Sharp estimates for perturbation errors in summations

When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.

PurposeIn a recent paper, Yoon and Lee (2019) (YL hereafter) propose a weighted Fama and MacBeth (FMB hereafter) two-step panel regression procedure and provide evidence that their weighted FMB procedure produces more efficient coefficient estimators than the usual unweighted FMB procedure. The purpose of this study is to supplement and improve their weighted FMB procedure, as they provide neither asymptotic results (i.e. consistency and asymptotic distribution) nor evidence on how close their standard error estimator is to the true standard error.Design/methodology/approachFirst, asymptotic results for the weighted FMB coefficient estimator are provided. Second, a finite-sample-adjusted standard error estimator is provided. Finally, the performance of the adjusted standard error estimator compared to the true standard error is assessed.FindingsIt is found that the standard error estimator proposed by Yoon and Lee (2019) is asymptotically consistent, although the finite-sample-adjusted standard error estimator proposed in this study works better and helps to reduce bias. The findings of Yoon and Lee (2019) are confirmed even when the average R2 over time is very small with about 1% or 0.1%.Originality/valueThe findings of this study strongly suggest that the weighted FMB regression procedure, in particular the finite-sample-adjusted procedure proposed here, is a computationally simple but more powerful alternative to the usual unweighted FMB procedure. In addition, to the best of the authors’ knowledge, this is the first study that presents a formal proof of the asymptotic distribution for the FMB coefficient estimator.

The bootstrap: Some large sample theory and connections with robustness

In social science researches, there may be cases where a category of the dependent variable is seen hundred times less (more) than the other category. Events like wars, mass migrations or coups in social sciences; an event of interest in binary variable(s) may have very low prevalence, resulting in low or even zero cell counts in one or two cells in the 2X2 tables of two factors. In this case, independent variable predict the dependent variable perfectly or almost perfectly, and this leads to an issue called complete or quasi-complete separation problem in statistical modelling. This study aims to compare three methods suggested in the literature for the quasi-complete separation in a real small dataset; penalized maximum likelihood (Firth-type), exact logistic regression and bayesian logistic regression. Methods were compared via odds ratios, odds’ standard error estimates, confidence intervals and statistical significance. Parameter estimates were obtained under three different models with binary and continuous variables. Results show that all methods can provide convergence in the presence of quasi-complete separation. In conclusion, bayesian logistic regression estimates tend to be superior than the other methods in terms of estimation of standard errors.

The energy consumption of supercomputers is one of the critical problems for the upcoming Exascale supercomputing era. The awareness of power an energy consumption is required on both software and hardware side. This poster deals with the energy consumption evaluation of the Total-Finite Element Tearing and Interconnect (TFETI) based solvers [2] of linear systems implemented in PERMON toolbox [1], which is an established method for solving real-world engineering problems, and with the energy consumption evaluation of the BLAS routines. The experiments performed in the poster deal with CPU frequency. This work is performed in the scope of the READEX project (Runtime Exploitation of Application Dynamism for Energy-efficient eXascale computing) [6]. The measurements were performed on the Intel Xeon E5-2680 (Intel Haswell micro-architecture) based Taurus system installed at TU Dresden. The system contains over 1400 nodes that have an FPGA-based power instrumentation called HDEEM (High Definition Energy Efficiency Monitoring), that allows for fine-grained and more accurate power and energy measurements. The measurements can be accessed through the HDEEM library, allowing developers to take energy measurements before and after the region of interest. We have evaluated the effect of the CPU frequency on the energy consumption of the TFETI solver for a linear elasticity 3D cube synthetic benchmark. On the dualized problem MPFX=MPd, we have evaluated the effect of frequency tuning on the energy consumption of the essential processing kernels of the TFETI method. There are two main phases in TFETI — preprocessing and solve. In preprocessing it is necessary to regularize the stiffness matrix K and factorize it and to assemble the G and GGT matrices and the second one to factorize. Both operations belong to the most time and also energy consuming operations. The solve employs the Preconditioned Conjugate Gradient (PCG) algorithm, which consists of sparse matrix-vector multiplications (by F, P, M L , M D matrices) and vector dot products and AXPY functions. In each iteration, we need to apply the direct solver twice, i.e., for forward and backward solves for the pseudoinverse K+ action and for the coarse problem solution, the (GGT)−1 action. The multiplication by the dense Schur complement matrix adds an additional operator with different computational characteristics, potentially increasing the exploitable dynamism. The poster provides results for two types of frequency tuning: (1) static tuning and (2) dynamic tuning. For static tuning experiments, the frequency is set before execution and kept constant during the runtime. For dynamic tuning, the frequency is changed during the program execution to adapt the system to the actual needs of the application. The poster shows that static tuning brings up 11.84% energy savings when compared to default CPU settings (the highest clock rate). The dynamic tuning improves this further by up to 2.68%. In total, the approach presented in this paper shows the potential to save up to 14.52% of energy for TFETI based solvers, see Table1. Another energy consumption evaluations were done with selected Sparse and Dense BLAS Level 1, 2 and 3 routines. For benchmarking we have used a set of matrices from University Florida collection [4]. We have employed AXPY, Sparse Matrix-Vector, Sparse MatrixMatrix, Dense Matrix-Vector, Dense Matrix-Matrix and Sparse Matrix-Dense Matrix multiplication routines from Intel Math Kernel Library (MKL) [3]. The measured characteristics illustrate the different energy consumption of BLAS routines, as some operations are memory-bounded and others are compute-bounded. Based on our recommendations one can explore dynamic frequency switching to achieve significant energy savings up to 23%, for more details see Table 2.

