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We consider the problem of estimating the parameter $\fnl$ in the standard local model of primordial CMB non-Gaussianity. We determine the properties of maximum likelihood (ML) estimates and show that the problem is not the typical ML estimation problem as there are subtle issues involved. In particular, The Cramer-Rao inequality is not applicable and the likelihood function is unbounded with several points of maxima. However, the particular characteristics of the likelihood function lead to ML estimates that are simple and easy to compute. We compare their performance to that of moment estimators. We find that ML is better than the latter for values $\fnl$ away from the origin. For small values of $\fnl$, the fifth order moment is better than ML and the other moment estimators. However, we show how for small $\fnl$, one can easily improve the estimators by a simple shrinkage procedure. This is clearly important when the goal is to estimate a very small $\fnl$. In the process of studying the inference problem, we address some basic issues regarding statistical estimation in general that were raised at the Workshop on Non-Gaussianity in Cosmology held in Trieste in July 2006., Comment: 15 pages including 6 figures

We consider the estimation of the regularization parameter for the simultaneous deblurring of multiple noisy images via Tikhonov regularization. We approach the problem in three ways. We first reduce the problem to a single-image deblurring for which the regularization parameter can be estimated through a classic generalized cross-validation (GCV) method. A modification of this function is used for correcting the undersmoothing typical of the original technique. With a second method, we minimize an average least-squares fit to the images and define a new GCV function. In the last approach, we use the classical $GCV$ on a single higher-dimensional image obtained by concatanating all the images into a single vector. With a reliable estimator of the regularization parameter, one can fully exploit the excellent computational characteristics typical of direct deblurring methods, which, especially for large images, makes them competitive with the more flexible but much slower iterative algorithms. The performance of the techniques is analyzed through numerical experiments. We find that under the independent homoscedastic and Gaussian assumptions made on the noise, the three approaches provide almost identical results with the first single image providing the practical advantage that no new software is required and the same image can be used with other deblurring algorithms., Comment: To appear in Astronomy & Astrophysics

We consider the simultaneous deblurring of a set of noisy images whose point spread functions are different but known and spatially invariant, and the noise is Gaussian. Currently available iterative algorithms that are typically used for this type of problem are computationally expensive, which makes their application for very large images impractical. We present a simple extension of a classical least-squares (LS) method where the multi-image deblurring is efficiently reduced to a computationally efficient single-image deblurring. In particular, we show that it is possible to remarkably improve the ill-conditioning of the LS problem by means of stable operations on the corresponding normal equations, which in turn speed up the convergence rate of the iterative algorithms. The performance and limitations of the method are analyzed through numerical simulations. Its connection with a column weighted least-squares approach is also considered in an appendix., Comment: 9 pages, 16 figures. High resolution figures available upon demand. To appear in A&A

In this second paper in a series dedicated to developing efficient numerical techniques for the deblurring Cosmic Microwave Background (CMB) maps, we consider the case of asymmetric point spread functions (PSF). Although conceptually this problem is not different from the symmetric case, there are important differences from the computational point of view because it is no longer possible to use some of the efficient numerical techniques that work with symmetric PSFs. We present procedures that permit the use of efficient techniques even when this condition is not met. In particular, two methods are considered: a procedure based on a Kronecker approximation technique that can be implemented with the numerical methods used with symmetric PSFs but that has the limitation of requiring only mildly asymmetric PSFs. The second is a variant of the classic Tikhonov technique that works even with very asymmetric PSFs but that requires discarding the edges of the maps. We provide details for efficient implementations of the algorithms. Their performance is tested on simulated CMB maps., Comment: 9 pages, 13 Figures

