Economical process adjustment with measurement error

Srivastava and Wu (1997) considered a random walk model with sampling interval and measurement error which was assumed to be white noise. In this paper, we consider the situation in which the measurement error is also a random walk. It is assumed that there is a sampling cost and an adjustment cost. The cost of deviating from the target value is assumed to be proportional to the square of the deviations. The long-run average cost rate is evaluated exactly in terms of the first four moments of a randomly stopped random walk. Using approximations of those moments, optimum, values of the control parameters are given.     

Some asymptotic results in the multivariate lognormal estimation theory

An asymptotic expansion of the variance of the uniformly minimum variance unbiased estimator of a class of parameters of the multivariate lognormal distribution is considered. It is obtained by some calculations of the zonal polynomials. Then it is compared with an asymptotic mean square error of the maximum likelihood estimator of the same parameter.