Asymptotic Sample Research Articles

Monte Carlo approaches to frequentist multiplicity-adjusted benefiting subgroup identification.

One common goal of subgroup analyses is to determine the subgroup of the population for which a given treatment is effective. Like most problems in subgroup analyses, this benefiting subgroup identification requires careful attention to multiple testing considerations, especially Type I error inflation. To partially address these concerns, the credible subgroups approach provides a pair of bounding subgroups for the benefiting subgroup, constructed so that with high posterior probability one is contained by the benefiting subgroup while the other contains the benefiting subgroup. To date, this approach has been presented within the Bayesian paradigm only, and requires sampling from the posterior of a Bayesian model. Additionally, in many cases, such as regulatory submission, guarantees of frequentist operating characteristics are helpful or necessary. We present Monte Carlo approaches to constructing confidence subgroups, frequentist analogues to credible subgroups that replace the posterior distribution with an estimate of the joint distribution of personalized treatment effect estimates, and yield frequentist interpretations and coverage guarantees. The estimated joint distribution is produced using either draws from asymptotic sampling distributions of estimated model parameters, or bootstrap resampling schemes. The approach is applied to a publicly available dataset from randomized trials of Alzheimer's disease treatments.

A Simple Mantel–Haenszel Type Test for Noninferiority

Randomized clinical trials designed to establish noninferiority of an experimental therapy as compared to a standard (active-control) therapy as measured by binomial proportions are being widely used. Randomization is often stratified by prognostic factors. We propose a Mantel–Haenszel (MH) type test to demonstrate noninferiority in terms of the differences in success proportions and when the noninferiority margin is not necessarily uniform in all strata. It is assumed that the efficacy of the active control is documented in at least one earlier placebo-controlled trial. Derivation of the new test originates from Wittes and Wallenstein, who have discussed asymptotic unconditional power and sample size for the MH test. We provide an easy to calculate formula for sample size. We call the new test as W-square test. It may be an alternative to the one proposed by Yanagawa et al. Yanagawa’s Neriproct suppository data are analyzed using the W-square test. A SAS code that calculates the p-value is provided.

A Comparative Study of the Bias Correction Methods for Differential Item Functioning Analysis in Logistic Regression with Rare Events Data.

The logistic regression (LR) model for assessing differential item functioning (DIF) is highly dependent on the asymptotic sampling distributions. However, for rare events data, the maximum likelihood estimation method may be biased and the asymptotic distributions may not be reliable. In this study, the performance of the regular maximum likelihood (ML) estimation is compared with two bias correction methods including weighted logistic regression (WLR) and Firth's penalized maximum likelihood (PML) to assess DIF for imbalanced or rare events data. The power and type I error rate of the LR model for detecting DIF were investigated under different combinations of sample size, moderate and severe magnitudes of uniform DIF (DIF = 0.4 and 0.8), sample size ratio, number of items, and the imbalanced degree (τ). Indeed, as compared with WLR and for severe imbalanced degree (τ = 0.069), there were reductions of approximately 30% and 24% under DIF = 0.4 and 27% and 23% under DIF = 0.8 in the power of the PML and ML, respectively. The present study revealed that the WLR outperforms both the ML and PML estimation methods when logistic regression is used to evaluate DIF for imbalanced or rare events data.

Rerandomization in $2^{K}$ factorial experiments

With many pretreatment covariates and treatment factors, the classical factorial experiment often fails to balance covariates across multiple factorial effects simultaneously. Therefore, it is intuitive to restrict the randomization of the treatment factors to satisfy certain covariate balance criteria, possibly conforming to the tiers of factorial effects and covariates based on their relative importances. This is rerandomization in factorial experiments. We study the asymptotic properties of this experimental design under the randomization inference framework without imposing any distributional or modeling assumptions of the covariates and outcomes. We derive the joint asymptotic sampling distribution of the usual estimators of the factorial effects, and show that it is symmetric, unimodal and more “concentrated” at the true factorial effects under rerandomization than under the classical factorial experiment. We quantify this advantage of rerandomization using the notions of “central convex unimodality” and “peakedness” of the joint asymptotic sampling distribution. We also construct conservative large-sample confidence sets for the factorial effects.

