Sieve-based confidence intervals and bands for L\'{e}vy densities

The estimation of the L\'{e}vy density, the infinite-dimensional parameter controlling the jump dynamics of a L\'{e}vy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the L\'{e}vy density based on Grenander's method of sieves was proposed in Figueroa-L\'{o}pez [IMS Lecture Notes 57 (2009) 117--146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the L\'{e}vy density. In the pointwise case, our estimators converge to the L\'{e}vy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth L\'{e}vy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{-1/2}(n)\cdot n^{-1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ286 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Optimal random sampling designs in random field sampling

A Horvitz-Thompson predictor is proposed for spatial sampling when the characteristic of interest is modeled as a random field. Optimal sampling designs are deduced under this context. Fixed and variable sample size are considered

Localization of Energy in General Relativity

In the framework of the teleparallel equivalent of general relativity the energy density of asymptoticaly flat gravitational fields can be naturally and unambiguously defined. Upon integration of the energy density over the whole three dimensional space we obtain the ADM energy. We use this energy density to calculate the energy inside a Schwarzschild black hole.Comment: 12 pages, LaTex file, no figure

Using a ‘wellbeing’ cost-effectiveness approach to improve resource allocation in social care

The promotion of wellbeing is the newly-stated guiding principle for the long-term care (social care) system in England. It signals a shift away from a focus on care need ‘deficits’ approach. Such a change in perspective has the potential to substantially alter how public care systems operate. The practical challenges are significant, both in the interpretation of wellbeing goals and in determining how the care system might be configured to achieve them. The main aim of this paper is to contrast a needs-led resource allocation system with one using a maximising wellbeing approach; that is, one based on: measuring the wellbeing consequences of using services and applying the principles of cost-effectiveness and opportunity cost. As a precursor, the paper also describes how a maximising wellbeing approach might be applied in the case of long-term care. We argue that in theory a maximising wellbeing approach with full information will produce greater total wellbeing improvement for the same budget than a needs-based system. In practice, the comparison will depend on: (a) whether we can actually measure wellbeing in a way that is consistent with the policy goals; (b) the availability of cost-effectiveness information; and (c) the decision rules used to implement a maximising wellbeing approach

Cramer-Rao bounds in the estimation of time of arrival in fading channels

This paper computes the Cramer-Rao bounds for the time of arrival estimation in a multipath Rice and Rayleigh fading scenario, conditioned to the previous estimation of a set of propagation channels, since these channel estimates (correlation between received signal and the pilot sequence) are sufficient statistics in the estimation of delays. Furthermore, channel estimation is a constitutive block in receivers, so we can take advantage of this information to improve timing estimation by using time and space diversity. The received signal is modeled as coming from a scattering environment that disperses the signal both in space and time. Spatial scattering is modeled with a Gaussian distribution and temporal dispersion as an exponential random variable. The impact of the sampling rate, the roll-off factor, the spatial and temporal correlation among channel estimates, the number of channel estimates, and the use of multiple sensors in the antenna at the receiver is studied and related to the mobile subscriber positioning issue. To our knowledge, this model is the only one of its kind as a result of the relationship between the space-time diversity and the accuracy of the timing estimation.Peer ReviewedPostprint (published version