Digital deblurring of images is an important problem that arises in multifrequency observations of the Cosmic Microwave Background (CMB) where, because of the width of the point spread functions (PSF), maps at different frequencies suffer a different loss of spatial resolution. Deblurring is useful for various reasons: first, it helps to restore high frequency components lost through the smoothing effect of the instrument's PSF; second, emissions at various frequencies observed with different resolutions can be better studied on a comparable resolution; third, some map-based component separation algorithms require maps with similar level of degradation. Because of computational efficiency, deblurring is usually done in the frequency domain. But this approach has some limitations as it requires spatial invariance of the PSF, stationarity of the noise, and is not flexible in the selection of more appropriate boundary conditions. Deblurring in real space is more flexible but usually not used because of its high computational cost. In this paper (the first in a series on the subject) we present new algorithms that allow the use of real space deblurring techniques even for very large images. In particular, we consider the use of Tikhonov deblurring of noisy maps with applications to {\it PLANCK}. We provide details for efficient implementations of the algorithms. Their performance is tested on Gaussian and non-Gaussian simulated CMB maps, and PSFs with both circular and elliptical symmetry. Matlab code is made available., Comment: 14 pages, 16, figures, A&A in press; high quality figures available upon request to the authors

We provide theoretical procedures and practical recipes to simulate non-Gaussian correlated, homogeneous random fields with prescribed marginal distributions and cross-correlation structure, either in a N-dimensional Cartesian space or on the celestial sphere. We illustrate our methods using far-infrared maps obtained with the Infrared Space Observatory. However, the methodology presented here can be used in other astrophysical applications that require modeling correlated features in sky maps, for example, the simulation of multifrequency sky maps where backgrounds, sources and noise are correlated and can be modeled by random fields., Comment: 24 pages, 4 figures. To appear in PASP

Point-source contamination in high-precision Cosmic Microwave Background (CMB) maps severely affects the precision of cosmological parameter estimates. Among the methods that have been proposed for source detection, wavelet techniques based on ``optimal'' filters have been proposed.In this paper we show that these filters are in fact only restrictive cases of a more general class of matched filters that optimize signal-to-noise ratio and that have, in general, better source detection capabilities, especially for lower amplitude sources. These conclusions are confirmed by some numerical experiments. \keywords{Methods: data analysis -- Methods: statistical, Comment: 6 pages, 3 figures

We consider wavelets as a tool to perform a variety of tasks in the context of analyzing cosmic microwave background (CMB) maps. Using Spherical Haar Wavelets we define a position and angular-scale-dependent measure of power that can be used to assess the existence of spatial structure. We apply planar Daubechies wavelets for the identification and removal of points sources from small sections of sky maps. Our technique can successfully identify virtually all point sources which are above 3 sigma and more than 80% of those above 1 sigma. We discuss the trade-offs between the levels of correct and false detections. We denoise and compress a 100,000 pixel CMB map by a factor of about 10 in 5 seconds achieving a noise reduction of about 35%. In contrast to Wiener filtering the compression process is model independent and very fast. We discuss the usefulness of wavelets for power spectrum and cosmological parameter estimation. We conclude that at present wavelet functions are most suitable for identifying localized sources., Comment: 10 pages, 6 figures. Submitted to MNRAS

The largest anisotropy in the cosmic microwave background (CMB) is the $\approx 3$ mK dipole assumed to be due to our velocity with respect to the CMB. Using the four year data set from all six channels of the COBE Differential Microwave Radiometers (DMR), we obtain a best-fit dipole amplitude $3.358 \pm 0.001 \pm 0.023$ mK in the direction $(l,b)=(264\deg.31 \pm 0\deg.04 \pm 0\deg.16, +48\deg.05 \pm 0\deg.02 \pm 0\deg.09)$, where the first uncertainties are statistical and the second include calibration and combined systematic uncertainties. This measurement is consistent with previous DMR and FIRAS results, Comment: New and improved version; to be published in ApJ next month

