Researching students’ strategies, episodes, and metacognitions in mathematical problem solving

The purpose of this study was to investigate the problem solving strategies, problem solving episodes, and metacognitions of five Turkish high school freshmen and explore the interplay of these on their problem solving success in mathematics. Participants worked on ten problems selected among the released mathematical literacy items used in Programme for International Student Assessment (PISA) 2003. The research data had been collected through clinical interviews and a self-monitoring questionnaire filled by the participants. The data were coded and analysed according to a framework adopted from various researchers in the field. Our results have confirmed that the problem solving success is too complex to be clarified by a unique property or behaviour of the solver. It requires overcoming various obstacles to reach a successful result. Therefore, not only the students should have the required mathematical knowledge and a good repertoire of different problem solving strategies, but also they should know when and how to use those strategies, along with monitoring and regulating their problem-solving processes using their metacognitive skills.     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

Ladislaus von Bortkiewicz—Statistician,Economist and a European Intellectual

Ladislaus von Bortkiewicz (1868–1931) was a European statistician. His scientific work covered theoretical economics, stochastics, mathematical statistics and radiology; today, we would call him a cross‐disciplinary scientist. With his clear views on mathematical principles with their applications in these fields, he stood in conflict with the mainstream economic schools in Germany at the dawn of the 20th century. He had many prominent students (Gumbel, Leontief and Freudenberg among them), and he carved out the path of modern statistical thinking. He was a true European intellectual with a career path from St. Petersburg via Göttingen to Straßburg and finally the Berliner Universität, now Humboldt‐Universität zu Berlin. He is known for the precise calibration of insurance claims applying the—at that time hardly known—Poisson distribution to Prussian horse kick and child suicide data. He proposed a simple solution to the Marxian transformation problem and wrote numerous articles and books on the mathematical treatment of statistical (including radiological physical) data. In this article, we sketch his life and work and point out the prominent role that he has in today's statistical thinking.     

Statistical Thinking in Computer-Based Learning Environments

The theme of technology is particularly important for statistics education because of the role of technology in changing views of statistical knowledge, pedagogy and learning. The interactive multimedia technology of today allows for the creation of powerful learning environments where the focus is on the process that produced the data and its associated variation and the statistical thinking required to solve the problem. However, despite this optimistic view, students' self‐regulation and intrinsic motivation seem to be key concepts for learning in complex interactive computer‐based learning environments. We conclude with observations of teachers and students working with applets. More of these evaluations should be published and/or made available to the statistical community so that both positive and negative teaching experiences can be shared.     

Industrial statistics and its recent contributions to total quality in the Netherlands

The use of statistical methods in quality management has a long history. Most of the pioneers, such as Walter A. Shewhart and W. Edwards Deming, refer to themselves as statisticians. Statistical thinking in industry means that all work is a series of interconnected processes, that all processes show variation, and that a reduction in variation is the key for continuous improvement. In the statistical literature we find several approaches to achieve this: quality by inspection, quality by process control and quality by design. In this paper we give a survey of the contributions of Dutch statisticians during the last five years.     

One Possible Frame for Thinking about Experiential Learning

I argue that teaching statistical thinking is harder than teaching mathematics, that experimental design is particularly well suited to teaching statistical thinking and that in teaching statistics, variation is good. We need a mix of archival data, simulations and activities, of varying degrees of complexity. Within this context, I applaud the important contributions to our profession represented by Darius et al. (2007) , and Nolan & Temple Lang (2007) , the first for showing us how to make simulation‐based learning simultaneously more flexible and more realistic than ever before, and the second for showing us a path‐breaking technology that can make archival data the basis for active learning at an impressively high level of sophistication, embedding statistical thinking within real scientific and practical investigations.     

