Small Sample Estimation Research Articles

A graphical model approach for inferring large-scale networks integrating gene expression and genetic polymorphism.

BackgroundGraphical models (e.g., Bayesian networks) have been used frequently to describe complex interaction patterns and dependent structures among genes and other phenotypes. Estimation of such networks has been a challenging problem when the genes considered greatly outnumber the samples, and the situation is exacerbated when one wishes to consider the impact of polymorphisms (SNPs) in genes.ResultsHere we describe a multistep approach to infer a gene-SNP network from gene expression and genotyped SNP data. Our approach is based on 1) construction of a graphical Gaussian model (GGM) based on small sample estimation of partial correlation and false-discovery rate multiple testing; 2) extraction of a subnetwork of genes directly linked to a target candidate gene of interest; 3) identification of cis-acting regulatory variants for the genes composing the subnetwork; and 4) evaluating the identified cis-acting variants for trans-acting regulatory effects of the target candidate gene. This approach identifies significant gene-gene and gene-SNP associations not solely on the basis of gene co-expression but rather through whole-network modeling. We demonstrate the method by building two complex gene-SNP networks around Interferon Receptor 12B2 (IL12RB2) and Interleukin 1B (IL1B), two biologic candidates in asthma pathogenesis, using 534,290 genotyped variants and gene expression data on 22,177 genes from total RNA derived from peripheral blood CD4+ lymphocytes from 154 asthmatics.ConclusionOur results suggest that graphical models based on integrative genomic data are computationally efficient, work well with small samples, and can describe complex interactions among genes and polymorphisms that could not be identified by pair-wise association testing.

Using Log‐Linear Smoothing to Improve Small‐Sample DIF Estimation

This study examined the extent to which log‐linear smoothing could improve the accuracy of differential item functioning (DIF) estimates in small samples of examinees. Examinee responses from a certification test were analyzed using White examinees in the reference group and African American examinees in the focal group. Using a simulation approach, separate DIF estimates for seven small‐sample‐size conditions were obtained using unsmoothed (U) and smoothed (S) score distributions. These small sample U and S DIF estimates were compared to a criterion (i.e., DIF estimates obtained using the unsmoothed total data) to assess their degree of variability (random error) and accuracy (bias). Results indicate that for most studied items smoothing the raw score distributions reduced random error and bias of the DIF estimates, especially in the small‐sample‐size conditions. Implications of these results for operational testing programs are discussed.

Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference

Structural vector autoregressions (SVARs) are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified. When identifying restrictions such as long-run restrictions are imposed on impulse responses, there have been no efficient algorithms for small-sample estimation and inference. To fill these important gaps in the literature, this paper makes four contributions. First, we establish general rank conditions for global identification of both overidentified and exactly identified models. Second, we show that these conditions can be checked as a simple matrix-filling exercise and that they apply to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we establish a very simple rank condition for exactly identified models that amounts to a straightforward counting exercise. Fourth, we develop a number of efficient algorithms for small-sample estimation and inference.

Should We Trust the Empirical Evidence from Present Value Models of the Current Account?

Abstract The present value model of the current account has been very popular, as it provides an optimal benchmark to which actual current account series have often been compared. We show why persistence in observed current account data makes the estimated optimal series very sensitive to small-sample estimation error, making it almost impossible to determine whether the consumption-smoothing current account tracks the actual current account closely, or not closely at all. Moreover, the standard Wald test of the model will falsely accept or reject the model with substantial probability. Monte Carlo simulations and estimations using annual and quarterly data from five OECD countries strongly support our predictions. In particular, we conclude that two important consensus results in the literature – that the optimal series is highly correlated with the actual series, but substantially less volatile – are not statistically robust.

