Automatic variance control and variance estimation loops

A closed loop servo approach is applied to the problem of controlling and estimating variance in nonstationary signals. The new circuit closely resembles but is not the same as, automatic gain control (AGC) which is common in radio and other circuits. The closed loop nature of the solution to this problem makes this approach highly accurate and can be used recursively in real time

Gini estimation under infinite variance

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution

Noise Variance Estimation In Signal Processing

We present a new method of estimating noise variance. The method is applicable for 1D and 2D signal processing. The essence of this method is estimation of the scatter of normally distributed data with high level of outliers. The method is applicable to data with the majority of the data points having no signal present. The method is based on the shortest half sample method. The mean of the shortest half sample (shorth) and the location of the least median of squares are among the most robust measures of the location of the mode. The length of the shortest half sample has been used as the measurement of the data scatter of uncontaminated data. We show that computing the length of several sub samples of varying sizes provides the necessary information to estimate both the scatter and the number of uncontaminated data points in a sample. We derive the system of equations to solve for the data scatter and the number of uncontaminated data points for the Gaussian distribution. The data scatter is the measure of the noise variance. The method can be extended to other distributions

Asymptotics for sliced average variance estimation

In this paper, we systematically study the consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve $\sqrt{n}$ consistency even when each slice contains only two data points. However, SAVE cannot be $\sqrt{n}$ consistent and it even turns out to be not consistent when each slice contains a fixed number of data points that do not depend on n, where n is the sample size. These results theoretically confirm the notion that SAVE is more sensitive to the number of slices than SIR. Taking this into account, a bias correction is recommended in order to allow SAVE to be $\sqrt{n}$ consistent. In contrast, when the response is discrete and takes finite values, $\sqrt{n}$ consistency can be achieved. Therefore, an approximation through discretization, which is commonly used in practice, is studied. A simulation study is carried out for the purposes of illustration.Comment: Published at http://dx.doi.org/10.1214/009053606000001091 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A note on multiple imputation for method of moments estimation

Multiple imputation is a popular imputation method for general purpose estimation. Rubin(1987) provided an easily applicable formula for the variance estimation of multiple imputation. However, the validity of the multiple imputation inference requires the congeniality condition of Meng(1994), which is not necessarily satisfied for method of moments estimation. This paper presents the asymptotic bias of Rubin's variance estimator when the method of moments estimator is used as a complete-sample estimator in the multiple imputation procedure. A new variance estimator based on over-imputation is proposed to provide asymptotically valid inference for method of moments estimation.Comment: 8 pages, 0 figur

Variance-type estimation of long memory

The aggregation procedure when a sample of length N is divided into blocks of length m = o(N), m ® ¥ and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu, Teverovsky and Willinger (1995), Teverovsky and Taqqu (1997) introduced an aggregate variance estimator of the long memory parameter of a stationary sequence with long range dependence and studied its empirial performance. With respect to autovariance structure and marginal distribution, the aggregated series is closer to Gaussian fractional noise than the initial series. However, the variance type estimator based on aggregated data is seriously biased. A refined estimator, which employs least squares regression across varying levels of aggregation, has much smaller bias, permitting derivation of limiting distributional properties of suitably centered estimates, as well as of a minimum mean squared error choice of bandwidth m. The results vary considerably with the actual value of the memory parameter

Variance estimation for a low-income proportion

Proportions below a given fraction of a quantile of an income distribution are often estimated from survey data in poverty comparisons. We consider the estimation of the variance of such a proportion, estimated from Family Expenditure Survey data. We show how a linearization method of variance estimation may be applied to this proportion, allowing for the effects of both a complex sampling design and weighting by a raking method to population controls. We show that, for 1998-99 data, the estimated variances are always increased when allowance is made for the design and raking weights, the principal effect arising from the design. We also study the properties of a simplified variance estimator and discuss extensions to a wider class of poverty measures