基于运筹优化的物流与设施规划课程教学研究

在传统教学中,物流与设施规划课程的物流分析、设施布局等内容的一些经典问题采用启发式方法或试验法来解决。为了更精确地解决这些问题,采用数学规划方法提供解决方案。主要围绕建模技巧、应用软件教学和解决物流与设施规划专业案例等方面培养学生运用优化原理与方法构建运筹优化模型的能力以及运用ILOG OPL优化软件解决实际优化问题的能力。以二次分配问题和多产品工艺过程图优化问题为案例阐述了数学规划建模和编程求解的过程。     

Sociology,history, and the abstract perspective

Our discussion is initiated as a response to the claim that sociologists should become “more historical” in their orientations. The issues are old, but every generation frames its own response. Our response is developed by appeal to intuitive convictions arising out of experience with mathematical models of social phenomena. We make a distinction between historical and sociological processes at a metaphysical level; that is, these two types of processes exemplify different categories of existence. Next we make this point of view concrete by using the idea of a model of social mobility as an example. The discussion then centers on problems related to the search for general laws. We frame a “fallacy of misplaced generality” and against this background discuss how the idea of scope conditions, used in conjunction with formal models, leads to a method for coping with the difficulties inherent in the effort to frame general sociological theories.     

Statistical Proofs of Some Matrix Theorems *

Books on linear models and multivariate analysis generally include a chapter on matrix algebra, quite rightly so, as matrix results are used in the discussion of statistical methods in these areas. During recent years a number of papers have appeared where statistical results derived without the use of matrix theorems have been used to prove some matrix results which are used to generate other statistical results. This may have some pedagogical value. It is not, however, suggested that prior knowledge of matrix theory is not necessary for studying statistics. It is intended to show that a judicious use of statistical and matrix results might be of help in providing elegant proofs of problems both in statistics and matrix algebra and make the study of both the subjects somewhat interesting. Some basic notions of vector spaces and matrices are, however, necessary and these are outlined in the introduction to this paper.     

Misspecification and Expectations Correction in New Keynesian DSGE Models

This paper focuses on the dynamic misspecification that characterizes the class of small‐scale New Keynesian models currently used in monetary and business cycle analysis, and provides a remedy for the typical difficulties these models have in accounting for the rich contemporaneous and dynamic correlation structure of the data. We suggest using a statistical model for the data as a device through which it is possible to adapt the econometric specification of the New Keynesian model such that the risk of omitting important propagation mechanisms is kept under control. A pseudo‐structural form is built from the baseline system of Euler equations by forcing the state vector of the system to have the same dimension as the state vector characterizing the statistical model. The pseudo‐structural form gives rise to a set of cross‐equation restrictions that do not penalize the autocorrelation structure and persistence of the data. Standard estimation and evaluation methods can be used. We provide an empirical illustration based on USA quarterly data and a small‐scale monetary New Keynesian model.     

Covariance functions that are stationary or nonstationary in space and stationary in time

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space–time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space–time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY ( Applied Statistics , 38 , 1989, 1).     

Hawkes process modeling of COVID-19 with mobility leading indicators and spatial covariates

Hawkes processes are used in statistical modeling for event clustering and causal inference, while they also can be viewed as stochastic versions of popular compartmental models used in epidemiology. Here we show how to develop accurate models of COVID-19 transmission using Hawkes processes with spatial-temporal covariates. We model the conditional intensity of new COVID-19 cases and deaths in the U.S. at the county level, estimating the dynamic reproduction number of the virus within an EM algorithm through a regression on Google mobility indices and demographic covariates in the maximization step. We validate the approach on both short-term and long-term forecasting tasks, showing that the Hawkes process outperforms several models currently used to track the pandemic, including an ensemble approach and an SEIR-variant. We also investigate which covariates and mobility indices are most important for building forecasts of COVID-19 in the U.S.     

