By G. Larry Bretthorst
This publication is basically a examine record at the program of likelihood concept to the parameter estimation challenge. the folks who could be drawn to this fabric are physicists, chemists, economists, and engineers who've to accommodate facts every day; for that reason, we now have incorporated loads of introductory and educational fabric. anyone with the identical of the math history required for the graduate-level examine of physics will be capable of keep on with the cloth contained during this ebook, although no longer with no attempt. during this paintings we follow chance conception to the matter of estimating parameters in quite normal types. specifically whilst the version involves a unmarried desk bound sinusoid we exhibit that the direct software of likelihood concept will yield frequency estimates an order of significance larger than a discrete Fourier remodel in signal-to-noise of 1. Latter, we generalize the matter and exhibit that likelihood idea can separate shut frequencies lengthy after the peaks in a discrete Fourier rework have merged.
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Extra info for Lecture Notes In Statistics Bayesian Spectrum Analysis And Parameter Estimation
To some, this result will be quite startling. Indeed, had we considered the periodogram itself to be a spectrum estimator, we would have calculated instead the width of its central peak. ^ would have a periodogram proportional to fN (! ) / sin sin f(! ^ )=2g thus the half-width at half amplitude is given by jN (^! )=2j = =4 or ! = =2N . 12) indicated by probability theory. 7), even for unit signal-to-noise ratio. 13) to be a little overoptimistic. The famous Rayleigh criterion  for resolving power of an optical instrument supposes that the minimum resolvable frequency dierence corresponds to the peak of the periodogram of one sinusoid coming at the rst zero of the periodogram of the second.
The second term in this equation is essentially an estimate of , but for small N it diers appreciably from the direct estimate found next. 3 The Estimated Noise Variance h2i One of the things that is of interest in an experiment is to estimate the noise power . This indicates how \good" the data appear to be in the light of the model, and can help one in making many judgments, such as whether to try a new model or build a new apparatus. g parameters; however, we can just as easily obtain the posterior moments hsi for any power s.
2N . 12) indicated by probability theory. 7), even for unit signal-to-noise ratio. 13) to be a little overoptimistic. The famous Rayleigh criterion  for resolving power of an optical instrument supposes that the minimum resolvable frequency dierence corresponds to the peak of the periodogram of one sinusoid coming at the rst zero of the periodogram of the second. 14) is a fundamental irreducible limit of resolution. Of course there is no such theorem, and workers in high resolution NMR have been routinely determining line positions to an accuracy that surpasses the Rayleigh limit by an order of magnitude, for thirty years.