Bayes and Empirical Bayes Two-Stage Procedures for Sampling Inspection

A batch of M items is inspected for defectives. Suppose there are d defective items in the batch. Let d ₀ be a given standard used to evaluate the quality of the population where 0 < d ₀ < M. The problem of testing H ₀: d < d ₀ versus H ₁: d ≥ d ₀ is considered. It is assumed that past observations are available when the current testing problem is considered. Accordingly, the empirical Bayes approach is employed. By using information obtained from the past data, an empirical Bayes two-stage testing procedure is developed. The associated asymptotic optimality is investigated. It is proved that the rate of convergence of the empirical Bayes two-stage testing procedure is of order O (exp(? c? n)), for some constant c? > 0, where n is the number of past observations at hand.     

Local Convergence Rate of Mean Squared Error in Density Estimation

The local convergence rate of a multivariate density estimators based on the certain delta-sequence is studied. In contrast to known results, the conditions on the density are formulated in terms of the modulus of continuity. The main contribution of this study is relaxing the corresponding smoothing conditions in terms of arbitrary modulus of continuity type majorant. In particular, when the density f ∈ L ^p (R ^d ) satisfies Lipschitz condition of order γ = 1 at x, the rate of convergency contains terms with logarithm, which is the best possible convergency rate.     

Estimation of a change point in the variance function based on the χ2-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.     

Co‐Op Advertising in Dynamic Retail Oligopolies

We study a supply chain in which a consumer goods manufacturer sells its product through a retailer. The retailer undertakes promotional expenditures, such as advertising, to increase sales and to compete against other retailer(s). The manufacturer supports the retailer’s promotional expenditure through a cooperative advertising program by reimbursing a portion (called the subsidy rate) of the retailer’s promotional expenditure. To determine the subsidy rate, we formulate a Stackelberg differential game between the manufacturer and the retailer, and a Nash differential subgame between the retailer and the competing retailer(s). We derive the optimal feedback promotional expenditures of the retailers and the optimal feedback subsidy rate of the manufacturer, and show how they are influenced by market parameters. An important finding is that the manufacturer should support its retailer only when a subsidy threshold is crossed. The impact of competition on this threshold is nonmonotone. Specifically, the manufacturer offers more support when its retailer competes with one other retailer but its support starts decreasing with the presence of additional retailers. In the case where the manufacturer sells through all retailers, we show under certain assumptions that it should support only one dominant retailer. We also describe how we can incorporate retail price competition into the model.     

螺旋藻添加剂在稚鳖生长中的作用研究

本试验利用螺旋藻添加剂对稚鳖在控温条件下（30－32℃）进行饲养生长对比研究，结果经统计表明：增重效果随添加量的增加而显著改善，各组之间稚鳖生长差异极显著（P＜0．01），试验组与对照组的成活率差异明显。