The Effect of Transformations on the Approximation of Univariate (Convex) Functions with Applications to Pareto Curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed.We study the effect of the use of increasing convex or increasing concave transformations on the approximation of univariate (convex) functions.In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions.Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations.We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.approximation theory;convexity;convex/concave transformation;Pareto curve

The Effect of Transformations on the Approximation of Univariate (Convex) Functions with Applications to Pareto Curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed.We study the effect of the use of increasing convex or increasing concave transformations on the approximation of univariate (convex) functions.In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions.Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations.We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem

A Method For Approximating Univariate Convex Functions Using Only Function Value Evaluations

In this paper, piecewise linear upper and lower bounds for univariate convex functions are derived that are only based on function value information. These upper and lower bounds can be used to approximate univariate convex functions. Furthermore, new Sandwich algo- rithms are proposed, that iteratively add new input data points in a systematic way, until a desired accuracy of the approximation is obtained. We show that our new algorithms that use only function-value evaluations converge quadratically under certain conditions on the derivatives. Under other conditions, linear convergence can be shown. Some numeri- cal examples, including a Strategic investment model, that illustrate the usefulness of the algorithm, are given.

Lorenz-based quantitative risk management

In this thesis, we address problems of quantitative risk management using a specific set of tools that go under the name of Lorenz curve and inequality indices, developed to describe the socio-economic variability of a random variable.Quantitative risk management deals with the estimation of the uncertainty that isembedded in the activities of banks and other financial players due, for example, tomarket fluctuations. Since the well-being of such financial players is fundamental for the correct functioning of the economic system, an accurate description and estimation of such uncertainty is crucial.Applied ProbabilityNumerical Analysi

Syvyysmittaan perustuvasta funktionaalisen aineiston luokittelusta

The purpose of this Master's Thesis is to discuss the application of functional statistical depth, a powerful nonparametric modeling tool, to supervised functional classification. With the recent rapid increase of the sophistication of measurement and storage tools, we have begun to encounter more and more complex datasets on all fields of research. This sudden explosion of very high dimensional complex data has brought with it an increasing need for inferential analytic tools for dealing with such data. However, developing methodology for functional data is far from straightforward due to the introduction of a wide range of important features unique to this type of data, most notably, shape and shape-outlyingness. The issue is furthermore complicated by the massive computational load many otherwise appealing approaches would impose. In this thesis, shape receptive depth based classification is considered. In particular, the focus is on Jth order kth moment integrated depth based classification. Receptiveness to shape features and shape-outlyingness of the Jth order kth moment integrated depth is discussed and important key-ideas related to its features are established. Then, the Jth order kth moment integrated depth is applied to supervised functional classification for two different real datasets. Performance of different functional depth approaches is compared. The real data examples illustrate excellent classification accuracy of the Jth order kth moment integrated depth. Finally, future work and improvement suggestions on the area are discussed.Työn tavoitteena on tarkastella funktionaalisen tilastollisen syvyyden soveltamista luokittelussa. Syvyysmitat ovat epäparametrisia mittareita, jotka kertovat havaintojen tilastollisesta poikkeavuudesta. Tänä päivänä pystymme tallentamaan valtavia määriä dataa. Tämä on mahdollistanut monimutkaisten ja korkeauloitteisten aineistojen keräämisen ja analysoinnin. Tästä on syntynyt tarve uusille menetelmille jotka soveltuvat korkeauloitteisen datan käsittelyyn. Funktionaaliset aineistot ovat ääretönuloitteisia. Menetelmien kehittäminen ääretönuloitteisten aineistojen analysointiin on vaikeaa. Erityisen haastavaa on huomioida funktionaalisten havaintojen muoto. Lisäksi laskennallinen taakka saattaa tuottaa ongelmia. Työssä tutkitaan funktioiden muotoa huomioivien syvyysmittarian käyttöä luokittelussa. Erityisesti, työssä tarkastellaan J:nnen asteen k:nnen momentin integroituun syvyysmittaan perustuvaa luokittelua. Työssä tarkastellaan J:nnen asteen k:nnen momentin integroidun syvyysmitan herkkyyttä funktioiden muodoille ja muodon suhteen poikkeaville havainnoille. J:nnen asteen k:nnen momentin integroitua syvyysmittaa käytetään kahden oikean aineiston luokitteluun. Menetelmän suorituskykyä verrataan muihin syvyysmittaan perustuviin luokittelijoihin. Aineistoesimerkit havainnollistavat J:nnen asteen k:nnen momentin integroidun syvyysmitan erinomaista luokittelukykyä. Työn lopussa esitetään ajatuksia mahdollisuuksista parantaa ja laajentaa tarkasteltua menetelmää

Monopoly Pricing in a Vertical Market with Demand Uncertainty

We study a vertical market with an upsteam supplier and multiple downstream retailers. Demand uncertainty falls to the supplier who acts first and sets a uniform wholesale price before the retailers observe the realized demand and engage in retail competition. Our focus is on the supplier's optimal pricing decision. We express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution. This allows for a closed form characterization of the points of unitary elasticity that maximize the supplier's profits and the derivation of a mild unimodality condition for the supplier's objective function that generalizes the widely used increasing generalized failure rate (IGFR) condition. A direct implication is that optimal prices between different markets can be ordered if the markets can be stochastically ordered according to their MRD functions or equivalently to their elasticities. Based on this, we apply the theory of stochastic orders to study the response of the supplier's optimal price to various features of the demand distribution. Our findings challenge previously established economic insights about the effects of market size, demand transformations and demand variability on wholesale prices and indicate that the conclusions largely depend on the exact notion that will be employed. We then turn to measure market performance and derive a distribution free and tight bound on the probability of no trade between the supplier and the retailers. If trade takes place, our findings indicate that ovarall performance depends on the interplay between demand uncertainty and level of retail competition