Probability Based Survey of Braking System: A Pareto-Optimal Approach

Active brake control systems are able to use more accurate control designs on the real-time knowledge of wheel slip such as tire braking forces and external momentums. A multi-objective Pareto type optimization approach is used to evaluate the inconsistent points of the optimization design namely those of minimizing the directional deviation while achieving maximum braking forces on wheels. Moreover, the optimal target slip values defined by the braking purposes like as shorter stopping distance and stability increment by keeping the vehicle in the straight line. This gets by the control of the longitudinal and lateral slip dynamics of each wheel concerning to the road conditions. A controller is optimally brought up by manipulating optimal control law with weights of two control inputs with mathematical and probabilistic characterization. The first-passage probability of critical response levels is used to directly control the vehicle directional stability. An impressive simulation technique based on Monte Carlo method named asymptotic sampling is applied to define the required first-passage probabilities and the Latin Hypercube sampling is used to design of experiments and to cover the design space. Afterwards, a Pareto-type optimization approach is applied to a trade-off between the inconsistent points of the optimization design. The simulation is conducted by the validated vehicle model and the results are indicated that the probability-based control design is represented as a preferable braking performance in comparison with the other systems for braking in sever conditions.

Approximate Nonlinear Discrete-Time Models Based on B-Spline Functions

We consider the discretization of continuous-time nonlinear systems described by normal forms. In particular, we consider the case when the input to the system is generated by a B-spline hold device to obtain an approximate discrete-time model. It is shown that the corresponding sampled-data model and its accuracy (measured in terms of the local truncation error) depend on the smoothness of the input and on the applied integration strategy, namely, the truncated Taylor series expansion. Moreover, the sampling zero dynamics of the discrete-time model are asymptotically characterized as the sampling period goes to zero, and it is shown that these zero dynamics converge to the asymptotic sampling zeros of the linear case.

Reliable Recovery of Hierarchically Sparse Signals for Gaussian and Kronecker Product Measurements

We propose and analyze a solution to the problem of recovering a block sparse signal with sparse blocks from linear measurements. Such problems naturally emerge inter alia in the context of mobile communication, in order to meet the scalability and low complexity requirements of massive antenna systems and massive machine-type communication. We introduce a new variant of the Hard Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a proof of convergence and a recovery guarantee for noisy Gaussian measurements that exhibit an improved asymptotic scaling in terms of the sampling complexity in comparison with the usual HTP algorithm. Furthermore, hierarchically sparse signals and Kronecker product structured measurements naturally arise together in a variety of applications. We establish the efficient reconstruction of hierarchically sparse signals from Kronecker product measurements using the HiHTP algorithm. Additionally, we provide analytical results that connect our recovery conditions to generalized coherence measures. Again, our recovery results exhibit substantial improvement in the asymptotic sampling complexity scaling over the standard setting. Finally, we validate in numerical experiments that for hierarchically sparse signals, HiHTP performs significantly better compared to HTP.

Towards a Simple Sampled-Data Control Law for Stably Invertible Linear Systems

A new high gain control law is proposed for stably invertible linear systems. The continuous-time case is first studied to set ideas. The extension to the sampled-data case is made difficult by the presence of sampling zeros. For continuous-time systems having relative degree greater than or equal to two, these zeros converge, as the sampling rate approaches zero, to either marginally stable or unstable locations. A methodology which specifically addresses the sampling zero issue is developed. The methodology uses an approximate model which includes, when appropriate, the asymptotic sampling zeros. The core idea is supported by simulation studies. Also, a preliminary theoretical analysis is provided for degree two, showing that the design based on the approximate model stabilizes the true system for the continuous and sampled-data cases.

Optimal design of overall yield-based variable repetitive sampling plans for processes with multiple characteristics

In this study, we developed two repetitive types of sampling plans for processes with multiple quality characteristics based on the overall yield index SpkT. These plans can be implemented for mutually independent and normally distributed characteristics. The plans are optimally designed based on the asymptotic sampling distribution of SpkT using an efficient nonlinear optimization algorithm. During the solution of optimization problems, the average sample number required for inspection and the contract requirements are treated as an objective function and constraints, respectively. The optimal parameters were determined for use in industrial environments with various combinations of requirements in tables. A simulation study was also conducted to show that the tabulated parameters based on the results obtained by large sample theory can guarantee the specified risks for finite sample sizes. Moreover, the limitations of the proposed plans with respect to the sample size were analyzed based on extensive simulations. Numerical calculations and graphical illustrations are presented to demonstrate the properties of the proposed plans. In addition, the advantages of the schemes are discussed compared with existing plans. Finally, the efficient plan is applied to two real industrial problems.

