Small Sample Properties Research Articles

Bayesian regression on non-parametric mixed-effect models with shape-restricted Bernstein polynomials

ABSTRACT We develop a Bayesian estimation method to non-parametric mixed-effect models under shape-constrains. The approach uses a hierarchical Bayesian framework and characterizations of shape-constrained Bernstein polynomials (BPs). We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of BPs to ensure the desired shape constraints. The small sample properties of the Bayesian shape-constrained estimators across a range of functions are provided via simulation studies. Two real data analysis are given to illustrate the application of the proposed method.

MEAT GOAT ENTERPRISE EFFICIENCY ANALYSIS IN THE SOUTHEASTERN UNITED STATES

AbstractMeat goat enterprise efficiency was estimated using an input distance function (IDF) by applying stochastic production frontier techniques for the southeastern U.S. region. We found increasing returns to scale and scope economies for southeastern U.S. meat goat enterprises. Mean technical efficiency was 0.81. Our results suggest southeastern U.S. meat goat enterprises can be scale efficient if their size of operation is >~60 goats or >40 breeding does. Cost and IDF analyses show input expenses decreased substantially with increasing scale of operations in southeastern U.S. meat goat production. Empirical Monte Carlo simulation techniques show consistency of small-sample properties for the IDF.

Non parametric portmanteau tests for detecting non linearities in high dimensions

ABSTRACTWe propose two non parametric portmanteau test statistics for serial dependence in high dimensions using the correlation integral. One test depends on a cutoff threshold value, while the other test is freed of this dependence. Although these tests may each be viewed as variants of the classical Brock, Dechert, and Scheinkman (BDS) test statistic, they avoid some of the major weaknesses of this test. We establish consistency and asymptotic normality of both portmanteau tests. Using Monte Carlo simulations, we investigate the small sample properties of the tests for a variety of data generating processes with normally and uniformly distributed innovations. We show that asymptotic theory provides accurate inference in finite samples and for relatively high dimensions. This is followed by a power comparison with the BDS test, and with several rank-based extensions of the BDS tests that have recently been proposed in the literature. Two real data examples are provided to illustrate the use of the test procedure.

Weighted Bootstrap Approach for the Variance Ratio Tests: A Test of Market Efficiency

By means of Monte Carlo experiments using the weighted bootstrap, we evaluate the size and power properties in small samples of Chow and Denning’s [1] multiple variance ratio test and the automatic variance ratio test of Choi [2]. Our results indicate that the weighted bootstrap tests exhibit desirable size properties and substantially higher power than corresponding conventional tests.

Econometric Analysis of Production Networks with Dominant Units

This paper builds on the work of Acemoglu et al. (2012) and considers a production network with unobserved common technological factor and establishes general conditions under which the network structure contributes to aggregate fluctuations. It introduces the notions of strongly and weakly dominant units, and shows that at most a finite number of units in the network can be strongly dominant, while the number of weakly dominant units can rise with N (the cross section dimension). This paper further establishes the equivalence between the highest degree of dominance in a network and the inverse of the shape parameter of the power law. A new extremum estimator for the degree of pervasiveness of individual units in the network is proposed, and is shown to be robust to the choice of the underlying distribution. Using Monte Carlo techniques, the proposed estimator is shown to have satisfactory small sample properties. Empirical applications to US input-output tables suggest the presence of production sectors with a high degree of pervasiveness, but their effects are not sufficiently pervasive to be considered as strongly dominant.

Inference in Partially Identified Heteroskedastic Simultaneous Equations Models

Identification through heteroskedasticity in heteroskedastic simultaneous equations models (HSEMs) is considered. The possibility that heteroskedasticity identifies the structural parameters only partially is explicitly allowed for. The asymptotic properties of the identified parameters are derived. Moreover, tests for identification through heteroskedasticity are developed and their asymptotic distributions are derived. Monte Carlo simulations are used to explore the small sample properties of the asymptotically valid methods. Finally, the approach is applied to investigate the relation between the extent of economic openness and inflation.

Econometric Analysis of Production Networks with Dominant Units

This paper builds on the work of Acemoglu et al. (2012) and considers a production network with unobserved common technological factor and establishes general conditions under which the network structure contributes to aggregate fluctuations. It introduces the notions of strongly and weakly dominant units, and shows that at most a finite number of units in the network can be strongly dominant, while the number of weakly dominant units can rise with N (the cross section dimension). This paper further establishes the equivalence between the highest degree of dominance in a network and the inverse of the shape parameter of the power law. A new extremum estimator for the degree of pervasiveness of individual units in the network is proposed, and is shown to be robust to the choice of the underlying distribution. Using Monte Carlo techniques, the proposed estimator is shown to have satisfactory small sample properties. Empirical applications to US input-output tables suggest the presence of production sectors with a high degree of pervasiveness, but their effects are not sufficiently pervasive to be considered as strongly dominant.

Testing Alternative Multi-Factor Models

A GMM-based system for two different linear factor pricing models is used to test if the pricing errors are the same. Simulations demonstrate the small sample properties. As an illustration, the test is applied to the Fama-French (1996, 2015) models.

