The axiomatic setting for **risk** **measures** has extensively been developed since seminal papers on coherent **risk** **measures** and **distortion** **risk** **measures**. Each set of axioms for **risk** **measures** corresponds to a particular behavior of decision makers under **risk**, as it has been shown, for instance, in Bleichrodt and Eeckhoudt (2006) and Denuit et al. (2006). Most often, articles on axiom-based **risk** measurement present the link to a theoretical foundation of human behavior explicitly. For example, Wang (1996) shows the **connection** **between** **distortion** **risk** **measures** and Yaari’s dual theory of choice under **risk**; Goovaerts et al. (2010b) investigate the additivity of **risk** **measures** in Quiggin’s rank-dependent utility theory; and Kaluszka and Krzeszowiec (2012) introduce the generalized Choquet integral premium principle and relate it to Kahneman and Tversky’s cumulative prospect theory.

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The axiomatic setting for **risk** **measures** has extensively been developed since seminal papers on coherent **risk** **measures** and **distortion** **risk** **measures**. Each set of axioms for **risk** **measures** corresponds to a particular behavior of decision makers under **risk**, as it has been shown, for instance, in Bleichrodt and Eeckhoudt (2006) and Denuit et al. (2006). Most often, articles on axiom-based **risk** measurement present the link to a theoretical foundation of human behavior explicitly. For example, Wang (1996) shows the **connection** **between** **distortion** **risk** **measures** and Yaari’s dual theory of choice under **risk**; Goovaerts et al. (2010b) investigate the additivity of **risk** **measures** in Quiggin’s rank-dependent utility theory; and Kaluszka and Krzeszowiec (2012) introduce the generalized Choquet integral premium principle and relate it to Kahneman and Tversky’s cumulative prospect theory.

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The problem of optimal portfolio selection is of paramount importance to investors, hedgers, fund managers, among others. Inspired by the seminal work of [3], the research on optimal portfolio selection has been growing rapidly. Researchers and practitioners are constantly seeking better and more sophisticated **risk** and reward tradeoff in constructing optimal portfolios. The classical Markowitz model used variance as the benchmark for **risk** measurement and this is perceived to be undesirable since it penalizes equally, regardless of downside **risk** or upside potential. Consequently, other **measures** of **risk** have been proposed in **connection** to portfolio optimization. These include semi-variance [4], partial moments [5], safety first principle [6], skewness and kurtosis ([7] and [8]), value- at-**risk** (VaR) [9] and conditional value-at-**risk** (CVaR) [1].

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There is a recent and growing literature on Bayesian model **averaging**. Examples of ap- plications of BMA can be found in a number of works (see, for example, Hoeting, Madigan, Raftery, and Volinsky (1999) and Steel (2011) for a recent overview). Our software package for parameter estimation and model comparison of linear regression models is based on Fer- nández, Ley, and Steel (2001a,b) and Koop (2003). We use the Markov Chain Monte Carlo Model Composition MC 3 sampling algorithm developed by Madigan, York, and Allard (1995) to select a representative subset of models.

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S(Y ) is called a **weighted** composition operator if there exist a complex- valued function u on Y , not necessarily continuous, and a map φ : Y → X such that T(f )(y) = u(y)f (φ(y)) for all f ∈ S(X) and y ∈ Y . Then T is denoted by uC φ and called the **weighted** composition operator induced

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However, a large new investment will not only increase the weight which the IRR calculation gives to future returns, it will also reduce the weight given to earlier returns (equation (3) shows that these weights sum to unity). This would be a retrospective adjustment which will boost the expected IRR even if (as in our simulations) there is no relationship **between** these intermediate cashflows and future returns. In this situation the IRR becomes a biased indicator of the profitability of this investment strategy, and we know that this bias is inherent in VA, since by construction disappointing returns are followed by larger net investments in order to raise the portfolio value to its target level.

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In recent years, many authors like Attele [12], Axler [13–16], Bercovici [17], Eschmeier [18], Luecking [19], Vukoti´c [20], and Zhu [21] have made a study of multiplication op- erators on Bergman spaces, whereas Campbell and Leach [22], Feldman [23], Lin [10], and Ohno and Takagi [24] have obtained a study of these **operators** on Hardy spaces. On Bloch spaces, these **operators** are studied by Arazy [25], Axler [15] and Brown and Shields [26]. Also, Axler and Shields [16] and Stegenga [27] have explored multiplication oper- ators on Dirichlet spaces. On BMOA, these **operators** are studied by Ortega and Fabreg´a [28]. Further, on Nevanlinna classes of analytic functions, these **operators** are studied by Jarchow et al. [29] and Yanagihara [30]. Besides these well-known analytic function spaces, a study of these **operators** on some other Banach spaces of analytic functions has also been pursued by Bonet et al. [31–34], Contreras and Hern´andez-D´ıaz [35], Ohno and Takagi [24], and Shields and Williams [36, 37].