Approximating Matrix Multiplication for Pattern Recognition Tasks

Improving importance estimation in covariate shift for providing accurate prediction error

The maximum likelihood estimators for the three distribution parameters of a log Pearson type 3 distribution are derived from the logarithmic likelihood function and a solution method is given.The T‐year event is then derived as a function of these parameters and the standard normal deviate t. These parameters are subject to sampling variances and covariances, whereas t is not. By using the logarithmic likelihood function the inverse dispersion matrix is derived, hence the variances and covariances of the parameters. Entering these in the general variance of the estimate equation of a function of three variables leads to an estimate of the asymptotic standard error of estimate of the T‐year event. Flood data from 37 hydrometric stations on Canadian rivers were analyzed by this method and compared with the more common moment analysis. In all cases the maximum likelihood analysis is markedly superior in terms of the estimate of standard error of estimate, but the asymptotic nature of the result must always be borne in mind. Possible sources of bias in the maximum likelihood estimate of the T‐year event are discussed.

The robustness of three multiple‐comparison procedures (multiple t test, Tukey WSD and Scheffé S test) to the violation of the homogeneous population variance assumption was investigated using three different standard error estimates. These estimates were (1) the square root of two mean square within/ n , (2) the standard error of the traditional t test and (3) the standard error of the Behrens‐Fisher t ′ statistic. The procedures were highly robust to variance heterogeneity using the Behrens‐Fisher standard error estimate. The procedures were considerably less robust using MSW. The universal use of the Behrens‐Fisher statistic with the Welch solution for critical values is recommended.

Background: The target of inference in air pollution cohort studies is the association between exposure and disease outcomes. Exposure is typically predicted from monitoring data; the resulting measurement error affects health effect estimates and their confidence intervals. Since standard Berkson and classical measurement error categories are not directly applicable, we present new characterizations and correction methods. Methods: We decompose the measurement error into Berkson-like and classical-like components. Berkson-like error is introduced by smoothing the exposure surface, while the classical-like error results from uncertainty in estimating exposure model parameters. We discuss their impacts in linear health models with exposure surfaces modeled in 1) a parametric framework (with a correctly specified mean model and spatially correlated residuals) and 2) a semi-parametric framework (where locations are random and the deterministic exposure can have a mis-specified mean model). We propose practical correction methods for each. Results: Both measurement error components change the variability of the health effect estimates. Classical-like error introduces bias, although this is frequently small. In the parametric framework Berkson-like error does not introduce bias, whereas in the semi-parametric framework it may introduce bias, depending on whether additional assumptions are satisfied. In the parametric framework, a parametric bootstrap (or a computationally less intensive alternative) can be used for bias correction and/or standard error estimation. In the semi-parametric framework, there are asymptotic methods for bias correction and a simple non-parametric bootstrap can be used to estimate standard errors. Conclusions: Measurement error in air pollution cohort studies is a complex mixture of Berkson-like and classical-like components. There are important implications for study design and for developing post-hoc correction methods.

