A generalized multiplicative directional distance function for efficiency measurement in DEA

For measuring technical efficiency relative to a log-linear technology, a generalized multiplicative directional distance function (GMDDF) is developed using the framework of multiplicative directional distance function (MDDF). Furthermore, a computational procedure is suggested for its estimation. The GMDDF serves as a comprehensive measure of efficiency in revealing Pareto-efficient targets as it accounts for all possible input and output slacks. This measure satisfies several desirable properties of an ideal efficiency measure such as strong monotonicity, unit invariance, translation invariance, and positive affine transformation invariance. This measure can be easily implemented in any standard DEA software and provides the decision makers with the option of specifying preferable direction vectors for incorporating their decision-making preferences. Finally, to demonstrate the ready applicability of our proposed measure, an illustrative empirical analysis is conducted based on real-life data set of 20 hardware computer companies in India.     

A class of transformation rate models for recurrent event data

Recurrent event data frequently occur in longitudinal studies, and it is often of interest to estimate the effects of covariates on the recurrent event rate. This paper considers a class of semiparametric transformation rate models for recurrent event data, which uses an additive Aalen model as its covariate dependent baseline. The new models are flexible in that they allow for both additive and multiplicative covariate effects, and some covariate effects are allowed to be nonparametric and time-varying. An estimating procedure is proposed for parameter estimation, and the resulting estimators are shown to be consistent and asymptotically normal. Simulation studies and a real data analysis demonstrate that the proposed method performs well and is appropriate for practical use.     

Exploratory Analysis and Modeling of Stock Returns

In this article, novel joint semiparametric spline-based modeling of conditional mean and volatility of financial time series is proposed and evaluated on daily stock return data. The modeling includes functions of lagged response variables and time as predictors. The latter can be viewed as a proxy for omitted economic variables contributing to the underlying dynamics. The conditional mean model is additive. The conditional volatility model is multiplicative and linearized with a logarithmic transformation. In addition, a cube-root power transformation is employed to symmetrize the lagged response variables. Using cubic splines, the model can be written as a multiple linear regression, thereby allowing predictions to be obtained in a simple manner. As outliers are often present in financial data, reliable estimation of the model parameters is achieved by trimmed least-square (TLS) estimation for which a reasonable amount of trimming is suggested. To obtain a parsimonious specification of the model, a new model selection criterion corresponding to TLS is derived. Moreover, the (three-parameter) generalized gamma distribution is identified as suitable for the absolute multiplicative errors and shown to work well for predictions and also for the calculation of quantiles, which is important to determine the value at risk. All model choices are motivated by a detailed analysis of IBM, HP, and SAP daily returns. The prediction performance is compared to the classical generalized autoregressive conditional heteroskedasticity (GARCH) and asymmetric power GARCH (APGARCH) models as well as to a nonstationary time-trend volatility model. The results suggest that the proposed model may possess a high predictive power for future conditional volatility. Supplementary materials for this article are available online.     

The duality property of the Discrete Fourier Transform based on Simpson's rule

The classical Discrete Fourier Transform (DFT) satisfies a duality property that transforms a discrete time signal to the frequency domain and back to the original domain. In doing so, the original signal is reversed to within a multiplicative factor, namely the dimension of the transformation matrix. In this paper, we prove that the DFT based on Simpson's method satisfies a similar property and illustrate its effect on a real discrete signal. The duality property is particularly useful in determining the components of the transformation matrix as well as components of its positive integral powers. Copyright © 2012 John Wiley & Sons, Ltd.     

Мультипликативный и нтеграл Фурье и его дискретные анал оги

An analogue of the Fourier integral with respect to a generalized multiplicative system of functions is constructed. It is proved that every absolutely integrable andg-continuous function can be represented as such a multiplicative Fourier integral. The multiplicative Fourier transformation (the spectral function of the multiplicative Fourier integral) is introduced, and the direct and inverse discrete multiplicative Fourier transformations are constructed. The relationship between the spectral function of the multiplicative Fourier integral and the discrete multiplicative Fourier transformation is illuminated. This enables us to elucidate the influence of the discretization of a given function on the properties of its multiplicative spectral characteristics.     

On a dimension reduction regression with covariate adjustment

In this paper, we consider a semiparametric modeling with multi-indices when neither the response nor the predictors can be directly observed and there are distortions from some multiplicative factors. In contrast to the existing methods in which the response distortion deteriorates estimation efficacy even for a simple linear model, the dimension reduction technique presented in this paper interestingly does not have to account for distortion of the response variable. The observed response can be used directly whether distortion is present or not. The resulting dimension reduction estimators are shown to be consistent and asymptotically normal. The results can be employed to test whether the central dimension reduction subspace has been estimated appropriately and whether the components in the basis directions in the space are significant. Thus, the method provides an alternative for determining the structural dimension of the subspace and for variable selection. A simulation study is carried out to assess the performance of the proposed method. The analysis of a real dataset demonstrates the potential usefulness of distortion removal.     

可加乘积模型的研究

文章将乘积模型推广为可加乘积模型,延伸了正测量数据的适用范围.方法具有更强的灵活性,同时也可避免维数祸根问题.文中主要采用B样条逼近技术、最小乘积相对误差(LPRE)方法,研究非参数函数的估计问题.并在一些条件下,证明了非参数函数具有最优收敛速度.最后,通过数值模拟和实例分析,比较了提出的方法和其它几种方法的有限样本表现,验证了新方法的有效性.     

Multiplicative Adams Bashforth–Moulton methods

The multiplicative version of Adams Bashforth–Moulton algorithms for the numerical solution of multiplicative differential equations is proposed. Truncation error estimation for these numerical algorithms is discussed. A specific problem is solved by methods defined in multiplicative sense. The stability properties of these methods are analyzed by using the standart test equation.     