Modeling large data sets in marketing

In the last two decades, marketing databases have grown significantly in terms of size and richness of available information. The analysis of these databases raises several information-related and statistical issues. We aim at providing an overview of a selection of issues related to the analysis of large data sets. We focus on the two important areas: single source databases and customer transaction databases. We discuss models that have been used to describe customer behavior in these fields. Among the issues discussed are the development of parsimonious models, estimation methods, aggregation of data, data-fusion and the optimization of customer-level profit functions. We conclude that problems related to the analysis of large databases are far from resolved, and will stimulate new research avenues in the near future.     

Quantile-regression estimates of cigarette demand elasticities for the United States

Using a simple model and state-level cross-section U.S. data from 1993 to 1999, quantile-regression estimates of price elasticity and income elasticity for cigarette demand are obtained. It is noted that price elasticity shows a sizable variation across the high and low quantity-quartiles. There is a similar variability in the income elasticity, but most of these estimates lack statistical significance. Besides providing an indication of the variation in the price (and income) elasticity for different consumption levels, the exercise suggests some interpretative caution in regard to estimates from constant-elasticity models.     

Some methods for assessing the need for non-linear models in business cycle analysis

It is often suggested that non-linear models are needed to capture business cycle features. In this paper, we subject this view to some critical analysis. We examine two types of non-linear models designed to capture the bounce-back effect in US expansions. This means that these non-linear models produce an improved explanation of the shape of expansions over that provided by linear models. But this is at the expense of making expansions last much longer than they do in reality. Interestingly, the fitted models seem to be influenced by a single point in 1958 when a large negative growth rate in GDP was followed by good positive growth in the next quarter. This seems to have become embedded as a population characteristic and results in overly long and strong expansions. That feature is likely to be a problem for forecasting if another large negative growth rate was observed.     

A Reduced Rank Regression Approach to Tests of Asset Pricing

Both the Arbitrage Pricing Theory (APT) and the Capital Asset Pricing Model (CAPM) place restrictions of the cross-sectional variation of conditional expectations of asset returns and of macro indicators. We show that these restrictions imposed on the reference statistical models lead to special cases of the reduced rank regression model. The maximum likelihood problem is solved by canonical correlation analysis. Likelihood ratio tests about the number of factors underlying stock returns are straightforward to calculate, thus allowing discrimination between competing financial theories. Moreover LR tests on the relevance of each macroeconomic indicator within a chosen model can be implemented. Some of the tests are illustrated by an application to Italian stock market data.     

Recent Developments of Statistical Approaches in Aspects of Accounting: a Review

We review and simultaneously introduce a convenient statistical concept for the mathematical representation of the Statistical Activity Cost Theory (SACT) introduced by Willett (1987, 1988). Further, we discuss, and present a critique of a variety of statistical models with respect to long debated accounting problems, such as the allocation of joint costs and depreciation. We finally propose that taking the effort to combine those models results in a novel statistical accounting system and this is discussed by means of the so-called virtual firm. Our approach is methodological rather than empirical. We endeavour to demonstrate that the statistical models presented herein have a more natural interpretation and applicability than their deterministic analogues: hence, this review presents promising outcomes and useful perspectives for the accounting profession.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Methods for Handling Dropouts in Longitudinal Clinical Trials

This paper focuses on the monotone missing data patterns produced by dropouts and presents a review of the statistical literature on approaches for handling dropouts in longitudinal clinical trials. A variety of ad hoc procedures for handling dropouts are widely used. The rationale for many of these procedures is not well-founded and they can result in biased estimates of treatment comparisons. A fundamentally difficult problem arises when the probability of dropout is thought to be related to the specific value that in principle should have been obtained; this is often referred to as informative or non-ignorable dropout. Joint models for the longitudinal outcomes and the dropout times have been proposed in order to make corrections for non-ignorable dropouts. Two broad classes of joint models are reviewed: selection models and pattern-mixture models. Finally, when there are dropouts in a longitudinal clinical trial the goals of the analysis need to be clearly specified. In this paper we review the main distinctions between a 'pragmatic' and an 'explanatory' analysis. We note that many of the procedures for handling dropouts that are widely used in practice come closest to producing an explanatory rather than a pragmatic analysis.