Identifying stationary series in panels: A Monte Carlo evaluation of sequential panel selection methods

Sequential panel selection methods (spsms — procedures that sequentially use conventional panel unit root tests to identify I(0) time series in panels) are increasingly used in the empirical literature. We check the reliability of spsms by using Monte Carlo simulations based on generating directly the individual asymptotic p values to be combined into the panel unit root tests, in this way isolating the classification abilities of the procedures from the small sample properties of the underlying univariate unit root tests. The simulations consider both independent and cross-dependent individual test statistics. Results suggest that spsms may offer advantages over time series tests only under special conditions.

Some small sample properties of the EWOC design in dose escalation trials

Model-based approaches to dose escalation trials such as CRM, EWOC and their variants are becoming increasingly popular in Phase I oncology trials. Although the literature includes a number of reasons why the practitioner may want to consider using a model-based design over a more traditional 3+3 design, there are still a number of reasons why they are not universally adopted. Theoretical reasons have often been provided for their superiority over 3+3 designs and these have generally relied on large sample theory, whereas Phase I trial sizes are usually considerably lower. Furthermore, when information external to the planned trial is available and may be used to construct informative prior distributions, the small sample properties of model-based designs are less well understood. By considering a planned trial for which only a discrete set of doses are available, we use simulation to show some small sample properties of the EWOC design (Babb et al, 1998). When the model recommends a dose between the available doses, we evaluate the strategy of always choosing the dose below that recommended by the model versus the closest available dose (even if that is higher). We also characterise two different parameterisations of the Bayesian 2-parameter logistic model that underpins the EWOC design and make some recommendations that could increase the acceptability of the EWOC design to those more comfortable with implementing a 3+3 design.

Group sequential testing of the predictive accuracy of a continuous biomarker with unknown prevalence.

Group sequential testing procedures have been proposed as an approach to conserving resources in biomarker validation studies. Previously, we derived the asymptotic properties of the sequential empirical positive predictive value (PPV) and negative predictive value (NPV) curves, which summarize the predictive accuracy of a continuous marker, under case-control sampling. A limitation of this approach is that the prevalence cannot be estimated from a case-control study and must be assumed known. In this paper, we consider group sequential testing of the predictive accuracy of a continuous biomarker with unknown prevalence. First, we develop asymptotic theory for the sequential empirical PPV and NPV curves when the prevalence must be estimated, rather than assumed known in a case-control study. We then discuss how our results can be combined with standard group sequential methods to develop group sequential testing procedures and bias-adjusted estimators for the PPV and NPV curve. The small sample properties of the proposed group sequential testing procedures and estimators are evaluated by simulation, and we illustrate our approach in the context of a study to validate a novel biomarker for prostate cancer.

On graphical tests for proportionality of hazards in two samples.

In this paper, we present a class of graphical tests of the proportional hazards hypothesis for two-sample censored survival data. The proposed tests are improvements over some existing tests based on asymptotic confidence bands of certain functions of the estimated cumulative hazard functions. The new methods are based on the comparison of unrestricted estimates of the said functions and their restricted versions under the hypothesis. They combine the rigour of analytical tests with the descriptive value of plots. Monte Carlo simulations suggest that the proposed asymptotic procedures have reasonable small sample properties. The power is much higher than existing graphical tests and comparable with existing analytical tests. The method is then illustrated through the analysis of a data set on bone marrow transplantation for Leukemia patients.

Empirical Tests for Stochastic Dominance Optimality

If a given risky prospect is compared with multiple choice alternatives, then a joint test for optimality is more appropriate than a series of pairwise Stochastic Dominance tests. We develop and implement a bootstrap empirical likelihood ratio test for this hypothesis. The test statistic and implied probabilities can be computed by searching over discrete distributions that obey a system of linear inequalities using quasi-Monte Carlo simulation and convex optimization methods. An extension of the Kroll-Levy simulation experiment shows favorable small-sample properties for data sets of realistic dimensions. In an application to Fama-French stock portfolios, pairwise tests classify a portfolio of small growth stocks as admissible, whereas our test classifies the portfolio as significantly non-optimal for every risk averter.

Conditional Systemic Risk with Penalized Copula

Financial contagion and systemic risk measures are commonly derived from conditional quantiles by using imposed model assumptions such as a linear parametrization. In this paper, we provide model free measures for contagion and systemic risk which are independent of the specification of conditional quantiles and simple to interpret. The proposed systemic risk measure relies on the contagion measure, whose tail behavior is theoretically studied. To emphasize contagion from extreme events, conditional quantiles are specified via hierarchical Archimedean copula. The parameters and structure of this copula are simultaneously estimated by imposing a non-concave penalty on the structure. Asymptotic properties of this sparse estimator are derived and small sample properties illustrated using simulations. We apply the proposed framework to investigate the interconnectedness between American, European and Australasian stock market indices, providing new and interesting insights into the relationship between systemic risk and contagion. In particular, our findings suggest that the systemic risk contribution from contagion in tail areas is typically lower during times of financial turmoil, while it can be significantly higher during periods of low volatility.