On the optimal strategy of stochastic-based reliability assessment of railway bridges for high-speed trains

The scope of this paper is to evaluate the performance and computational efficiency of various stochastic simulation methods for a stochastic based reliability assessment of railway bridges subjected to high-speed trains. Depending on the degree of sophistication, application of crude Monte Carlo simulation to a realistic mechanical model of the uncertain bridge-train interacting dynamical system can be prohibitively expensive. Thus, three alternative stochastic methods, i.e. line sampling, subset simulation, and asymptotic sampling, are tested on two example problems. These examples represent two classes of bridges with different dynamic response characteristics. While in the one class of bridges distinctive resonance peaks govern the dynamic peak response, the random response amplification of the second group of bridges is primarily induced by track irregularities. The studies are conducted on a simplified mechanical model, composed of a plain beam representing the bridge and a planar mass-spring-damper system representing the train. This modeling strategy captures the fundamental characteristics of dynamic bridge-train interaction, and thus, facilitates the desired assessment of the stochastic methods with reasonable computational effort. It is shown that both line sampling and subset simulation reduce significantly the computational expense for the first class of bridges, while maintaining the accuracy of the predicted bridge reliability. To ensure accuracy and efficiency, these methods need to be modified when applied to systems where track irregularities dominate the random response. For the latter class of bridges, subset simulation proved to be a suitable method for assessing the reliability of this dynamic interacting system when appropriately modified.

Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements

We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $\ell^1$ minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the $L^2$-norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity $s$ and at most logarithmic in the dimension $d$, thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.

Optimal reliability design of pure friction isolators using asymptotic sampling

Probabilistic methods are widely being used in structural design due to the vital effects on the structural safety prediction. In the present study, reliability-based analysis of a shear frame supported on pure friction base isolators equipped with a restoring device is performed and changing of friction coefficient of the base isolator, the mass of the base raft ratio and the damping ratio of the restoring device are investigated. The earthquake herein is treated as a non-stationary random process. The probability of failure is computed using first-passage probability which is determined utilizing a new sampling technique called asymptotic sampling. This study introduces the safe range for each variable on the design space based on reliability indices. The assessment is done for different circumstances in which there are one determined variable and two uncertain. In each circumstance, a Pareto-type optimization is done in order to achieve the optimum design variables. Finally, a total optimization is done for all three design variables to determine the optimum values. Moreover, the reliability analysis of the isolated frame without a restoring device is conducted to show the substantial impact of adding the restoring device to the isolator by achieving considerably higher safety indices.

On the relationship between spline interpolation, sampling zeros and numerical integration in sampled-data models

Real systems are usually modeled in continuous-time by differential equations. In practice, however, we have to deal with them using discretized models and sampled data. These models, however, have more zeros than the continuous-time model. In this paper we show that there is a specific relation between these sampling zeros and the smoothness of the continuous-time input to the plant generated by a hold device using spline interpolation. On the other hand, in numerical analysis, ordinary differential equations are solved using numerical integration methods such as Runge–Kutta. In this paper we also characterize the asymptotic sampling zeros of approximate sampled-data models when using a Runge–Kutta method of a given order under uniform sampling and the input signal is obtained by spline interpolation. These results show the strong connections between the presence and characterization of sampling zeros, spline interpolation and numerical integration techniques. Moreover, the results presented in the paper provide additional insights about the impact of the details of the sampling process on the resulting discrete-time model.

Realized Genome Sharing in Heritability Estimation Using Random Effects Models.

For heritability estimation using a two-component random effects model, we provided formulas for the limiting distribution of the maximum likelihood estimate. These formulas are applicable even when the wrong measure of kinship is used to capture additive genetic correlation. When the model is correctly specified, we showed that the asymptotic sampling variance of heritability estimate is determined by both the study design and the extent of variation in the kinship measure that constitutes the additive genetic correlation matrix. When the correlation matrix is mis-specified, the extent of asymptotic bias depends additionally on how the fitted correlation matrix differs from the truth. In particular, we showed in a simulation study that estimating heritability using a population-based design and the classic GRM as the fitted correlation matrix can potentially contribute to the ”missing heritability” problem.

An efficient method for reliability estimation using the combination of asymptotic sampling and weighted simulation

In this paper, an efficient reliability method is proposed. The Asymptotic Sampling (AS) and Weighted Simulation (WS) are two main basic tools of the presented method. In AS, the standard deviation of the distributions are amplified at several levels to find an adequate number of failed samples, then by using a simple regression technique, the reliability index is determined. The WS is another method which uses the uniform distribution for sampling, where the information about the distributions of the variables is taken into account through the weight indexes. The WS provides interesting flexibility where a sample generated for a specific standard deviation can be used as a sample for another standard deviation without having to reevaluate the limit state function. In AS the deviations of variables are scaled in each step, where one can use the flexibility of the WS to decrease the required calls of limit state function. Using this technique results in a new efficient method so-called Asymptotic Weighted Simulation (AWS). In addition, using the strengths of both AS and WS can be considered another superiority of the hybrid version. Performance of the presented method is investigated by solving several mathematical and engineering examples.

