Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.Comment: 24 page

A split questionnaire survey design applied to German media and consumer surveys

On the basis of real data sets it is shown that splitting a questionnaire survey according to technical rather than qualitative criteria can reduce costs and respondent burden remarkably. Household interview surveys about media and consuming behavior are analyzed and splitted into components. Following the matrix sampling approach, respondents are asked only the varying subsets of the components inducing missing data by design. These missing data are imputed afterwards to create a complete data set. In an iterative algorithm every variable with missing values is regressed on all other variables which either are originally complete or contain actual imputations. The imputation procedure itself is based on the socalled predictive mean matching. In this contribution the validity of split and imputation is discussed based on the preservation of empirical distributions, bivariate associations, conditional associations and on regression inference. Finally, we find that many empirical distributions of the complete data are reproduced well in the imputed data sets. Concerning these long media and consumer questionnaires we like to conclude that nearly the same inference can be achieved by means of such a split design with reduced costs and minor respondent burden --

Fractional Operators, Dirichlet Averages, and Splines

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented

Pitfalls in modeling loss given default of bank loans

The parameter loss given default (LGD) of loans plays a crucial role for risk-based decision making of banks including risk-adjusted pricing. Depending on the quality of the estimation of LGDs, banks can gain significant competitive advantage. For bank loans, the estimation is usually based on discounted recovery cash flows, leading to workout LGDs. In this paper, we reveal several problems that may occur when modeling workout LGDs, leading to LGD estimates which are biased or have low explanatory power. Based on a data set of 71,463 defaulted bank loans, we analyze these issues and derive recommendations for action in order to avoid these problems. Due to the restricted observation period of recovery cash flows the problem of length-biased sampling occurs, where long workout processes are underrepresented in the sample, leading to an underestimation of LGDs. Write-offs and recoveries are often driven by different influencing factors, which is ignored by the empirical literature on LGD modeling. We propose a two-step approach for modeling LGDs of non-defaulted loans which accounts for these differences leading to an improved explanatory power. For LGDs of defaulted loans, the type of default and the length of the default period have high explanatory power, but estimates relying on these variables can lead to a significant underestimation of LGDs. We propose a model for defaulted loans which makes use of these influence factors and leads to consistent LGD estimates. --Credit risk,Bank loans,Loss given default,Forecasting

On the valuation of fader and discrete barrier options in Heston's Stochastic Volatility Model

We focus on closed-form option pricing in Hestons stochastic volatility model, in which closed-form formulas exist only for few option types. Most of these closed-form solutions are constructed from characteristic functions. We follow this approach and derive multivariate characteristic functions depending on at least two spot values for different points in time. The derived characteristic functions are used as building blocks to set up (semi-) analytical pricing formulas for exotic options with payoffs depending on finitely many spot values such as fader options and discretely monitored barrier options. We compare our result with different numerical methods and examine accuracy and computational times. --exotic options,Heston Model,Characteristic Function,Multidimensional Fast Fourier Transforms

How important is cultural background for the level of intergenerational mobility?

Using results on brother correlations of different groups of second generation immigrants based on administrative data from Denmark, this note analyzes the role of cultural background in the determination of the level of intergenerational mobility. The estimated correlations indicate that cultural background is not an important factor for the level of intergenerational mobility. --Intergenerational mobility,Sibling correlations

Factoring bivariate lacunary polynomials without heights

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap Theorem which allows to test whether a polynomial of the form P(X,X+1) is identically zero in time polynomial in the number of terms of P(X,Y). The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.Comment: 25 pages, 1 appendi

Convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces

The computation of robust regression estimates often relies on minimization of a convex functional on a convex set. In this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization-minimization algorithms. Our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case. We prove convergence on convex function spaces for general coercive and convex functionals F and derive geometric convergence in certain unconstrained settings. The algorithm is applied to TV penalized quantile regression and is compared with a step size corrected Newton-Raphson algorithm. It is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better, whereas the Newton type method outpaces the former only after many iterations. Finally, in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows to utilize highly efficient algorithms for special quadratic programs in more complex settings. --regression analysis,monotone regression,quantile regression,shape constraints,L1 regression,nonparametric regression,total variation semi-norm,reweighted least squares,Fermat's problem,convex approximation,quadratic approximation,pool adjacent violators algorithm