The Differential Microwave Radiometers (DMR) instrument aboard the Cosmic Background Explorer (COBE) has mapped the full microwave sky to mean sensitivity 26 uK per 7 deg field of view. The absolute calibration is determined to 0.7% with drifts smaller than 0.2% per year. We have analyzed both the raw differential data and the pixelized sky maps for evidence of contaminating sources such as solar system foregrounds, instrumental susceptibilities, and artifacts from data recovery and processing. Most systematic effects couple only weakly to the sky maps. The largest uncertainties in the maps result from the instrument susceptibility to the Earth's magnetic field, microwave emission from the Earth, and upper limits to potential effects at the spacecraft spin period. Systematic effects in the maps are small compared to either the noise or the celestial signal: the 95% confidence upper limit for the pixel-pixel rms from all identified systematics is less than 6 uK in the worst channel. A power spectrum analysis of the (A-B)/2 difference maps shows no evidence for additional undetected systematic effects., Comment: 47 pages including 13 encapsulated PostScript figures. Submitted to The Astrophysical Journal

The sky-RMS is the simplest model-independent characterization of a cosmological anisotropy signal. The RMS temperature fluctuations determined from the $COBE$-DMR four-year sky maps are frequency independent, consistent with the hypothesis that they are cosmological in origin, with a typical amplitude at 7 degrees of ~ 35 \pm\ 2 uK and at 10 degrees of ~29 \pm\ 1 uK. A joint analysis of the 7 deg and 10 deg "cross"-RMS derived from the data in both Galactic and Ecliptic coordinates is used to determine the RMS quadrupole normalization, $Q_{rms-PS}$, for a scale-invariant Harrison-Zel'dovich power law model. The difference in the inferred normalizations either including or excluding the quadrupole is a consequence of the sensitivity of the method to the Galaxy contaminated observed sky quadrupole. Whilst there are variations depending on the data selection, all results are consistent with an inferred \qrms\ normalization of ~18 \pm\ 2 uK., Comment: 14 pages, submitted to The Astrophysical Journal (Letters)

The frequency-independent RMS temperature fluctuations determined from the COBE-DMR two year sky maps are used to infer the parameter Q_{rms-PS}, which characterizes the normalization of power law models of primordial cosmological temperature anisotropy. In particular, a 'cross'-RMS statistic is used to determine Q_{rms-PS} for a forced fit to a scale-invariant Harrison-Zel'dovich (n = 1) spectral model. Using a joint analysis of the 7 degree and 10 degree RMS temperature derived from both the 53 and 90 GHz sky maps, we find Q_{rms-PS} = 17.0^{+2.5}_{-2.1} uK when the low quadrupole is included, and Q_{rms-PS} = 19.4^{+2.3}_{-2.1} uK excluding the quadrupole. These results are consistent with the n = 1 fits from more sensitive methods (e.g. power spectrum, correlation function). The effect of the low quadrupole derived from the COBE-DMR data on the inferred Q_{rms-PS} normalization is investigated. A bias to lower Q_{rms-PS} is found when the quadrupole is included. The higher normalization for a forced n = 1 fit is then favored by the cross-RMS technique. As initially pointed out in Wright et al. (1994a) and further discussed here, analytic formulae for the RMS sky temperature fluctuations will NOT provide the correct normalization amplitude., Comment: 15 pages including 3 tables and 3 figures, uuencoded Postscript; Accepted for publication in The Astrophysical Journal; COBE Preprint 94-16

The {\it COBE} DMR sky maps contain low-level correlated noise. We obtain estimates of the amplitude and pattern of the correlated noise from three techniques: angular averages of the covariance matrix, Monte Carlo simulations of two-point correlation functions, and direct analysis of the DMR maps. The results from the three methods are mutually consistent. The noise covariance matrix of a DMR sky map is diagonal to an accuracy of better than 1\%. For a given sky pixel, the dominant noise covariance occurs with the ring of pixels at an angular separation of $60 \deg$ due to the $60 \deg$ separation of the DMR horns. The mean covariance at $60 \deg$ is $0.45\% ^{+0.18}_{-0.14}$ of the mean variance. Additionally, the variance in a given pixel is $0.7\%$ greater than would be expected from a single beam experiment with the same noise properties. Auto-correlation functions suffer from a $\sim 1.5\; \sigma$ positive bias at $60 \deg$ while cross-correlations have no bias. Published {\it COBE} DMR results are not significantly affected by correlated noise. COBE pre-print 94-, Comment: 11 pages + 3 figures, post-script file

