Small Sample Variability Research Articles

Kernel-based covariate functional balancing for observational studies

Covariate balance is often advocated for objective causal inference since it mimics randomization in observational data. Unlike methods that balance specific moments of covariates, our proposal attains uniform approximate balance for covariate functions in a reproducing-kernel Hilbert space. The corresponding infinite-dimensional optimization problem is shown to have a finite-dimensional representation in terms of an eigenvalue optimization problem. Large-sample results are studied, and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods.

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.

Efficient Simulation of (Log)Normal Random Fields for Hydrogeological Applications

Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.

Portfolio Efficiency and Discount Factor Bounds with Conditioning Information: An Empirical Study

In this paper, we study the properties of portfolios and discount factor bounds in the presence of conditioning information. The main contribution of this paper is to provide a detailed comparison between various stochastic discount factor bounds with conditioning information. We do this by exploiting the explicit link between the stochastic discount factor approach and portfolio efficiency in the presence of conditioning information. For common choices of base assets and conditioning instruments, we find that the unconditionally efficient bounds of Ferson and Siegel (2002) are statistically indistinguishable from the (theoretically) optimal bounds of Gallant, Hansen, and Tauchen (1990), while having smaller sampling variability. We demonstrate that the difference in sampling variability of the UE and GHT bounds is due to the different behavior of the portfolio weights underlying their construction. Our work is closely related to and extends Ferson and Siegel (2001), Ferson and Siegel (2002) and Bekaert and Liu (2001).

Estimation with weak instruments: Accuracy of higher‐order bias and MSE approximations

In this paper, we consider parameter estimation in a linear simultaneous equations model. It is well known that two‐stage least squares (2SLS) estimators may perform poorly when the instruments are weak. In this case 2SLS tends to suffer from the substantial small sample biases. It is also known that LIML and Nagar‐type estimators are less biased than 2SLS but suffer from large small sample variability. We construct a bias‐corrected version of 2SLS based on the Jackknife principle. Using higher‐order expansions we show that the MSE of our Jackknife 2SLS estimator is approximately the same as the MSE of the Nagar‐type estimator. We also compare the Jackknife 2SLS with an estimator suggested by Fuller (Econometrica 45, 933–54) that significantly decreases the small sample variability of LIML. Monte Carlo simulations show that even in relatively large samples the MSE of LIML and Nagar can be substantially larger than for Jackknife 2SLS. The Jackknife 2SLS estimator and Fuller's estimator give the best overall performance. Based on our Monte Carlo experiments we conduct informal statistical tests of the accuracy of approximate bias and MSE formulas. We find that higher‐order expansions traditionally used to rank LIML, 2SLS and other IV estimators are unreliable when identification of the model is weak. Overall, our results show that only estimators with well‐defined finite sample moments should be used when identification of the model is weak.

Gene ranking using bootstrapped P-values

Recent research has shown that it is possible to find genes involved in the pathogenesis of a particular condition on the basis of microarray experiments. Genes which are differentially expressed, for example between healthy and diseased tissues, are likely to be relevant to the disease under study. Some of the properties of microarray datasets make the task of finding these genes a challenging one. This paper proposes a gene-ranking algorithm whose main novelty is the use of bootstrapped P-values. We present an analysis of the algorithm, showing how it takes account of small-sample variability in observed values of the test statistic, in a way conventional statistical tests cannot. Experimental results show that our algorithm outperforms the widely-used two-sample t -test on challenging artificial data. Gene ranking is then performed on two well-known microarray datasets, with encouraging results. For example, a number of genes from one of the datasets, whose differential expression was subsequently confirmed by a more reliable biochemical analysis, are found to be ranked higher by the bootstrapped algorithm than by the conventional t -test, suggesting that the proposed algorithm may be better able to exploit the limited data available to infer biologically useful information.

A New Approach to Random Utility Modeling using the Dirichlet Multinomial Distribution

In this paper we introduce a new econometricapproach to analyzing recreational site choicedata, the Dirichlet multinomial model. Thismodel, which nests the standard conditionalmultinomial logit model, can accommodateover-dispersed data and may provide moreefficient estimators of coefficients andconsequent welfare measures than the standardconditional logit model, which is so widelyused in the Random Utility Model approach torecreation demand. We illustrate thisDirichlet approach using a data set of rockclimbers in Scotland, and study the impacts onper-trip consumers surplus of alternativemanagement strategies for popular rock climbingsites. Results show that the Dirichletmultinomial approach produces coefficient andwelfare estimates having smaller samplingvariability in this case. We also compareclassical welfare measures with their posteriorequivalents, which allow for welfare changes tobe dis-aggregated.

