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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.

Process capability indices are routinely used to estimate the mean-variability performance of industrial products with respect to both targets and specification limits. However, when the target variable is defined over a planar surface of a manufact, it is relevant to assess the capability of the production process locally, that is, at any spatial location of the surface, in particular if the manufact has to be split into pieces to obtain single production items. In this article, focusing on the specification introduced by Clements [Qual Prog., 22, 95–100], we suggest an approach based on additive quantile models to estimate, in a Bayesian paradigm, the index locally. We demonstrate its use in the context of the etching phase of the integrated circuit fabrication process. Since capability of etching processes is typically assessed for batches of wafers, we also propose two algorithms based on resampling to perform local capability analysis at the lot level.

In this article, we investigate the occurrence of defects in integrated circuit fabrication and show how spatial analysis can be effective in grasping and representing spatial regularities in defect patterns on the silicon supports, called wafers, used to produce microchips. Defects occurring on the wafer surface are the main cause of yield loss in the semiconductor industry; hence, to promptly detect an excess of defects and identify their spatial structure is crucial to the entire fabrication process. To address this hard-to-solve problem, this article proposes a concatenation of different methods, namely, a control chart, a clustering algorithm, and a graphical tool. First, a control chart based on the p-value of an appropriate test recognizes spatially structured defects on the wafer area. Then a clustering procedure grounded on the minimum spanning tree algorithm is adopted to identify those regions more prone to defect occurrences. Finally, alpha-shapes are employed to display their shape effectively. The suggested procedure proves to be extremely fast and effective, allowing its implementation in-line during the fabrication process. This provides a great advantage in modern microelectronics where items tend to be highly specialized and often produced in small lots. In particular, due to the Monte Carlo nature of the procedure, the control chart proposed hereafter does not require gold standard data to be set. This is particularly advantageous for small lot production, which is typically limited in time and does not permit to collect long time series of the charting statistics.

Worldwide, thousands of people die annually in highway-related crashes and millions are injured. Hence, car wrecks have very high direct social costs but also relevant indirect economic effects such as an adverse impact on the burden of hospitalization and an increased health expenditure. The analysis of car crash data has long been used as a basis for influencing highway and vehicle designs but also, and perhaps more importantly, to support local authorities in allocating resources aimed at improving road safety and making political decisions to mitigate road risks in the most exposed areas. In this paper, we show how a range of information collected from open data sources concerning the structure of the road network (road typology, traffic lights, pedestrian crossings, etc.), socio-demographical dimensions and crash history can be proficiently used for this aim. We adopt a dynamic Zero Inflated Poisson (ZIP) regression model to define two indexes. The first index, derived from the counting component of the ZIP model, measures how prone to crash risk a segment is. The other, derived by the zero component of the ZIP model, represents a measure of the likelihood of segments to not be exposed to crashes. Focussing on the city of Milan (Northern Italy), we found that the most relevant determinant of road risk proneness is crash history and that structural characteristics of the road are much more relevant than demographic information. Finally, we show how this information can be spatialized to produce maps of crash proneness and predict future spatial risk indexes.

We propose a model to extract significant risk spatial interactions between countries adopting the Graphical Lasso algorithm, used in graph theory to sort out spurious conditional correlations. In this context, the major issue is the definition of the penalization parameter. We propose a search algorithm aimed at the best separation of the variables (expressed in terms of conditional dependence) given an a priori desired partition. The case study focuses on Credit Default Swap (CDS) returns over the period 2009–2017. The proposed algorithm is used to estimate the spatial systemic risk relationship between Peripheral and Core Countries in the Euro Area.

In general insurance, measuring the uncertainty of future loss payments and estimating the claims reserve are primary goals of actuaries. To deal with these tricky tasks, a broad literature is available on deterministic and stochastic approaches, most of which aims at straightforwardly modelling the overall claims reserve. In this paper by an extended, very general and reproducible case-study, we analyze the reserving process by attributing to each cell of the lower part of the run-off triangle a Compound mixed Poisson Process, calibrated upon both the numbers of claims and future average costs and considering as well the dependence among incremental claims. We provide analytically the moments of both incremental payments and the total reserve. Furthermore, we accordingly consider the probability distribution of the claims reserve, which is necessary for the assessment of the Risk Reserve capital requirement in a Solvency II framework. To test the impact of the model under different scenarios, insurers and lines of business, the case study is thoroughly analyzed by exploiting the Fisher-Lange average cost method.

In this paper we propose a methodology to estimate the probability that a car accident occurs in urban roads. Our approach is based on logistic regression and takes into account the particular nature of the data which conforms to a spatial point pattern on a network. Using the open data on street networks provided within the OpenStreetMap project, we estimate the probability of car accidents for every street in the municipality of Milan.

This study investigates the assumption that stock riskiness, captured by the market global beta, is constant over the market returns domain. To relax this assumption we propose to model stock returns through disjoint truncated normal distributions fit by means of the Minimum Distance Approach. This provides a set of disjoint conditional regions where normality still holds but it allows to decompose the unconditional beta into local ones referred to each region. In the case study we show that this approach, while preserving the simplicity of describing data by normal distributions, significantly improves the accuracy of the fit, compensating the well-known inadequacy of the standard normal distribution to fit returns in the tails. An extensive out of sample returns predictive test shows that the quality of prediction obtained with our methodology globally has similar statistical properties as for the standard global beta, but it outperforms the latter especially when the negative returns' domain is considered.

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 Non-Life 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 naïve 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.

