All Admissible Linear Estimators under Quadratic Loss in Multivariate Model

For multivariate linear model Y=XΘ+ε, ~N(0, σ 2ΣV), this paper is concerned with the admissibility of linear estimators of estimable function SXΘ in the class of all estimators. All admissible linear estimators of SXΘ are given under each of four definitions of admissibility.

ADMISSIBILITY OF LINEAR ESTIMATORS OF REGRESSION COEFFICIENTS UNDER QUADRATIC LOSS

For the general fixed effects linear model: Y = X_T+ε, ε～N(0, V), V≥0, weobtain the necessary and sufficient conditions for LY +a to be admissible for a linear estimablefunction S_r in the class of all estimators under the loss function (d -- Sr)'D(d --Sr), whereD≥0 is known. For the general random effects linear model: Y = Xβ+ε,(βε)～N((Aα 0), (V_(11)V_(12)V_(21)V_(22))), ∧= XV_(11)X'+XV_(12)+ V_(21)X+V_(22)≥0, we also get the necessaryand sufficient conditions for LY+a to be admissible for a linear estimable function Sα+Qβin the class of all estimators under the loss function (d-Sα-Qβ)'D(d-Sα-Qβ).whereD≥0 is known.

New Hybrid EWMA Charts for Efficient Process Dispersion Monitoring with Application in Automobile Industry

The EWMAcharts are thewell-knownmemory-type charts used formonitoring the small-to-intermediate shifts in the process parameters(location and/or dispersion).The hybrid EWMA(HEWMA)charts are enhanced version of the EWMA charts,which effectivelymonitor the process parameters.This paper aims to develop two new uppersided HEWMAcharts for monitoring shifts in process variance,i.e.,HEWMA1 and HEWMA2 charts.The design structures of the proposed HEWMA1 and HEWMA2 charts are based on the concept of integrating the features of two EWMAcharts.TheHEWMA1 and HEWMA2charts plotting statistics are developed using one EWMAstatistic as input for the other EWMAstatistic.AMonte Carlo simulations method is used as a computational technique to determine the numerical results for the performance characteristics,such as average run length(ARL),median run length,and standard deviation run length(SDRL)for assessing the performance of the proposed HEWMA1 and HEWMA2 charts.In addition,to evaluate the overall performance of the proposed HEWMA1 and HEWMA2 charts,other numerical measures consisting of the extra quadratic loss(EQL),relative average run length(RARL),and performance comparison index(PCI)are also computed.The proposed HEWMA1 and HEWMA2 charts are compared to some existing charts,such as CH,CEWMA,HEWMA,AEWMAHHW1,HHW2,AIB-EWMA-I,and AIB-EWMA-II charts,on the basis aforementioned numerical measures.The comparison reveals that the proposed HEWMA1 and HEWMA2 charts achieve better detection ability against the existing charts.In the end,a real-life data application is also provided to enhance the implementation of the proposed HEWMA1 and HEWMA2 charts practically.

A Semi-Analytical Potential Solution for Wave Resonance in Gap Between Floating Box and Vertical Wall

Based on potential flow theory, a dissipative semi-analytical solution is developed for the wave resonance in the narrow gap between a fixed floating box and a vertical wall by using velocity potential decompositions and matched eigenfunction expansions. The energy dissipation near the box is modelled in the potential flow solution by introducing a quadratic pressure loss condition on the gap entrance. Such a treatment is inspired by the classical local head loss formula for the sudden change of cross section in channel flow, where the energy dissipation is assumed to be proportional to the square of local velocity for high Reynolds number flows. The dimensionless energy loss coefficient is calibrated based on experimental data. And it is found to be insensitive to the incident wave height and wave frequency. With the calibrated energy loss coefficient, the resonant wave height in gap and the reflection coefficient are calculated by the present dissipative semi-analytical solution. The predictions are in good agreement with experimental data. Case studies suggest that the maximum relative energy dissipation occurs near the resonant frequency, which leads to the minimum reflection coefficient. The horizontal wave forces on the box and the vertical wall attain also maximum values near the resonant frequency, while the vertical wave force on the box decreases abruptly there to a small value.