Predictive Regressions: A Reduced-Bias Estimation Method

Standard predictive regressions produce biased coefficient estimates in small samples when the regressors are Gaussian first-order autoregressive with errors that are correlated with the error series of the dependent variable; see Stambaugh (1999) for the single-regressor model. This paper proposes a direct and convenient method to obtain reduced-bias estimators for single and multiple regressor models by employing an augmented regression, adding a proxy for the errors in the autoregressive model. We derive bias expressions for both the ordinary least squares and our reduced-bias estimated coefficients. For the standard errors of the estimated predictive coefficients we develop a heuristic estimator which performs well in simulations, for both the single-predictor model and an important specification of the multiple-predictor model. The effectiveness of our method is demonstrated by simulations and by empirical estimates of common predictive models in finance. Our empirical results show that some of the predictive variables that were significant under ordinary least squares become insignificant under our estimation procedure.

Estimation of the Mean of Truncated Exponential Distribution

Problem statement: In this study, the researcher considers the problem of estimation of the mean of the truncated exponential distribution. Approach: This study contracted with maximum likelihood and unique minimum variance unbiased estimators and gives a modification for the maximum likelihood estimator, asymptotic variances and asymptotic confidence intervals for the estimators. The properties of these estimators in small, moderate and large samples were investigated via asymptotic theory and computer simulation. Results: It turns out that the modified maximum likelihood estimator was more efficient than the others and exists with probability 1. Conclusion: The modified maximum likelihood estimator was always exist, fast and straightforward to compute and more likely to yield feasible values than the unique minimum variance unbiased estimator. Its variance was well approximated by the large sample variance of the other estimators.

A comparative study of the bias corrected estimates in logistic regression

Logistic regression is frequently used in many areas of applied statistics. The maximum likelihood estimates (MLE) of the logistic regression parameters are usually computed using the iterative Newton-Raphson method. It is well known that these estimates are biased. Several methods are proposed to correct the bias of these estimates. Among them Firth (1993) and Cordeiro and McCullagh (1991) proposed two promising methods. The conditional exact method (CMLE) is popular for small-sample estimates, and is available in many software packages. In this article we compare these methods in terms of their bias. In general, our extensive simulations show that the methods proposed by Cordeiro and McCullagh and by Firth work well, though Cordeiro and McCullagh is slightly better in our simulations. In case of separation, Firth or CMLE can be used; however, a judicious approach is required when there is a wide variation in results. Two real data analyses are given exhibiting these properties. The data analysis also includes bootstrap results.

On Estimation of Covariance Matrices With Kronecker Product Structure

The estimation of signal covariance matrices is a crucial part of many signal processing algorithms. In some applications, the structure of the problem suggests that the underlying, true covariance matrix is the Kronecker product of two valid covariance matrices. Examples of such problems are channel modeling for multiple-input multiple-output (MIMO) communications and signal modeling of EEG data. In applications, it may also be that the Kronecker factors in turn can be assumed to possess additional, linear structure. The maximum-likelihood (ML) method for the associated estimation problem has been proposed previously. It is asymptotically efficient but has the drawback of requiring an iterative search for the maximum of the likelihood function. Two methods that are fast and noniterative are proposed in this paper. Both methods are shown to be asymptotically efficient. The first method is a noniterative variant of a well-known alternating maximization technique for the likelihood function. It performs on par with ML in simulations but has the drawback of not allowing for extra structure in addition to the Kronecker structure. The second method is based on covariance matching principles and does not suffer from this drawback. However, while the large sample performance is the same, it performs somewhat worse than the first estimator in small samples. In addition, the Cramer-Rao lower bound for the problem is derived in a compact form. The problem of estimating the Kronecker factors and the problem of detecting if the Kronecker structure is a good model for the covariance matrix of a set of samples are related. Therefore, the problem of detecting the dimensions of the Kronecker factors based on the minimum values of the criterion functions corresponding to the two proposed estimation methods is also treated in this work.

A New Family of Nonparametric Quantile Estimators

A new core methodology for creating nonparametric L-quantile estimators is introduced and three new quantile L-estimators (SV1 p , SV2 p , and SV3 p ) are constructed using the new methodology. Monte Carlo simulation was used in order to investigate the performance of the new estimators for small and large samples under normal distribution and a variety of light and heavy-tailed symmetric and asymmetric distributions. The new estimators outperform, in most of the cases studied, the Harrell–Davis quantile estimator and the weighted average at X ([np]) quantile estimator.

