Estimation from quantized Gaussian measurements: when and how to use dither

Subtractive dither is a powerful method for removing the signal dependence of quantization noise for coarsely quantized signals. However, estimation from dithered measurements often naively applies the sample mean or midrange, even when the total noise is not well described with a Gaussian or uniform distribution. We show that the generalized Gaussian distribution approximately describes subtractively dithered, quantized samples of a Gaussian signal. Furthermore, a generalized Gaussian fit leads to simple estimators based on order statistics that match the performance of more complicated maximum likelihood estimators requiring iterative solvers. The order statistics-based estimators outperform both the sample mean and midrange for nontrivial sums of Gaussian and uniform noise. Additional analysis of the generalized Gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements. Specifically, we find subtractive dither to be beneficial when the ratio between the Gaussian standard deviation and quantization interval length is roughly less than one-third. When that ratio is also greater than 0.822/K^0.930 for the number of measurements K > 20, estimators we present are more efficient than the midrange.https://arxiv.org/abs/1811.06856Accepted manuscrip

Labor supply models: unobserved heterogeneity, nonparticipation and dynamics

This chapter is concerned with the identification and estimation of models of labor supply. The focus is on the key issues that arise from unobserved heterogeneity, nonparticipation and dynamics. We examine the simple ‘static’ labor supply model with proportional taxes and highlight the problems surrounding nonparticipation and missing wages. The difference in differences approach to estimation and identification is developed within the context of the labour supply model. We also consider the impact of incorporating nonlinear taxation and welfare programme participation. Family labor supply is looked at from botht e unitary and collective persepctives. Finally we consider intertemporal models focusing on the difficulties that arise with participation and heterogeneity

Optimal estimates of the diffusion coefficient of a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.Comment: 8 pages, 5 figure

Entropy and information in neural spike trains: Progress on the sampling problem

The major problem in information theoretic analysis of neural responses and other biological data is the reliable estimation of entropy--like quantities from small samples. We apply a recently introduced Bayesian entropy estimator to synthetic data inspired by experiments, and to real experimental spike trains. The estimator performs admirably even very deep in the undersampled regime, where other techniques fail. This opens new possibilities for the information theoretic analysis of experiments, and may be of general interest as an example of learning from limited data.Comment: 7 pages, 4 figures; referee suggested changes, accepted versio

Shrinkage Estimation of the Power Spectrum Covariance Matrix

We seek to improve estimates of the power spectrum covariance matrix from a limited number of simulations by employing a novel statistical technique known as shrinkage estimation. The shrinkage technique optimally combines an empirical estimate of the covariance with a model (the target) to minimize the total mean squared error compared to the true underlying covariance. We test this technique on N-body simulations and evaluate its performance by estimating cosmological parameters. Using a simple diagonal target, we show that the shrinkage estimator significantly outperforms both the empirical covariance and the target individually when using a small number of simulations. We find that reducing noise in the covariance estimate is essential for properly estimating the values of cosmological parameters as well as their confidence intervals. We extend our method to the jackknife covariance estimator and again find significant improvement, though simulations give better results. Even for thousands of simulations we still find evidence that our method improves estimation of the covariance matrix. Because our method is simple, requires negligible additional numerical effort, and produces superior results, we always advocate shrinkage estimation for the covariance of the power spectrum and other large-scale structure measurements when purely theoretical modeling of the covariance is insufficient.Comment: 9 pages, 7 figures (1 new), MNRAS, accepted. Changes to match accepted version, including an additional explanatory section with 1 figur

Optimal Correlation Estimators for Quantized Signals

Using a maximum-likelihood criterion, we derive optimal correlation strategies for signals with and without digitization. We assume that the signals are drawn from zero-mean Gaussian distributions, as is expected in radio-astronomical applications, and we present correlation estimators both with and without a priori knowledge of the signal variances. We demonstrate that traditional estimators of correlation, which rely on averaging products, exhibit large and paradoxical noise when the correlation is strong. However, we also show that these estimators are fully optimal in the limit of vanishing correlation. We calculate the bias and noise in each of these estimators and discuss their suitability for implementation in modern digital correlators.Comment: 8 Pages, 3 Figures, Submitted to Ap

Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime

Power-law tail behavior and the summation scheme of Levy-stable distributions is the basis for their frequent use as models when fat tails above a Gaussian distribution are observed. However, recent studies suggest that financial asset returns exhibit tail exponents well above the Levy-stable regime ($0<\alpha\le 2$). In this paper we illustrate that widely used tail index estimates (log-log linear regression and Hill) can give exponents well above the asymptotic limit for $\alpha$ close to 2, resulting in overestimation of the tail exponent in finite samples. The reported value of the tail exponent $\alpha$ around 3 may very well indicate a Levy-stable distribution with $\alpha\approx 1.8$.Comment: To be published in Int. J. Modern Physics C (2001) vol. 12 no.

Monte Carlo-based tail exponent estimator

In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under {\alpha}-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can hardly be used on samples with small length. Utilizing our results, we introduce a Monte Carlo-based method of estimation for the tail exponent. Our proposed method is not sensitive to the choice of tail size and works well also on small data samples. The new estimator also gives unbiased results with symmetrical confidence intervals. Finally, we demonstrate the power of our estimator on the international world stock market indices. On the two separate periods of 2002-2005 and 2006-2009, we estimate the tail exponent