DEVELOPMENT OF REALISTIC QUALITY LOSS FUNCTIONS FOR INDUSTRIAL APPLICATIONS

A number of quality loss functions, most recently the Taguchi loss function, have been developed to quantify the loss due to the deviation of product performance from the desired target value. All these loss functions assume the same loss at the specified specification limits. In many real life industrial applications, however, the losses at the two different specifications limits are often not the same. Further, current loss functions assume a product should be reworked or scrapped if product performance falls outside the specification limits. It is a common practice in many industries to replace a defective item rather than spending resources to repair it, especially if considerable amount of time is required. To rectify these two potential problems, this paper proposes more realistic quality loss functions for proper applications to real-world industrial problems. This paper also carries out a comparison studies of all the loss functions it considers.

The Minimax Estimator of Stochastic Regression Coefficients and Parameters in the Class of All Estimators

In this paper, the authors address the problem of the minimax estimator of linear combinations of stochastic regression coefficients and parameters in the general normal linear model with random effects. Under a quadratic loss function, the minimax property of linear estimators is investigated. In the class of all estimators, the minimax estimator of estimable functions, which is unique with probability 1, is obtained under a multivariate normal distribution.

ESTIMATION OF SCATTER MATRIX BASED ON i．i．d． SAMPLE FROM ELLIPTICAL DISTRIBUTIONS

In this paper, we consider the estimation of a scatter matrix under entropy loss, quadratic loss, when the samples x(1),…,x(n) are i.i.d. and x(1)￣ECp(μ,Σ,f). With respect to entropy and quadratic losses, we obtain the best estimator of Σ having the form αSx as well as having the form Tx△Tx' , where Sx,Tx and △ are given in the text, and obtain the minimax estimator of Σ and the best equivariant estimator of Σ with respect to the triangular transformations group LT+ (p)(the group consisting of lower triangular matrices with positive diagonal elements). Some related discussion are given as its generalizations.

Simultaneous determination of manufacturer’s process mean and production run length,and retailer’s order quantity

In Chen and Liu’s optimum profit model with a traditional production system,they did not consider the effect of product quality on the customer’s demand order quantity,and also ignored the used cost of customers for product.In fact,the customer’s demand quantity is always seriously related to product quality.Hence,in the present paper,we modify Chen and Liu’s model to address the determination of the optimal process parameters by employing the idea of quality loss and single sampling rectifying inspection plan.Assuming that the quality characteristic of the product is normally distributed,Taguchi’s symmetric quadratic quality loss function is applied in evaluating the product quality.Three decision variables,i.e.,the mean of the process characteristic,the production run length of the product and the retailer’s order quantity,are jointly determined in our modified model to maximize the expected total profit of society,which includes both the manufacturer and the retailer.A heuristic solution procedure is developed for this optimization problem,and a numerical example is provided for illustration.From the numerical results,it can be seen that both the sale price per unit and the intercept of the mean demand of the customer are two major(or significant)parameters in the model and should be more accurately estimated in practice.Finally,the quality investment policy is provided to compare its effect on the optimum profit model with quality improvement.

Simultaneous settings of order quantity,wholesale price,production run length,process mean,and warranty period

In this article,the authors propose a modified version of S.L.Chen and Liu’s model with a two-stage production system.Assume that the retailer’s order quantity is concerned with the manufacturer’s selling price and the warranty period of product.The used cost of the customer is measured under the Taguchi’s quadratic quality loss function and concluded in the retailer’s profit function.The quality of the lot for the manufacturer is determined by adopting a two-stage single sampling rectifying inspection plan.The modified economic manufacturing quantity(EMQ)model is addressed in formulating the manufacturer’s expected profit.The retailer’s order quantity,manufacturer’s wholesale price,production run length,process mean,and warranty period of product will be jointly determined by maximizing the total expected profit of the supply chain system including the manufacturer and the retailer.Finally,the quality investment policy is introduced to illustrate the profit improvement for the supply chain system.