Robust estimation of the Pareto tail index: a Monte Carlo analysis

The Pareto distribution is often used in many areas of economics to model the right tail of heavy-tailed distributions. However, the standard method of estimating the shape parameter (the Pareto tail index) of this distribution—the maximum likelihood estimator (MLE), also known as the Hill estimator—is non-robust, in the sense that it is very sensitive to extreme observations, data contamination or model deviation. In recent years, a number of robust estimators for the Pareto tail index have been proposed, which correct the deficiency of the MLE. However, little is known about the performance of these estimators in small-sample setting, which often occurs in practice. This paper investigates the small-sample properties of the most popular robust estimators for the Pareto tail index, including the optimal B-robust estimator (Victoria-Feser and Ronchetti in Can J Stat 22:247–258, 1994), the weighted maximum likelihood estimator (Dupuis and Victoria-Feser in Can J Stat 34:639–658, 2006), the generalized median estimator (Brazauskas and Serfling in Extremes 3:231–249, 2001), the partial density component estimator (Vandewalle et al. in Comput Stat Data Anal 51:6252–6268, 2007), and the probability integral transform statistic estimator (PITSE) (Finkelstein et al. in N Am Actuar J 10:1–10, 2006). Monte Carlo simulations show that the PITSE offers the desired compromise between ease of use and power to protect against outliers in the small-sample setting.

Measuring the flow properties of small powder samples using an avalanche tester

ABSTRACTThe feasibility of using a small-scale avalanche tester to measure the flow properties of pharmaceutical lactose powders was examined. The modes of behavior observed in larger systems were displayed and showed a clear distinction between angular, free-flowing particles and more spherical particles of similar flow characteristics. Angular Lactohale LH100 particles showed slumping behavior at a rotational frequency of 0.33 Hz that disappeared at higher frequencies. Spherical lactose powder with a similar flow function to LH100 only showed rolling behavior under the same conditions, as did more cohesive powders LH200 and LH300. Further investigation of the LH100 data using fast Fourier analysis showed that the slumping frequency was one tenth of the rotational frequency.

The mean squared error of the likelihood based estimating equations in the general linear model

Mahmood and King (17) revealed that the marginal likelihood has the inherent property of unbiased estimating equa- tions amongst a range of modified likelihood methods. In this paper we extend our investigation to deriving the mean squared error of the scores of the profile likelihood, the marginal likelihood, the conditional profile likelihood and the conditional profile restricted likelihood. In terms of minimum mean squared error, the estimating equation from the conditional profile restricted likelihood emerged as the preferred method. This provides further support to the implications of thefindings of poor small-sample properties of Lagrange Multiplier (LM) tests in the literature which are based on biased estimating equations or having a larger mean squared error of the scores. We demonstrate that the relative error of the mean squared error between the conditional profile restricted and the marginal likelihood methods is negligible for increasingly larger samples. Amongst the unbiased estimating equations the minimum mean squared error criteria provides a clear choice of selecting the estimating equation for the purpose of estimation and testing.

Fractional Frequency Flexible Fourier Form to approximate smooth breaks in unit root testing

In this study, a Fractional Frequency Flexible Fourier Form DF-type unit root test is proposed. The small sample properties of the proposed test are found to be better than that of the integer frequency counterpart.

Quasi Maximum Likelihood Estimation of Spatial Models with Heterogeneous Coefficients

This paper considers spatial autoregressive panel data models and extends their analysis to the case where the spatial coefficients differ across the spatial units. It derives conditions under which the spatial coefficients are identified and develops a quasi maximum likelihood (QML) estimation procedure. Under certain regularity conditions, it is shown that the QML estimators of individual spatial coefficients are consistent and asymptotically normally distributed when both the time and cross section dimensions of the panel are large. It derives the asymptotic covariance matrix of the QML estimators allowing for the possibility of non-Gaussian error processes. Small sample properties of the proposed estimators are investigated by Monte Carlo simulations for Gaussian and non-Gaussian errors, and with spatial weight matrices of differing degree of sparseness. The simulation results are in line with the paper’s key theoretical findings and show that the QML estimators have satisfactory small sample properties for panels with moderate time dimensions and irrespective of the number of cross section units in the panel, under certain sparsity conditions on the spatial weight matrix.

Conditional quantile estimation based on optimal quantization: From theory to practice

Small-sample properties of a nonparametric estimator of conditional quantiles based on optimal quantization, that was recently introduced (Charlier et al., 2015), are investigated. More precisely, (i) the practical implementation of this estimator is discussed (by proposing in particular a method to properly select the corresponding smoothing parameter, namely the number of quantizers) and (ii) its finite-sample performances are compared to those of classical competitors. Monte Carlo studies reveal that the quantization-based estimator competes well in all cases and sometimes dominates its competitors, particularly when the regression function is quite complex. A real data set is also treated. While the main focus is on the case of a univariate covariate, simulations are also conducted in the bivariate case.