General Linear Function Research Articles

Convex Loss Applied to Design in Regression Problems

Summary A general linear regression function is to be observed at n points in order to estimate a known linear combination of the unknown parameters. The n points and the estimator are to be optimum in some sense and in this paper the main criterion for optimality involves uniformly minimizing certain convex loss functions. Three main sets of results are obtained, followed by some further results for normally distributed errors.

Cournot Oligopoly and Competitive Behaviour

This paper shows that the Cournot oligopoly model need not converge to perfect competition, regardless of the quasi-competitiveness of the model. Convergence takes place if, and only if, there are no scale economies. Without convergence, interest centres on the other properties of the Cournot model! It is shown, for example, that both quasicompetitiveness and stability may break down with large numbers of firms. But an example suggests that the numbers involved, though not impossible, might be higher than one could expect in practice. Throughout the analysis we examine the strong case in which all firms are identical. This strips away the non-essentials and considerably simplifies the analysis. Section II describes the model and demonstrates the uniqueness of the solution. Section III is devoted to the limiting behaviour of the Cournot model. Section IV examines the quasi-competitiveness and stability of the model. Section V offers an illustrative and suggestive example based on a general linear demand function and a general cubic cost function. Finally, Section VI makes some concluding remarks on the relation between numbers of firms and competitive behaviour.

A “Variable s” Inventory Model

This paper develops a linear periodic review inventory model in which the reorder rule is an extension of the “order up to S” or (S, T) policy where S varies as a general linear function of demand and net inventory. Stationary and non-stationary demand processes are considered. Lead time is assumed to be an integral multiple of the review period and, at most, an independent and identically distributed random variable. Analysis of the model includes stability, transient response, steady-state response, and the effects of common nonlinearities.

A Note on a Functional Equation

and determine the conditions on a polynomial F (x, y) in order that (1) can have a continuous strictly monotone (say increasing) solution. If F(x, y) is x + y or xy, we have the classical cases of Cauchy's functional equations. The cases when F(x, y) is a general linear function,1 and also the case when it is x + y + nxy, have been treated.2 In this paper the case when F(x, y) is a polynomial of degree greater than unity is settled, as follows:

Linear Functionals on Certain Spaces of Abstractly-Valued Functions

This paper is concerned with the study of certain classes of abstractly-valued functions which arise naturally in connection with the theory of integration of functions with values in a Banach space. These classes correspond to the usual classes C, LP, M, etc. which are well known in analysis. The subject matter of the paper is divided into seven sections as follows: ?2 contains most of the technical apparatus, which consists of the Lebesgue integral of abstract functions as defined by Bochner,2 and several types of integrals based on the notion of the Riemann-Stieltjes integral. Since, in the case of abstractly-valued functions, there may be a distinction between an indefinite integral and an absolutely continuous function,3 we have found it important to introduce the class of functions of finite p-variation, corresponding to the classical criterion for a function's being the indefinite integral of a function of class LV. For certain Banach spaces it is known that the distinction mentioned above does not arise.4 Such a space is said to satisfy condition (D) [for the precise statement see ?21. We have obtained some information about the implications of this condition and its relation to other properties of the space [see especially ??4, 5 and 7]. ?3 contains the determination of the most general linear functionals on the various function classes under discussion. ?4 is devoted to a discussion of weak convergence and its relation to condition (D). In ?5 the interrelations of condition (D), weak completeness, and reflexiveness of Banach spaces is taken up. In ?6 we have results on the Parseval relation and properties of the Fourier coefficients of abstract functions. Counter examples to show the divergence from classical theory are given. In ?7 we return to the question of reflexivity and condition (D), obtaining a criterion for spaces which satisfy this condition. Finally, in ?8 it is shown how by the methods of Fourier series it is possible to characterize