Process capability indices are widely used to check quality standards both at the production level and for business activity. They consider the location and the deviation from specification limits and targets. The literature contains many contributions on this topic both in the univariate and the multivariate context. Motivated by a real semiconductor case study, we discuss the role of rational subgroups and the challenge they present in the computation of capability indices, especially when data refer to lots of products. In addition, our context involves a mix of problems: unilateral specification limit, nonsymmetric distribution of the data, evidence of data from a mixture of distributions, and the need to filter one component of the mixture. After solving the previous issues and because of the peculiar characteristics of semiconductor processes based on the so called "wafers," we contribute to the literature a proposal on how to compute capability indices in the case of heteroscedastic spatial processes. With a generalized additive model, we show that it is possible to estimate a capability surface that allows the identification of regions expected to not be fully compliant with the desired quality standards. K E Y W O R D S dry-etching semiconductor processes, mixture distributions, process capability indices, sampling network 1 | INTRODUCTION Process capability indices (PCIs) are well-known tools to estimate the mean-deviance performance of key product characteristics with respect to both targets and specification limits. The main aim of PCIs was originally to evaluate the instant quality of a process given specification limits and prior to manufacturing. Next, because of their simple interpretation, they have also become a fundamental tool for commercial activity. Nowadays, PCIs are unavoidable instruments for managers and engineers; hence, when assumptions upon which these indices are grounded can be violated, much effort has been made to extend and adapt them to such situations (see Montgomery 1). A flow of contributions is available in the literature for both univariate and multivariate cases. The capability indices that researchers have proposed are so many that, since the masterful work of Kotz and Johnson, 2 the word "avalanche" has been coined. A detailed and somewhat critical review on PCIs is available in Kotz and Lovelace, 3 whereas Vännman 4 proposes a unified approach. A thorough review of contributions is reported in de-Felipe and Benedito 5 and Bong-Jin and Kwan-Woo 6 among others). Most of the papers focus on the assumption of normality of the random components, but how capability indices could be efficiently estimated when non normal distributions are involved remains

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The monitoring of spatial production processes typically involves sampling network to gather information about the status of the process. Sampling costs are often not marginal, and once the process has been accurately calibrated, it might be appropriate to reduce the dimension of the sampling grid. This aim is often achieved through the allocation of a brand new network of less dimension. In some cases that is not possible and it might be necessary the selection of a subgrid extracted from the original network. Motivated by a real semiconductor problem, we propose a method to extract a monitoring subgrid from a given one, based upon grid representativeness, accuracy, and spatial coverage of the subgrid and, if available, by expert knowledge of the weights to be assigned to those areas where production may need greater precision. Discussion is mainly focused on circular spatial domain, since, in microelectronics, the basic production support, called wafer, is a circle. Straightforward generalizations to different spatial domains are possible. Furthermore, conditionally upon the availability of experimental data, we check the loss of accuracy by fitting a dual mean‐variance response surface on the reduced grid. Joining the latter information and the criteria used to select the subgrid, we provide additional guidelines on how to fine‐tune the subgrid selection. Real case studies are used to show the effectiveness of the proposal.

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This paper introduces a special issue entirely devoted to the topic 'Statistics for Microelectronics'. The relevance of the theme for this journal is due to both the primary role that microelectronics has in many business activities of the modern society and to the complexity of the production process of integrated circuits that are obtained by several different steps performed on a wafer (i.e., a thin silicon slice of a few inches of diameter). Advanced statistical methods are necessary to monitor and improve this process because the yield of this industrial manufacturing is typically tiny and requires a very high precision. After a brief review of the semiconductor manufacturing process, we point out how statistical methodologies can contribute to this. Finally, we introduce the papers that discuss some key aspects of this subject. Copyright © 2013 John Wiley & Sons, Ltd.Keywords: semiconductor manufacturing; integrated circuit; process control Microelectronics industry faces a relevant role in modern society. Microchips, which control electronic devices, enable systems and products that we use to work, communicate, travel, entertain, harness energy, treat illness, and make scientific discoveries. They are present in several different kinds of objects, ranging from computers, tablets and smartphones to cars and planes, household appliances, and military systems.Also, the huge advances in medical devices are being driven by most recent developments in semiconductors that permit to reduce size, cost, and power consumption of medical electronic devices, yet boasting significant improvement in overall performance. A variety of not invasive, portable, and wearable devices are now usable to monitor the condition of a patient, such as the contact lens sensor for continuous monitoring of intra-ocular pressure. Moreover, several implantable and invasive medical devices are available as well, such as insulin pump for drug delivery and defibrillators.Over the last 50 years, the semiconductors content in electronic equipment has grown up significantly, raising from 2% in 1965 to 23% in 2012, when the total worldwide semiconductor revenue reached $299.9 bn.

New risk-based solvency requirements for insurance companies across European markets have been introduced by Solvency II and will come in force from 1 January 2016. These requirements, derived by a Standard Formula or an Internal Model, will be by far more risk-sensitive than the required solvency margin provided by the current legislation. In this regard, a Partial Internal Model for Premium Risk is developed here for a multi-line NonLife insurer. We follow a classical approach based on a Collective Risk Model properly extended in order to consider not only the volatility of aggregate claim amounts but also expense volatility. To measure the effect of risk mitigation, suitable reinsurance strategies are pursued. We analyze how naï ve coverage as conventional Quota Share and Excess of Loss reinsurance may modify the exact moments of the distribution of technical results. Furthermore, we investigate how alternative choices of commission rates in proportional treaties may affect the variability of distribution. Numerical results are also figured out in the last part of the paper with evidence of different effects for small and large companies. The main reasons for these differences are pointed out.