A domain-specific processor for energy-efficient execution of Recognition and Data Mining (RM) workloads is presented. The processor consists of a 2-D array of processing elements and a streaming memory hierarchy and interconnect network that are customized to efficiently execute dominant computational kernels (matrix-vector multiplication, vector dot product, L1 norm, and L2 norm) from a wide range of RM algorithms. To achieve further energy efficiency, the RM processor utilizes scalable effort design, a technique that exploits the inherent resilience of algorithms to inexactness in their constituent computations. The scalable effort RM processor adopts a cross-layer approach by combining scaling mechanisms at the algorithm, architecture, and circuit levels, to create a desirable trade off between energy consumption and output quality. Measurements from the implemented chip in 65nm CMOS indicate processing efficiencies of 569 GOPS/W-4.68 TOPS/W. The use of scalable effort design achieves energy savings of 1.2-2.3X with no loss in output quality, and 2X-20X with modest reduction in quality.

Applying single-level statistical models to multilevel data typically produces underestimated standard errors, which may result in misleading conclusions. This study examined the impact of ignoring multilevel data structure on the estimation of item parameters and their standard errors of the Rasch, two-, and three- parameter logistic models in item response theory (IRT) to demonstrate the degree of such underestimation in IRT. Also, the Lord’s chi-square test using the underestimated standard errors was used to test differential item functioning (DIF) to show the impact of such underestimation on the practical applications of IRT. The results of simulation studies showed that, in the most severe case of multilevel data, the standard error estimate from the standard single-level IRT models was about half of the minimal asymptotic standard error, and the type I error rate of the Lord’s chi-square test was inflated up to .35. The results of this study suggest that standard single-level IRT models may seriously mislead our conclusions in the presence of multilevel data, and therefore multilevel IRT models need to be considered as alternatives.

Fundamental to most classical data collection sampling theory development is the random drawings assumption requiring that each targeted population member has a known sample selection (i.e., inclusion) probability. Frequently, however, unrestricted random sampling of spatially autocorrelated data is impractical and/or inefficient. Instead, randomly choosing a population subset accounts for its exhibited spatial pattern by utilizing a grid, which often provides improved parameter estimates, such as the geographic landscape mean, at least via its precision. Unfortunately, spatial autocorrelation latent in these data can produce a questionable mean and/or standard error estimate because each sampled population member contains information about its nearby members, a data feature explicitly acknowledged in model-based inference, but ignored in design-based inference. This autocorrelation effect prompted the development of formulae for calculating an effective sample size (i.e., the equivalent number of sample selections from a geographically randomly distributed population that would yield the same sampling error) estimate. Some researchers recently challenged this and other aspects of spatial statistics as being incorrect/invalid/misleading. This paper seeks to address this category of misconceptions, demonstrating that the effective geographic sample size is a valid and useful concept regardless of the inferential basis invoked. Its spatial statistical methodology builds upon the preceding ingredients.

This paper shows how to bootstrap hypothesis tests in the context of the Parks’s (1967) Feasible Generalized Least Squares estimator. It then demonstrates that the bootstrap outperforms FGLS(Parks)’s top competitor. The FGLS(Parks) estimator has been a workhorse for the analysis of panel data and seemingly unrelated regression equation systems because it allows the incorporation of cross-sectional correlation together with heteroskedasticity and serial correlation. Unfortunately, the associated, asymptotic standard error estimates are biased downward, often severely. To address this problem, Beck and Katz (1995) developed an approach that uses the Prais-Winsten estimator together with “panel corrected standard errors” (PCSE). While PCSE produces standard error estimates that are less biased than FGLS(Parks), it forces the user to sacrifice efficiency for accuracy in hypothesis testing. The PCSE approach has been, and continues to be, widely used. This paper develops an alternative: a nonparametric bootstrapping procedure to be used in conjunction with the FGLS(Parks) estimator. We demonstrate its effectiveness using an experimental approach that creates artificial panel datasets modelled after actual panel datasets. Our approach provides a superior alternative to existing estimation options by allowing researchers to retain the efficiency of the FGLS(Parks) estimator while producing more accurate hypothesis test results than the PCSE.

We introduce Clover, a new library for efficient computation using low-precision data, providing mathematical routines required by fundamental methods in optimization and sparse recovery. Our library faithfully implements variants of stochastic quantization that guarantee convergence at low precision, and supports data formats from 4-bit quantized to 32-bit IEEE-754 on current Intel processors. In particular, we show that 4-bit can be implemented efficiently using Intel AVX despite the lack of native support for this data format. Experimental results with dot product, matrix-vector multiplication (MVM), gradient descent (GD), and iterative hard thresholding (IHT) demonstrate that the attainable speedups are in many cases close to linear with respect to the reduction of precision due to reduced data movement. Finally, for GD and IHT, we show examples of absolute speedup achieved by 4-bit versus 32-bit, by iterating until a given target error is achieved.