Estimation of trend in state-space models: Asymptotic mean square error and rate of convergence

The focus of this paper is on trend estimation for a general state-space model $Y_t=\mu_t+\varepsilon_t$, where the $d$th difference of the trend $\{\mu_t\}$ is assumed to be i.i.d., and the error sequence $\{\varepsilon_t\}$ is assumed to be a mean zero stationary process. A fairly precise asymptotic expression of the mean square error is derived for the estimator obtained by penalizing the $d$th order differences. Optimal rate of convergence is obtained, and it is shown to be "asymptotically equivalent" to a nonparametric estimator of a fixed trend model of smoothness of order $d-0.5$. The results of this paper show that the optimal rate of convergence for the stochastic and nonstochastic cases are different. A criterion for selecting the penalty parameter and degree of difference $d$ is given, along with an application to the global temperature data, which shows that a longer term history has nonlinearities that are important to take into consideration.Comment: Published in at http://dx.doi.org/10.1214/08-AOS675 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The evolution of faint AGN between z~1 and z~5 from the COMBO-17 survey

We present a determination of the optical/UV AGN luminosity function and its evolution, based on a large sample of faint (R<24) QSOs identified in the COMBO-17 survey. Using multi-band photometry in 17 filters within 350nm < lambda_obs < 930nm, we could simultaneously determine photometric redshifts with an accuracy of sigma_z<0.03 and obtain spectral energy distributions. The redshift range covered by the sample is 1.2<z<4.8, which implies that even at z~3, the sample reaches below luminosities corresponding to M_B = -23, conventionally employed to distinguish between Seyfert galaxies and quasars. We clearly detect a broad plateau-like maximum of quasar activity around z~2 and map out the smooth turnover between z~1 and z~4. The shape of the LF is characterised by some mild curvature, but no sharp `break' is present within the range of luminosities covered. Using only the COMBO-17 data, the evolving LF can be adequately described by either a pure density evolution (PDE) or a pure luminosity evolution (PLE) model. However, the absence of a strong L*-like feature in the shape of the LF inhibits a robust distinction between these modes. We present a robust estimate for the integrated UV luminosity generation by AGN as a function of redshift. We find that the LF continues to rise even at the lowest luminosities probed by our survey, but that the slope is sufficiently shallow that the contribution of low-luminosity AGN to the UV luminosity density is negligible. Although our sample reaches much fainter flux levels than previous data sets, our results on space densities and LF slopes are completely consistent with extrapolations from recent major surveys such as SDSS and 2QZ.Comment: 17 pages, 14 figures, Astronomy & Astrophysics, in print, revised versio