A model-based concordance-type index for evaluating the added predictive ability of novel risk factors and markers in the logistic regression models

ABSTRACTThe Concordance statistic (C-statistic) is commonly used to assess the predictive performance (discriminatory ability) of logistic regression model. Although there are several approaches for the C-statistic, their performance in quantifying the subsequent improvement in predictive accuracy due to inclusion of novel risk factors or biomarkers in the model has been extremely criticized in literature. This paper proposed a model-based concordance-type index, CK, for use with logistic regression model. The CK and its asymptotic sampling distribution is derived following Gonen and Heller's approach for Cox PH model for survival data but taking necessary modifications for use with binary data. Unlike the existing C-statistics for logistic model, it quantifies the concordance probability by taking the difference in the predicted risks between two subjects in a pair rather than ranking them and hence is able to quantify the equivalent incremental value from the new risk factor or marker. The simulation study revealed that the CK performed well when the model parameters are correctly estimated for large sample and showed greater improvement in quantifying the additional predictive value from the new risk factor or marker than the existing C-statistics. Furthermore, the illustration using three datasets supports the findings from simulation study.

Asymptotic Inference for Optimal Rerandomization Designs

AbstractRecently a computational-based experimental design strategy called rerandomization has been proposed as an alternative or complement to traditional blocked designs. The idea of rerandomization is to remove, from consideration, those allocations with large imbalances in observed covariates according to a balance criterion, and then randomize within the set of acceptable allocations. Based on the Mahalanobis distance criterion for balancing the covariates, we show that asymptotic inference to the population, from which the units in the sample are randomly drawn, is possible using only the set of best, or ‘optimal’, allocations. Finally, we show that for the optimal and near optimal designs, the quite complex asymptotic sampling distribution derived by Li et al. (2018), is well approximated by a normal distribution.

An $${{\varvec{L}}}^{2}$$L2-norm-based test for equality of several covariance functions: a further study

For the multi-sample equal covariance function (ECF) testing problem, Zhang (Analysis of variance for functional data, CRC Press, Boca Raton, 2013) proposed an $$L^{2}$$-norm-based test. However, its asymptotic power and finite sample performance have not been studied. In this paper, its asymptotic power is investigated under some mild conditions. It is shown that the $$L^2$$-norm-based test is root-n consistent. In addition, intensive simulation studies are conducted to evaluate its finite sample performance against several existing competitors. In particular, they demonstrate that in terms of size control and power, the $$L^{2}$$-norm-based test outperforms the dimension-reduction-based tests proposed by Panaretos et al. (J Am Stat Assoc 105(490):670–682, 2010) and Fremdt et al. (Scand J Stat 40(1):138–152, 2013), respectively, when functional data are less correlated or when the covariance function differences of functional data contain mainly high-frequency signals. Two real data applications are presented to illustrate the $$L^2$$-norm-based test.

The Local Time Method for Targeting and Selection

We propose a framework for targeting and selection (T&S), a new problem class in simulation optimization where the objective is to select a simulation alternative whose mean performance matches a prespecified target as closely as possible. T&S resembles the more well-known problem of ranking and selection but presents unexpected challenges: for example, a one-step look-ahead method may produce statistically inconsistent estimates of the values, even under very standard normality assumptions. We create a new and fundamentally different approach, based on a Brownian local time model, that exhibits characteristics of two widely studied methodologies, namely expected value of information and optimal computing budget allocation. We characterize the asymptotic sampling rates of this method and relate them to the convergence rates of metrics of interest. The local time method outperforms benchmarks in experiments, including problems where the modeling assumptions of T&S are violated. The online appendix is available at https://doi.org/10.1287/opre.2018.1731 .

Accelerated orthogonal least-squares for large-scale sparse reconstruction

We study the problem of inferring a sparse vector from random linear combinations of its components. We propose the Accelerated Orthogonal Least-Squares (AOLS) algorithm that improves performance of the well-known Orthogonal Least-Squares (OLS) algorithm while requiring significantly lower computational costs. While OLS greedily selects columns of the coefficient matrix that correspond to non-zero components of the sparse vector, AOLS employs a novel computationally efficient procedure that speeds up the search by anticipating future selections via choosing L columns in each step, where L is an adjustable hyper-parameter. We analyze the performance of AOLS and establish lower bounds on the probability of exact recovery for both noiseless and noisy random linear measurements. In the noiseless scenario, it is shown that when the coefficients are samples from a Gaussian distribution, AOLS with high probability recovers a k-sparse m-dimensional sparse vector using O(klog⁡mk+L−1) measurements. Similar result is established for the bounded-noise scenario where an additional condition on the smallest nonzero element of the unknown vector is required. The asymptotic sampling complexity of AOLS is lower than the asymptotic sampling complexity of the existing sparse reconstruction algorithms. In simulations, AOLS is compared to state-of-the-art sparse recovery techniques and shown to provide better performance in terms of accuracy, running time, or both. Finally, we consider an application of AOLS to clustering high-dimensional data lying on the union of low-dimensional subspaces and demonstrate its superiority over existing methods.