The Equity Premium Puzzle: New Zealand Evidence

This paper examines the implications of standard barter models of market equilibrium for financial security returns in New Zealand. The key question addressed is: does the ‘equity premium puzzle’ of Mehra and Prescott (1985) found in the U.S. also hold in ?ew Zealand? To examine the existence of the equity premium puzzle, quarterly financial security returns and consumption data are examined from 1965 to 1997 to calibrate parameters in the Consumption Based Asset Pricing Model. Unlike much of the existing international evidence, this paper corrects for durable goods consumption following the assumptions of the model that all consumption be consumed in a given period. Numerical analyses indicate that the class of models examined are unable to generate equity premia consistent with historical estimates of the equity premium in New Zealand. Due to small sample variability however, while this discrepancy is material in size, the result is not statistically significant.

PREDICTING PROBABILITIES: INHERENT AND SAMPLING VARIABILITY IN THE ESTIMATION OF DISCRETE‐CHOICE MODELS

In this paper, we address two issues: the small sample variability in estimation of the parameters of female labor force participation models with dichotomous dependent variables due to random variation across small samples; and whether large-sample estimates of discrete-choice models yield the same results. In the first case, both logit and profit models exhibit similar patterns of parameter instability in 'small' samples and that a sample size of over 10,000 observations is required to reduce the issue of sampling variability to acceptable levels. In the second case, both models generate similar qualitative results in large samples; however, even in the largest samples, the quantitative results generated by these models diverge significantly. Coauthors are Zhengxi Lin, Lars Osberg, and Shelley Phipps. Copyright 1994 by Blackwell Publishing Ltd

Bivariate Normal Swimbladder Size Allometry Models and Allometric Exponents for 38 Mesopelagic Swimbladdered Fish Species Commonly Found in the North Sargasso Sea

With x and y log-transformed fish standard length/and equivalent swimbladder radius R, respectively, probability density distributions p(x,y) for mesopelagic swimbladdered fish species are assumed to be bivariate normal. From archival measurements, models are developed for 36 nonregressive species, 2 regressive species, and selected species groups found off Bermuda. Statistical tests suggest the hypothesis is valid for many of these species, and that failures are due largely to small sample variability and bias in the available data. Marked differences between regression, major axis, and reduced major axis allometric exponents, which sometimes occur when x,y correlation is low, are explained. A dimensionless allometric law R′* = k* (l′*)m involving structural variables l′* and R′* scaled to lmax is used to compare specific swimbladder growth trends. Major axis estimates of m* and k* for the nonregressive species group are each lognormally distributed with respective means [Formula: see text] and [Formula: see text]. A structural relation model for this group based on the hypothesis that swimbladder wall area grows approximately as the first power of fish volume or mass is verified. Last, detailed buoyancy properties are calculated for 6 nonregressive species with bivariate normal (x,w*)-distributions, where w* is the logarithm of percentage swimbladder volume.

Estimation of Breast Doses and Breast Cancer Risk Associated with Repeated Fluoroscopic Chest Examinations of Women with Tuberculosis

A methodology is presented to estimate cumulative breast dose and breast cancer risk for women exposed to repeated fluoroscopic chest examinations during air collapse therapy for pulmonary tuberculosis. Medical record abstraction, physician interview, patient contact, machine exposure measurements, and absorbed dose computations were combined to estimate average breast doses for 1047 Massachusetts women who were treated between 1930 and 1954. The methodology presented considers breast size and composition, patient orientation, x-ray field size and location, beam quality, type of examination, machine exposure rate, and exposure time during fluoroscopic examinations. The best estimate for the risk of radiation-induced cancer for the women living longer than 10 years after initial fluoroscopic exposure is 6.2 excess breast cancers per million woman-year-rad with 90% confidence limits of 2.8 and 10.7 cancers/10/sup 6/ WY-rad. When breast cancer risk is considered as a function of absorbed dose in the breast, instead of as a function of the number of fluoroscopic examinations, a linear dose--response relationship over the range of estimated doses is consistent with the data. However, because of the uncertainty due to small-sample variability and because of the wide range of assumptions regarding certain fluoroscopy conditions, other dose--response relationships are compatible with the data.