Mean square convergence rates for maximum quasi-likelihood estimators

In this note we study the behavior of maximum quasilikelihood estimators (MQLEs) for a class of statistical models, in which only knowledge about the first two moments of the response variable is assumed. This class includes, but is not restricted to, generalized linear models with general link function. Our main results are related to guarantees on existence, strong consistency and mean square convergence rates of MQLEs. The rates are obtained from first principles and are stronger than known a.s. rates. Our results find important application in sequential decision problems with parametric uncertainty arising in dynamic pricing

Dynamic pricing and learning with finite inventories

We study a dynamic pricing problem with finite inventory and parametric uncertainty on the demand distribution. Products are sold during selling seasons of finite length, and inventory that is unsold at the end of a selling season, perishes. The goal of the seller is to determine a pricing strategy that maximizes the expected revenue. Inference on the unknown parameters is made by maximum likelihood estimation. We propose a pricing strategy for this problem, and show that the Regret - which is the expected revenue loss due to not using the optimal prices - after T selling seasons is O(log2(T)). Apart from a small modification, our pricing strategy is a certainty equivalent pricing strategy, which means that at each moment, the price is chosen that is optimal w.r.t. the current parameter estimates. The good performance of our strategy is caused by an endogenous-learning property: using a pricing policy that is optimal w.r.t. a certain parameter sufficiently close to the optimal one, leads to a.s. convergence of the parameter estimates to the true, unknown parameter. We also show an instance in which the regret for all pricing policies grows as log(T). This shows that ourupper bound on the growth rate of the regret is close to the best achievable growth rate

Mean square convergence rates for maximum quasi-likelihood estimators

In this note we study the behavior of maximum quasilikelihood estimators (MQLEs) for a class of statistical models, in which only knowledge about the first two moments of the response variable is assumed. This class includes, but is not restricted to, generalized linear models with general link function. Our main results are related to guarantees on existence, strong consistency and mean square convergence rates of MQLEs. The rates are obtained from first principles and are stronger than known a.s. rates. Our results find important application in sequential decision problems with parametric uncertainty arising in dynamic pricing

On hardware for generating routes in Kautz digraphs

In this paper we present a hardware implementation of an algorithm for generating node disjoint routes in a Kautz network. Kautz networks are based on a family of digraphs described by W.H. Kautz[Kautz 68]. A Kautz network with in-degree and out-degree d has N = dk + dk¿1 nodes (for any cardinals d, k>0). The diameter is at most k, the degree is fixed and independent of the network size. Moreover, it is fault-tolerant, the connectivity is d and the mapping of standard computation graphs such as a linear array, a ring and a tree on a Kautz network is straightforward.\ud The network has a simple routing mechanism, even when nodes or links are faulty. Imase et al. [Imase 86] showed the existence of d node disjoint paths between any pair of vertices. In Smit et al. [Smit 91] an algorithm is described that generates d node disjoint routes between two arbitrary nodes in the network. In this paper we present a simple and fast hardware implementation of this algorithm. It can be realized with standard components (Field Programmable Gate Arrays)

Dynamic Pricing and Learning with Finite Inventories

We study a dynamic pricing problem with finite inventory and parametric uncertainty on the demand distribution. Products are sold during selling seasons of finite length, and inventory that is unsold at the end of a selling season perishes. The goal of the seller is to determine a pricing strategy that maximizes the expected revenue. Inference on the unknown parameters is made by maximum likelihood estimation. We show that this problem satisfies an endogenous-learning property, which means that the unknown parameters are learned on-the-fly if the chosen selling prices are sufficiently close to the optimal ones. We show that a small modification to the certainty equivalent pricing strategy - which always chooses the optimal price w.r.t. current parameter estimates - satisfies Regret(T) = O(log2(T)), where Regret(T) measures the expected cumulative revenue loss w.r.t. a clairvoyant who knows the demand distribution. We complement this upper bound by showing an instance for which the regret of any pricing policy satisfies Ω(logT)

Dynamic Pricing and Learning with Finite Inventories

We study a dynamic pricing problem with finite inventory and parametric uncertainty on the demand distribution. Products are sold during selling seasons of finite length, and inventory that is unsold at the end of a selling season perishes. The goal of the seller is to determine a pricing strategy that maximizes the expected revenue. Inference on the unknown parameters is made by maximum likelihood estimation. We show that this problem satisfies an endogenous-learning property, which means that the unknown parameters are learned on-the-fly if the chosen selling prices are sufficiently close to the optimal ones. We show that a small modification to the certainty equivalent pricing strategy - which always chooses the optimal price w.r.t. current parameter estimates - satisfies Regret(T) = O(log2(T)), where Regret(T) measures the expected cumulative revenue loss w.r.t. a clairvoyant who knows the demand distribution. We complement this upper bound by showing an instance for which the regret of any pricing policy satisfies Ω(logT)