Estimation of IBNR claims by credibility theory

We develop a stochastic multiplicative model for the forecasting of IBNR claims. The factor depending on the accident year is credibility adjusted.The title of this note also suits for the papers by Straub (1971) and Kramreiter and Straub (1973). We made stronger assumptions simplifying drastically the numerical calculations and the parameter estimation problem.As showed in the numerical illustrations, the developed method is also applicable in case of scarce irregular data.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.     

Continuous Contour Monte Carlo for Marginal Density Estimation With an Application to a Spatial Statistical Model

The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this article, we propose a technique, the so-called continuous Contour Monte Carlo (CCMC) algorithm, for solving this problem. CCMC can be viewed as a continuous version of the contour Monte Carlo (CMC) algorithm recently proposed in the literature. CCMC abandons the use of sample space partitioning and incorporates the techniques of kernel density estimation into its simulations. CCMC is more general than other marginal density estimation algorithms. First, it works for any density functions, even for those having a rugged or unbalanced energy landscape. Second, it works for any transformation λ(x) regardless of the availability of the analytical form of the inverse transformation. In this article, CCMC is applied to estimate the unknown normalizing constant function for a spatial autologistic model, and the estimate is then used in a Bayesian analysis for the spatial autologistic model in place of the true normalizing constant function. Numerical results on the U.S. cancer mortality data indicate that the Bayesian method can produce much more accurate estimates than the MPLE and MCMLE methods for the parameters of the spatial autologistic model.     

Gravitational clustering and additive coalescence

We investigate a gravitational system in dimension one, started from some “uniform” random initial data. In Section 2, a connection is established with the additive coalescent. An hydrodynamic limit is obtained in Section 3 and it suggests a new construction of the standard additive coalescent. The latter is given in Section 3.2. An infinite system is considered in Section 4, and is shown to be closely related to the Smoluchowski additive and multiplicative equations.     

Additive–multiplicative hazards model with current status data

The additive–multiplicative hazards (AMH) regression model specifies an additive and multiplicative form on the hazard function for the counting process associated with a multidimensional covariate process, which contains the Cox proportional hazards model and the additive hazards model as its special cases. In this paper, we study the AMH model with current status data, where the cumulative hazard hazard function is assumed to be nonparametric and is estimated using B-splines with monotonicity constraint on the functional, while a simultaneous sieve maximum likelihood estimation is proposed to estimate regression parameters. The proposed estimator for the parameter vector is shown to be asymptotically normal and semiparametric efficient. The B-splines estimator of the functional of the cumulative hazard function is shown to achieve the optimal nonparametric rate of convergence. A simulation study is conducted to examine the finite sample performance of the proposed estimators and algorithm, and a real data example is presented for illustration.     

Attractors for a Damped Stochastic Wave Equation of Sine–Gordon Type with Sublinear Multiplicative Noise

Abstract

The existence of compact random attractors is proved for a damped stochastic wave equation of Sine–Gordon type with sublinear multiplicative noise under homogeneous Dirichlet boundary condition. To be important, in this note a precise estimate of upper bound of Hausdorff dimension of the random attractors is obtained in lower dimension.     

Integrating accounting and multiplicative calculus: an effective estimation of learning curve

Numerical interpolation methods are essential for the estimation of nonlinear functions and they have a wide range of applications in economics and accounting. In this regard, the idea of using interpolation methods based on multiplicative calculus for suitable accounting problems is self-evident. The purpose of this study, therefore, is to develop a way to better estimate the learning curve, which is an exponentially decreasing function, based on multiplicative Lagrange interpolation. The results of this study show that the proposed multiplicative method of learning curve provides more accurate estimates of labour costs when compared to the conventional methods. This is because the exponential functions are linear in multiplicative calculus. Furthermore, the results reveal that using the proposed method enables cost and managerial accountants to better calculate both cost of unused capacity and product cost in a cumulative production represented by a nonlinear function. The results of this study are also expected to help researchers, practitioners, economists, business managers, and cost and managerial accountants to understand how to construct a multiplicative based learning curve to improve such decisions as pricing, profit planning, capacity management, and budgeting.     

Algebra over estimation algorithms: Normalization with respect to the interval

Algebra over estimation algorithms with addition, multiplication by a constant, and normalization operations is investigated. Normalization is interpreted as a linear (with respect to each row) transformation of the matrix of estimates that takes the maximum entry of the row to unity and the minimum entry to zero. The algebraic closure is described, a formula for its dimension is obtained, and correctness criteria are formulated.     

Gelfand-Kirillov dimension in Jordan Algebras

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.

     

Multiperson decision-making based on multiplicative preference relations

A multiperson decision-making problem, where the information about the alternatives provided by the experts can be presented by means of different preference representation structures (preference orderings, utility functions and multiplicative preference relations) is studied. Assuming the multiplicative preference relation as the uniform element of the preference representation, a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives. In this decision model, several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations. The decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives, quantifier guided dominance degree and quantifier guided non-dominance degree. The consistency of the model is analysed to prove that it acts coherently.     

Relation between rank and multiplicative complexity of a bilinear form over a commutative Noetherian ring

The concept of multiplicative complexity of a bilinear form is introduced for a commutative Noetherian ring. Rings are described for which the multiplicative complexity coincides with the rank for all forms. It is shown that for regular rings of dimension 3 the multiplicative complexity can exceed the rank by an arbitrarily large number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 86, pp. 66–81, 1979.     

A diffusion-approximation theorem in navier-stokes equation

A convergence in distribution theorem is proved for the solution of the stochastic Navier-Stokes equation with multiplicative noise in dimension 2 or 3 when the noise is a mixing process. This result generalizes previous diffusionapproximation theorems to a non-linear case