Lorens A. Imhof

Matrix Algebra and Its Applications to Statistics and Econometrics

Vector spaces unitary and Euclidean spaces linear transformations and matrices some characteristics of matrices factorization of matrices operations on matrices projectors and idempotent operators generalized inverses majorization inequalities for Eigenvalues matrix approximations optimization problems in statistics and econometrics quadratic subspaces inequalities in statistics and econometrics total least squares and regression.

Power control, capacity, and duality of uplink and downlink in cellular CDMA systems

Accurate power control is an essential requirement in the design of cellular code-division multiple-access (CDMA) systems. In this paper, we contribute three main themes to the power control problem. First, we derive an efficient algorithm for computing minimal power levels for large-scale networks within seconds. Nice and intuitive conditions for the existence of feasible power solutions follow from this approach. Second, we define the capacity region of a network by the set of effective spreading gains, or data rates, respectively, which can be supplied by the network. This is achieved by bounding the spectral radius of a certain matrix containing system parameters and mutual transmission gain information. It is shown that the capacity region is a convex set. Finally, we reveal an interesting duality between the uplink and downlink capacity region. In a clear-cut analytical way, it substantiates the fact that the uplink is the more restricting factor in cellular radio networks. The same methods carry over to certain models of soft handover. In the case that the channel gains are subject to log-normal shadowing, we introduce the concept of level-/spl alpha/ capacity regions. Despite the complicated structure, it can still be shown that this set is sandwiched by two convex sets coming arbitrarily close as variance decreases.

Monotone Imitation Dynamics in Large Populations

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235-265] and the related processes proposed by BenaI¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873-903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352-363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoners dilemma when players are restricted to a small set of simple strategies.

Pricing and Hedging of American Knock-In Options

Barrier options are some of the most popular among the myriad of “exotic” derivative products. Although they are path-dependent, “out” options are easy to value in a standard binomial or other lattice model. The path-dependence of “in” options can not be handled using a tree, but for European-style contracts, by “in-out parity” an out option is equal to the value of the plain vanilla option minus the price of the matching out option. Unfortunately, the parity relation is broken by American exercise, since the complementary in and out contracts will not be rationally exercised early at the same time. In this article, AitSahlia, Imhof, and Lai show how to price an American knock-in option by decomposing it into corresponding European knock-in plus the early exercise premium, which can be approximated separately.

Random Matching in Adaptive Dynamics

This paper studies the effect of randomness in per-period matching on the long-run outcome of non-equilibrium adaptive processes. If there are many matchings between each strategy revision, the randomness due to matching will be small; our question is when a very small noise due to matching has a negligible effect. We study two different senses of this idea, and provide sufficient conditions for each. The less demanding sense corresponds to sending the matching noise to zero while holding fixed all other aspects of the adaptive process. The second sense in which matching noise can be negligible is that it doesnt alter the limit distribution obtained as the limit of the invariant distributions as an exogeneous mutation rate goes to zero. When applied to a model with mutations, the difference between these two senses is in the order of limits: the first sense asks for continuity of e.g. the ergodic distribution in the matching noise holding the mutation rate fixed, whereas the second sense asks for continuity of the limit distribution in the matching noise.

Exact Designs Minimising the Integrated Variance in Quadratic Regression

Exact designs are given which minimise the average variance of a quadratic regression polynomial fitted by the method of least squares.

A counterexample in experimental design showing that the G-criterion function is not necessarily strictly decreasing

It is shown that, in the case of polynomial regression, there exist experimental designs x, y such that y is not better than x in the sense of the G-criterion although it is uniformly better than x. Furthermore, a simple proof of Jungs theorem on G-optimal exact designs for straight-line regression is given.