Identification of temporal fundamental economic structure (FES) of India: An input–output and cross-entropy analysis

This study provides an understanding of the economic structure and structural changes in the Indian economy utilizing the fundamental economic structure (FES) approach. The FES construct states that certain selected characteristics of an economy will vary predictably with economic size, as measured by gross national product, population and total gross output. The main problem addressed in this study concerns the question whether identifiable patterns of relations between various macro-aggregates and economic transactions can be revealed via input–output tables. Jensen, West and Hewings have discussed the tiered, partitioned, and temporal approaches to the identification of FES using input–output tables. This study addresses the following four research questions: Does a temporal FES exist for the Indian economy? What proportions of the cells are predictable? Can the 1968–1990 temporal FES predict 1993–1994 table? Does a temporal FES manifest an enhanced understanding of the Indian economic structure? Regression analyses are used to identify the FES and non-FES cells for the Indian economy. Small sample estimation issues are addressed applying the cross-entropy approach. The input–output tables for 1968–1969, 1973–1974, 1978–1979, 1983–1984, 1989–1990 and 1993–1994 provide data for the analysis. Analysis reveals that temporal FES includes primary, secondary and tertiary sectors as components. This research has extended the notion of FES to include weak, moderate and strong FES cells.

Modeling and Calibration Errors in Measures of Portfolio Credit Risk

This paper develops an empirical procedure for analyzing the impact of model misspecification and calibration errors on measures of portfolio credit risk. When applied to large simulated portfolios with realistic characteristics, this procedure reveals that violations of key assumptions of the well-known Asymptotic Single-Risk Factor (ASRF) model are virtually inconsequential. By contrast, flaws in the calibrated inter-dependence of credit risk across exposures, which are driven by plausible small-sample estimation errors or popular rule-of-thumb values of asset return correlations, can lead to significant inaccuracies in measures of portfolio credit risk. Similar inaccuracies arise under erroneous, albeit standard, assumptions regarding the tails of the distribution of asset returns.

Small-sample estimation of negative binomial dispersion, with applications to SAGE data

We derive a quantile-adjusted conditional maximum likelihood estimator for the dispersion parameter of the negative binomial distribution and compare its performance, in terms of bias, to various other methods. Our estimation scheme outperforms all other methods in very small samples, typical of those from serial analysis of gene expression studies, the motivating data for this study. The impact of dispersion estimation on hypothesis testing is studied. We derive an "exact" test that outperforms the standard approximate asymptotic tests.

Identification and Small Sample Estimation of Thurstone's Unrestricted Model for Paired Comparisons Data

The interpretation of a Thurstonian model for paired comparisons where the utilities' covariance matrix is unrestricted proved to be difficult due to the comparative nature of the data. We show that under a suitable constraint the utilities' correlation matrix can be estimated, yielding a readily interpretable solution. This set of identification constraints can recover any true utilities' covariance matrix, but it is not unique. Indeed, we show how to transform the estimated correlation matrix into alternative correlation matrices that are equally consistent with the data but may be more consistent with substantive theory. Also, we show how researchers can investigate the sample size needed to estimate a particular model by exploiting the simulation capabilities of a popular structural equation modeling statistical package.

Probit Models with Dummy Endogenous Regressors

This study considers heckit-type approximations useful for a number of different trivariate probit models. They are simple to use and have no convergence problems like full maximum likelihood. Simulations show that a heckit and a least squares approximation perform as well as the trivariate probit estimator in small samples when the degree of endogeneity is not too severe. A simple double-heckit and a heteroskedasticity corrected heckit approximation seem particularly robust and promising for testing exogeneity. The methods are used to estimate the impact of physician advice on physical activity, where the heckit approximations work as well as full maximum likelihood.

