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"variance"
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Using the negative binomial distribution to model overdispersion in ecological count data
A Poisson process is a commonly used starting point for modeling stochastic variation of ecological count data around a theoretical expectation. However, data typically show more variation than implied by the Poisson distribution. Such overdispersion is often accounted for by using models with different assumptions about how the variance changes with the expectation. The choice of these assumptions can naturally have apparent consequences for statistical inference. We propose a parameterization of the negative binomial distribution, where two overdispersion parameters are introduced to allow for various quadratic mean-–variance relationships, including the ones assumed in the most commonly used approaches. Using bird migration as an example, we present hypothetical scenarios on how overdispersion can arise due to sampling, flocking behavior or aggregation, environmental variability, or combinations of these factors. For all considered scenarios, mean-–variance relationships can be appropriately described by the negative binomial distribution with two overdispersion parameters. To illustrate, we apply the model to empirical migration data with a high level of overdispersion, gaining clearly different model fits with different assumptions about mean-–variance relationships. The proposed framework can be a useful approximation for modeling marginal distributions of independent count data in likelihood-based analyses.
Journal Article
Toward more accurate contextualization of the CEO effect on firm performance
We introduce multiple refinements to the standard method for assessing CEO effects on performance, variance partitioning methodology, more accurately contextualizing CEOs' contributions. Based on a large 20-year sample, our new 'CEO in Context' technique points to a much larger aggregate CEO effect than is obtained from typical approaches. As a validation test, we show that our technique yields estimates of CEO effects more in line with what would be expected from accepted theory about CEO influence on performance. We do this by examining the CEO effects in subsamples of low-, medium-, and high-discretion industries. Finally, we show that our technique generates substantially different—and we argue more logical—estimates of the effects of many individual CEOs than are obtained through customary analyses.
Journal Article
Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise
This paper shows how to use realized kernels to carry out efficient feasible inference on the ex post variation of underlying equity prices in the presence of simple models of market frictions. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which equals that of the maximum likelihood estimator in the parametric version of this problem. Realized kernels can also be selected to (i) be analyzed using endogenously spaced data such as that in data bases on transactions, (ii) allow for market frictions which are endogenous, and (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice.
Journal Article
Temporal fluctuation scaling in populations and communities
Taylor's law, one of the most widely accepted generalizations in ecology, states that the variance of a population abundance time series scales as a power law of its mean. Here we reexamine this law and the empirical evidence presented in support of it. Specifically, we show that the exponent generally depends on the length of the time series, and its value reflects the combined effect of many underlying mechanisms. Moreover, sampling errors alone, when presented on a double logarithmic scale, are sufficient to produce an apparent power law. This raises questions regarding the usefulness of Taylor's law for understanding ecological processes. As an alternative approach, we focus on short-term fluctuations and derive a generic null model for the variance-to-mean ratio in population time series from a demographic model that incorporates the combined effects of demographic and environmental stochasticity. After comparing the predictions of the proposed null model with the fluctuations observed in empirical data sets, we suggest an alternative expression for fluctuation scaling in population time series. Analyzing population fluctuations as we have proposed here may provide new applied (e.g., estimation of species persistence times) and theoretical (e.g., the neutral theory of biodiversity) insights that can be derived from more generally available short-term monitoring data.
Journal Article
On the meta-analysis of response ratios for studies with correlated and multi-group designs
A common effect size metric used to quantify the outcome of experiments for ecological meta-analysis is the response ratio (RR): the log proportional change in the means of a treatment and control group. Estimates of the variance of RR are also important for meta-analysis because they serve as weights when effect sizes are averaged and compared. The variance of an effect size is typically a function of sampling error; however, it can also be influenced by study design. Here, I derive new variances and covariances for RR for several often-encountered experimental designs: when the treatment and control means are correlated; when multiple treatments have a common control; when means are based on repeated measures; and when the study has a correlated factorial design, or is multivariate. These developments are useful for improving the quality of data extracted from studies for meta-analysis and help address some of the common challenges meta-analysts face when quantifying a diversity of experimental designs with the response ratio.
Journal Article
Variance estimation using refitted cross‐validation in ultrahigh dimensional regression
Variance estimation is a fundamental problem in statistical modelling. In ultrahigh dimensional linear regression where the dimensionality is much larger than the sample size, traditional variance estimation techniques are not applicable. Recent advances in variable selection in ultrahigh dimensional linear regression make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are selected, leading to a serious underestimate of the level of noise. We propose a two‐stage refitted procedure via a data splitting technique, called refitted cross‐validation, to attenuate the influence of irrelevant variables with high spurious correlations. Our asymptotic results show that the resulting procedure performs as well as the oracle estimator, which knows in advance the mean regression function. The simulation studies lend further support to our theoretical claims. The naive two‐stage estimator and the plug‐in one‐stage estimators using the lasso and smoothly clipped absolute deviation are also studied and compared. Their performances can be improved by the refitted cross‐validation method proposed.
Journal Article