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In this paper, we derived a useful relation **between** the predictiveness curve and the AUC and showed that this relation still holds when considering extensions of these concepts to survival outcomes. This relation enabled us to propose new estimators for the cumulative/dynamic AUC, relying on primary estimators of the conditional absolute **risk** function. These estimators are similar, in spirit, to one of the two estimators formerly proposed by Chambless and Diao (2006). Through an empirical study, we further showed that our estimation procedure attained performances similar to that reached by existing estimates. This simulation study also highlighted that much attention had to be paid when selecting the form of the model used to estimate the conditional **risk** function. Working under an appropriate parametric model usually yields more accurate estimates (for both the conditional **risk** function and AUC C,D (t)) than those obtained from purely nonparametric approaches, but misspecifying the model generally leads to dramatically biased estimates. This observation leads us to recommend to always use nonparametric estimators of the conditional absolute **risk** function at least to visually check the goodness-of-fit of parametric models, for instance by comparing estimates of the predictiveness curve. It is noteworthy that the proposed estimators of AUC C,D (t) are straightforward to implement: standard survival packages indeed return estimates of the conditional absolute **risk** function from which estimates of AUC C ,D (t) are readily obtained in view of Equation (7). Moreover, because of their ”plug-in” nature, their theoretical properties should follow from those established for estimators of the conditional absolute **risk** function. Closed form expressions might further be obtained for confidence intervals, but sub-sampling techniques (bootstrap for instance) can already be used to provide such intervals.

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Research Article Weighted Composition Operators between Mixed Norm Spaces and Hα∞ Spaces in the Unit Ball Stevo Stević Received 15 March 2007; Accepted 1 November 2007 Recommended by Ul[r]

Lindstr ¨om and Wolf 17 generalized Nieminen’s results on more general **weighted** Banach spaces. Furthermore, they estimated the essential norm of diﬀerences of two **weighted** composition **operators**. These works concerned with diﬀerences of **weighted** composition **operators** mainly focused on the setting of one variable. Recently, Toews 18, Gorkin et al. 19, and Aron et al. 20 extended the results of 12 to the case of several variables, respectively. In this paper, we study the boundedness and compactness of diﬀerences of **weighted** composition **operators** on **weighted** Banach spaces in the setting of several variables and extend some results of 16, 17. Due to the diﬀerence **between** one variable and several variables, some special constructive techniques are applied. After collecting some preliminary results in the next section, we give an elegant inequality see Lemma 3.2 which is useful to characterize the boundedness of diﬀerences of **weighted** composition **operators** on **weighted** Banach spaces in Section 3. In Section 4, we continue to describe the compactness of diﬀerences of **weighted** composition **operators** on these spaces and obtain some interesting corollaries.

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Actually, the main question that we will give the aﬃrmative answer to states that the solvability of the dynamic equation (3.2) not only is necessary for the validity of the **weighted** dynamic Hardy-type inequality (3.3) but also is suﬃcient. The next result will guarantee the ﬁrst direction, which emphasizes the need to achieve the equation in order to prove the legitimacy of the inequality. In the rest of the paper, we will assume that the function v(x) satisﬁes the condition

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Let X be a Banach space, we study Riemann-Stieltjes **operators** **between** X-valued **weighted** Bloch spaces. Some necessary and suﬃcient conditions for these **operators** induced by holomorphic functions to be weakly compact and weakly conditionally compact are given by certain growth properties of the inducing symbols and some structural properties of the abstract Banach space, which extend some previous results.

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Abstract. In this paper, we characterize the bonudedness and compactness of **weighted** composition **operators** from **weighted** Bergman spaces to **weighted** Bloch spaces. Also, we investigate **weighted** composition **operators** on **weighted** Bergman spaces and extend the obtained results in the unit ball of C n .

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The proposed OWA-PSSM approach is introduced in the following subsections. The OWA-PSSM method is designed based on the PSSM (Position Specific Scoring Matrix) which is a popular technique in the prediction of MHC binding [9,10,17,25-28]. In general, the lengths of MHC II binding cores are nine amino acids. Every position at the binding core is related to a specific pocket. The PSSM is employed to specify the binding strengths **between** twenty basic amino acids with these nine pockets, such that the binding specificities of HLA- DR molecules could be quantified.

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Analyzing the results in the articles cited above, one can see some common features that lead to explore the possibility of giving a general theorem characterizing the boundedness in **weighted** amalgams of a wide family of positive **operators**, and provid- ing, in such a way, a unified approach to the subject. This is the purpose of this article. 2. The results

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result to deﬁne the **weighted** Atangana–Baleanu fractional integral. In Sect. 3, we present the **weighted** Atangana–Baleanu **operators** in terms of the well-known Riemann–Liouville fractional integral, and investigate several properties of them. Finally, we end with some concluding remarks in Sect. 4.

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Main Results We will investigate the property of Hypercyclicity Criterion for a linear operator and in the special case, we will give suﬃcient conditions for the adjoint of a weighted co[r]

Here the leading coeﬃcients are locally VMO functions, while the hypotheses on the other coeﬃcients and the boundary conditions involve a suitable weight function.. Copyright © 2006 Lore[r]

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Manhas, Composition operators on function spaces, North Holland Math. Takagi, Compact weighted composition operators on L p -spaces, Proc.[r]

In the late 1960’s, E.A. Nordgren and J.V. Ryff studied composition opera- tors on the Hardy space H 2 . They provided upper and lower bounds on the norms of general composition **operators** and gave the exact norm in the case where the symbol map is an inner function.

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