SMALL‐SAMPLE DIF ESTIMATION USING LOG‐LINEAR SMOOTHING: A SIBTEST APPLICATION

ABSTRACTThe purpose of the current study was to examine whether log‐linear smoothing of observed score distributions in small samples results in more accurate differential item functioning (DIF) estimates under the simultaneous item bias test (SIBTEST) framework. Data from a teacher certification test were analyzed using White candidates in the reference group and African American candidates in the focal group. Smoothed and raw DIF estimates from 100 replications under seven different sample‐size conditions were compared to a criterion to determine the effect of smoothing on small‐sample DIF estimation. Root‐mean‐squared deviation and bias were used to evaluate the accuracy of DIF detection in the smoothed versus raw data conditions. Results indicate that, for most studied items, smoothing the raw score distributions reduced variability and bias of the DIF estimates especially in the small‐sample‐size conditions. Implications of these results for actual testing programs and future directions for research are discussed.

Estimating the rate of re-expansion of spontaneous pneumothorax by a formula derived from computed tomography volumetry studies

Background: Available evidence suggesting that the rate of re-expansion of spontaneous pneumothorax is 1.25%/day is based on a small sample and mathematical modelling-based estimates. Aim: To estimate the rate of...

Predictive Regressions: Method of Least Autocorrelation

Conventional predictive regressions produce biased and inefficient coefficient estimates in small samples when the predicting variable is Gaussian first order persistent and its innovations are highly correlated with the error series of the return. We propose a new estimation method (the method of least autocorrelation) conditional on the assumption of serially uncorrelated innovations to the dependent variable. The new estimator minimizes the weighted sum of squared autocorrelations of residuals obtained from fitting a return series through estimated parameters. The asymptotic properties of the estimator and a refined weighting matrix are discussed. Finite sample behavior of the estimator is compared with existing estimation methods in the simulation study. The new method produces more accurate and efficient point estimate of the true parameter and improves performance in the power and size tests. Our empirical results provide some fresh evidence on the predictability of stock returns during the post-war time period.

Análise de eventos hidrológicos extremos, usando-se a distribuição GEV e momentos LH

A distribuição de probabilidade generalizada de valores extremos (GEV), tem facilitado muitas aplicações em hidrologia, utilizada na modelação de eventos extremos naturais. Estudos sobre o assunto mostram que estimadores de máxima verossimilhança dos parâmetros da GEV são instáveis em pequenas amostras, podendo fornecer valores absurdos do parâmetro de forma, quando então são recomendados estimadores de momentos LH, baseados na combinação linear de estatísticas de altas ordens, introduzidas para caracterizar a parte mais alta da distribuição e os valores extremos dos dados; contudo, não se dispõe de programas computacionais para PC, que modelem eventos extremos via momentos LH. Objetivou-se, com este trabalho, apresentar a modelação de eventos hidrológicos extremos através da distribuição GEV, utilizando-se momentos LH para estimar seus parâmetros e o teste estatístico proposto por Wang (1998) para verificação da qualidade dos ajustes desenvolvidos no ambiente Matlab. Como resultados, são apresentados as estimativas dos parâmetros da GEV, os valores das taxas de momentos LH: coeficientes de variação, assimetria e curtose, e os valores do teste de qualidade de ajuste, em aplicações com dados de vazão de rios do Paraná.

Monte Carlo uncertainty calculations with small-sample estimates of complex quantities

Three statistical distributions have been tested as candidates to represent the uncertainty of complex-valued quantities in Monte Carlo measurement uncertainty calculations. Two candidates are Bayesian multivariate t-distributions with ‘non-informative’ priors. The other is the distribution of a ‘generalized pivotal quantity’ (GPQ) for a multivariate mean. The best performance observed was from the GPQ: the two multivariate t-distributions were unsatisfactory in terms of coverage. The testing methodology is general and can be used to check the validity of other uncertainty calculation procedures.

On some aspects of estimation of a common mean of two independent normal populations

In this paper, we study some aspects of the problem of estimation of a common mean of two normal populations from an asymptotic point of view. The Bayes estimate of the common mean under Jeffrey's prior is also considered. A simulation study is carried out to compare several competing estimates in small samples.