Minimax Estimation of Nonregular Parameters and Discontinuity in Minimax Risk

When a parameter of interest is nondifferentiable in the probability, the existing theory of semiparametric efficient estimation is not applicable, as it does not have an influence function. Song (2014) recently developed a local asymptotic minimax estimation theory for a parameter that is a nondifferentiable transform of a regular parameter, where the nondifferentiable transform is a composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map. The contribution of this paper is two fold. First, this paper extends the local asymptotic minimax theory to nondifferentiable transforms that are a composite map of a Lipschitz continuous map having a finite set of nondifferentiability points and a translation-scale equivariant map. Second, this paper investigates the discontinuity of the local asymptotic minimax risk in the true probability and shows that the proposed estimator remains to be optimal even when the risk is locally robustified not only over the scores at the true probability, but also over the true probability itself. However, the local robustification does not resolve the issue of discontinuity in the local asymptotic minimax risk

Comparison of Information Structures and Completely Positive Maps

A theorem of Blackwell about comparison between information structures in classical statistics is given an analogue in the quantum probabilistic setup. The theorem provides an operational interpretation for trace-preserving completely positive maps, which are the natural quantum analogue of classical stochastic maps. The proof of the theorem relies on the separation theorem for convex sets and on quantum teleportation.Comment: 12 pages. Substantial changes. Accepted for publication in Journal of Physics

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Fisher information and asymptotic normality in system identification for quantum Markov chains

This paper deals with the problem of estimating the coupling constant $\theta$ of a mixing quantum Markov chain. For a repeated measurement on the chain's output we show that the outcomes' time average has an asymptotically normal (Gaussian) distribution, and we give the explicit expressions of its mean and variance. In particular we obtain a simple estimator of $\theta$ whose classical Fisher information can be optimized over different choices of measured observables. We then show that the quantum state of the output together with the system, is itself asymptotically Gaussian and compute its quantum Fisher information which sets an absolute bound to the estimation error. The classical and quantum Fisher informations are compared in a simple example. In the vicinity of $\theta=0$ we find that the quantum Fisher information has a quadratic rather than linear scaling in output size, and asymptotically the Fisher information is localised in the system, while the output is independent of the parameter.Comment: 10 pages, 2 figures. final versio

Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states

We investigate the problem of optimally reversing the action of an arbitrary quantum channel C which acts independently on each component of an ensemble of n identically prepared d-dimensional quantum systems. In the limit of large ensembles, we construct the optimal reversing channel R* which has to be applied at the output ensemble state, to retrieve a smaller ensemble of m systems prepared in the input state, with the highest possible rate m/n. The solution is found by mapping the problem into the optimal reversal of Gaussian channels on quantum-classical continuous variable systems, which is here solved as well. Our general results can be readily applied to improve the implementation of robust long-distance quantum communication. As an example, we investigate the optimal reversal rate of phase flip channels acting on a multi-qubit register.Comment: 17 pages, 3 figure

Differential selection pressures exerted by host resistance quantitative trait loci on a pathogen population: a case study in an apple × Venturia inaequalis pathosystem

Understanding how pathogens evolve according to pressures exerted by their plant hosts is essential for the derivation of strategies aimed at the durable management of resistant cultivars. The spectrum of action of the resistance factors in the partially resistant cultivars is thought to be an important determinant of resistance durability. However, it has not yet been demonstrated whether the pressures exerted by quantitative resistance are different according to their spectrum of action.To investigate selection pressures exerted by apple genotypes harbouring various resistance quantitative trait loci (QTLs) on a mixed inoculum of the scab disease agent, Venturia inaequalis, we monitored V. inaequalis isolate proportions on diseased apple leaves of an F1 progeny using quantitative pyrosequencing technology and QTL mapping. Broad-spectrum resistances did not exert any differential selection pressures on the mixed inoculum, whereas narrow-spectrum resistances decreased the frequencies of some isolates in the mixture relative to the susceptible host genotypes. Our results suggest that the management of resistant cultivars should be different according to the spectrum of action of their resistance factors. The pyramiding of broad-spectrum factors or the use of a mixture of apple genotypes that carry narrow-spectrum resistance factors are two possible strategies for the minimization of resistance erosion

The threat of wild habitat to scab resistant apple cultivars

Evaluations of plant resistance to pathogens are rarely made using isolates from wild habitats, although the heterogeneity of such habitats may generate pathogen diversity which could be a source of new virulence in cultivated habitats. The aim of this study was to investigate whether scab resistance factors, identified and characterized in apples using isolates of Venturia inaequalis from a cultivated habitat, remained effective against isolates from a wild habitat. Three V. inaequalis core collections originating from the cultivated apple Malus × domestica and from two wild species, M. sieversii and M. sylvestris, were established to maximize pathogen diversity. For each core collection, 10 isolates were inoculated in mixtures onto 51 genotypes from an apple progeny segregating for two qualitative resistance genes and six quantitative resistance loci (QRL). On each apple genotype, isolates that contributed to the scab symptoms were identified within the mixture using microsatellite markers. The most frequently detected isolates were inoculated singly to compare their aggressiveness according to their host origin. The results showed that isolates from a wild habitat were able to infect the susceptible apple genotypes. However, these isolates were never more aggressive than isolates from the cultivated habitat on the resistance factors tested. It can therefore be concluded that the resistance factors used in this study, identified with V. inaequalis isolates from a cultivated habitat, remained effective against isolates from M. sylvestris and M. sieversii