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.

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 (c) 2013 John Wiley & Sons, Ltd.

Deposition of silicon dioxide (SiO2) is a critical step of integrated circuit manufacturing; hence it is monitored during the manufacturing process at a grid of points defined on the wafer area. Since collecting thickness measurements is expensive, it is a compelling issue to investigate how a sub grid can be identified. A strategy based on spatial prediction and simulating annealing is proposed to tackle the problem which proved to be effective when applied to a real process. A diagnostic device for monitoring the deposition process is also discussed which can be usefully adopted in the day-to-day activity by practitioners acting in process control of a microelectronics fab. © 2012 Elsevier B.V. All rights reserved.

Measurement uncertainty in experiments is receiving increasing interest among practitioners both for quantitative and qualitative evaluations. Applications may be found, for example, to compare experiments run in different laboratories, to certify the precision of gauges used in the masurement process or to measure uncertainty in a classification process, i.e. when objects are qualitatively evaluated by appraisers. In this paper, after having briefly presented how to measure uncertainty using gauge repeatability and reproducibility studies, we focus on a problem quite relevant for practitioners: how to deal with the presence of correlation among ‘replications’. This issue is common to many food/agriculture experiments. By a modification of a nested mixed effect design, we describe the impact of dependencies among replications on measurement capability. Copyright © 2011 John Wiley & Sons, Ltd.

We compare the approach to measure uncertainties proposed in ISO 5725 and GUM from a statistician point of view. In particular we give some warnings to the application of the expanded uncertainty introduced in GUM when the input variables are few and we report some considerations on the relevant role of the interactions among the input variables in the measurement equation as well as the role of statistical design of experiments to measure uncertainties.

We propose a novel usage of CUB models in order to evaluate Repeatability and Reproducibility (R&R) for ordinal data in business and industrial experiments. This is a context where there is a small group of appraisers who have to evaluate a sample of objects classifying them according to ordinal categories. By comparing the cumulative distribution functions obtained fitting CUB models to judgments given by appraisers, we give both graphical and analytical instruments to assess R&R for an ordinal measurement system. The approach is applied to the real-life example reported in de Mast and van Wieringen (2010).

The R-square of the market model is a very popular measure of stock price efficiency. However, its interpretation is far from being unambiguous. Some scholars argue that the R-square is a direct measure of efficiency, others argue that the R-square is an indirect measure of efficiency. This paper contributes to the literature in two ways. First, we model the relationship between the market model R-square and the delay in the price discovery process and, second, we find that the correlation between R-square and delay is consistently negative. We conclude that the R-square is a direct measure of price efficiency.

In Measurement System Analysis a relevant issue is how to find confidence intervals for the parameters used to evaluate the capability of a gauge. In literature approximate solutions are available but they produce so wide intervals that they are often not effective in the decision process. In this article we introduce a new approach and, with particular reference to the parameter γR, i.e., the ratio of the variance due to the process and the variance due to the instrument, we show that, under quite realistic assumptions, we obtain confidence intervals narrower than other methods. An application to a real microelectronic case study is reported.

Evaluation of measurement systems is necessary in many industrial contexts. The literature on this topic is mainly focused on how to measure uncertainties for systems that yield continuous output. Few references are available for categorical data and they are briefly recalled in this paper. Finally a new proposal to measure uncertainty when the output is bounded ordinal is introduced.

The literature on multivariate calibration shows an increasing interest in non-parametric or semiparametric methods. Using Empirical Likelihood (EL). we present a semiparametric approach to find multivariate calibration confidence regions and we show how a unique optimum calibration point may be found weighting the EL profile function. In addition, a freeware VBA for Excel© program has been implemented to solve the many relevant computational problems. An example taken from a process of a semiconductor industry is presented.

Most of the papers on calibration are based on either classic or bayesian parametric context. In addition to the typical problems of the parametric approach (choice of the distribution for the measurement errors, choice of the model that links the sets of variables, etc.), a relevant problem in calibration is the construction of confidence region for the unknown levels of the explanatory variables. In this paper we propose a semiparametric approach, based on simplicial depth, to test the hypothesis of linearity of the link function and then how to find calibration depth confidence regions.

La maggior parte dei lavori sulla taratura statistica univariata e multivariata propongono contributi in ambito parametrico inquadrato in un contesto statistico classico o bayesiano. Oltre alla presenza dei problemi tipici dell'approccio parametrico (scelta della legge distribuzionale, del modello, ecc.), uno dei problemi piu rilevanti e la costruzione di regioni di confidenza per gli incogniti livelli delle variabili esplicative. Al fine di superare alcuni di tali problemi, nel presente lavoro si propone un approccio di tipo semiparametrico, basato sulla nozione di funzione depth: l'obiettivo e, dopo aver verificato con un apposito test se il modello di taratura puo essere di tipo lineare, costruire regioni depth di taratura nell'intorno degli incogniti livelli delle vere misure dell'esperimento.

Up to the 60s the design criteria of industrial production systems were mainly based on the new findings achieved in that period in technical and physical environments and generally on techniques having deterministic nature. Examples can be typically found in the field of structural project that must support mechanic, dielectric and thermic stress. The functional forms of the relationship between stress (cause), strain and performance changes (effects) were supposed to be known using variables or parameters related to the size of the item under consideration. This kind of design enabled the collection and control of the unknown and random components by using the so-called safety coefficients. As a consequence of this approach, the existence of random effects in industrial work was ignored and a deterministic point of view was accepted.