We use statistical and topological quantities to test the COBE-DMR first year sky maps against the hypothesis that the observed temperature fluctuations reflect Gaussian initial density perturbations with random phases. Recent papers discuss specific quantities as discriminators between Gaussian and non-Gaussian behavior, but the treatment of instrumental noise on the data is largely ignored. The presence of noise in the data biases many statistical quantities in a manner dependent on both the noise properties and the unknown CMB temperature field. Appropriate weighting schemes can minimize this effect, but it cannot be completely eliminated. Analytic expressions are presented for these biases, and Monte Carlo simulations used to assess the best strategy for determining cosmologically interesting information from noisy data. The genus is a robust discriminator that can be used to estimate the power law quadrupole-normalized amplitude independently of the 2-point correlation function. The genus of the DMR data are consistent with Gaussian initial fluctuations with Q_rms = 15.7 +/- 2.2 - (6.6 +/- 0.3)(n - 1) uK where n is the power law index. Fitting the rms temperature variations at various smoothing angles gives Q_rms = 13.2 +/- 2.5 uK and n = 1.7 +0.3 -0.6. While consistent with Gaussian fluctuations, the first year data are only sufficient to rule out strongly non-Gaussian distributions of fluctuations., Comment: 17 pages post script

The frequency-independent rms temperature fluctuations determined from the Cosmic Background Explorer-Differential Microwave Radiometer (COBE-DMR) two-year sky maps are used to infer the parameter Q(sub rms-PS), which characterizes the normalization of power-law models of primordial cosmological temperature anisotropy, for a forced fit to a scale-invariant Harrison-Zel'dovich (n = 1) spectral model. Using a joint analysis of the 7 deg and 10 deg 'cross'-rms derived from both the 53 and 90 GHz sky maps, we find Q(sub rms-PS) = 17.0(sub -2.1 sup +2.5) micro Kelvin when the low quadrupole is included, and Q(sub rms-PS) = 19.4(sub -2.1 sup +2.3) micro Kelvin excluding the quadrupole. These results are consistent with the n = 1 fits from more sensitive methods. The effect of the low quadrupole derived from the COBE-DMR data on the inferred Q(sub rms-PS) normalization is investigated. A bias to lower Q(sub rms-PS) is found when the quadrupole is included. The higher normalization for a forced n = 1 fit is then favored by the cross-rms technique.

Cosmic anisotrophy produces an excess variance sq sigma(sub sky) in the Delta maps produced by the Differential Microwave Radiometer (DMR) on cosmic background explorer (COBE) that is over and above the instrument noise. After smoothing to an effective resolution of 10 deg, this excess sigma(sub sky)(10 deg), provides an estimate for the amplitude of the primordial density perturbation power spectrum with a cosmic uncertainty of only 12%. We employ detailed Monte Carlo techniques to express the amplitude derived from this statistic in terms of the universal root mean square (rms) quadrupole amplitude, (Q sq/RMS)(exp 0.5). The effects of monopole and dipole subtraction and the non-Gaussian shape of the DMR beam cause the derived (Q sq/RMS)(exp 0.5) to be 5%-10% larger than would be derived using simplified analytic approximations. We also investigate the properties of two other map statistics: the actual quadrupole and the Boughn-Cottingham statistic. Both the sigma(sub sky)(10 deg) statistic and the Boughn-Cottingham statistic are consistent with the (Q sq/RMS)(exp 0.5) = 17 +/- 5 micro K reported by Smoot et al. (1992) and Wright et al. (1992).