Expressions For Asymptotic Variance Research Articles

Central limit theorem for the number of records in Fα-scheme

Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.

Note on the delta method for finite population inference with applications to causal inference

This work derives a finite population delta method. The delta method creates more general inference results when coupled with central limit theorem results for the finite population. This opens up a range of new estimators for which we can find finite population asymptotic properties. We focus on the use of this method to derive asymptotic distributional results and variance expressions for causal estimators. We illustrate the use of the method by obtaining a finite population asymptotic distribution for a causal ratio estimator.

Inference for conditional value-at-risk of a predictive regression

Conditional value-at-risk is a popular risk measure in risk management. We study the inference problem of conditional value-at-risk under a linear predictive regression model. We derive the asymptotic distribution of the least squares estimator for the conditional value-at-risk. Our results relax the model assumptions made in (Oper. Res. 60 (2012) 739–756) and correct their mistake in the asymptotic variance expression. We show that the asymptotic variance depends on the quantile density function of the unobserved error and whether the model has a predictor with infinite variance, which makes it challenging to actually quantify the uncertainty of the conditional risk measure. To make the inference feasible, we then propose a smooth empirical likelihood based method for constructing a confidence interval for the conditional value-at-risk based on either independent errors or GARCH errors. Our approach not only bypasses the challenge of directly estimating the asymptotic variance but also does not need to know whether there exists an infinite variance predictor in the predictive model. Furthermore, we apply the same idea to the quantile regression method, which allows infinite variance predictors and generalizes the parameter estimation in (Econometric Theory 22 (2006) 173–205) to conditional value-at-risk in the Supplementary Material. We demonstrate the finite sample performance of the derived confidence intervals through numerical studies before applying them to real data.

Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo

AbstractWe consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo‐marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelization and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time‐discretized diffusions. Our experimental results are promising and show that the IS‐type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelization.

Extreme-cum-median ranked set sampling

A mixture of Extreme Ranked Set Sampling (ERSS) and Median Ranked Set Sampling (MRSS) is introduced to obtain a more representative sample using three out of five number summary statistics [i.e., Minimum, Median and Maximum]. The proposed sampling scheme provides unbiased estimator of mean for symmetric population and gives moderate efficiency for both symmetric and asymmetric populations under perfect as well as imperfect rankings. Expressions for bias and asymptotic variance are presented. A simulation study is also conducted to observe the performance of the proposed estimator. Application of proposed sampling scheme is illustrated through a real life example.

Asymptotics of bivariate penalised splines

ABSTRACTWe study the class of bivariate penalised splines that use tensor product splines and a smoothness penalty. Similar to Claeskens, G., Krivobokova, T., and Opsomer, J.D. [(2009), ‘Asymptotic Properties of Penalised Spline Estimators’, Biometrika, 96(3), 529–544] for the univariate penalised splines, we show that, depending on the number of knots and penalty, the global asymptotic convergence rate of bivariate penalised splines is either similar to that of tensor product regression splines or to that of thin plate splines. In each scenario, the bivariate penalised splines are found rate optimal in the sense of Stone, C.J. [(12, 1982), ‘Optimal Global Rates of Convergence for Nonparametric Regression’, The Annals of Statistics, 10(4), 1040–1053] for a corresponding class of functions with appropriate smoothness. For the scenario where a small number of knots is used, we obtain expressions for the local asymptotic bias and variance and derive the point-wise and uniform asymptotic normality. The theoretical results are applicable to tensor product regression splines.

Analysis of over-sampling based identification

In this work, firstly, an asymptotic variance expression is derived for the transfer function estimates in the over-sampling based identification scheme. Then the result is used to analyze the over-sampling scheme. The asymptotic variance expression says that the joint covariance matrix of the transfer function estimates is proportional to the “generalized” noise-to-signal ratio. It is an application of the result in Ljung (1985); and it covers both open-loop and closed-loop tests. Using the result, one can show that informativity of the closed-loop tests without external excitation can be attained by using the over-sampling scheme, but only when the output disturbance has high frequency content beyond the bandwidth of the plant. Moreover, when the output disturbance has high frequency content and when test signal is used, the result points out that the over-sampling scheme can increase model accuracy, and, it will outperform conventional method with anti-aliasing filtering. Numerical examples are used to illustrate the asymptotic variance expression and the analysis about over-sampling scheme of identification.

Identification of errors‐in‐variables systems: An asymptotic approach

SummaryThis work studies the identification of errors‐in‐variables (EIV) systems. An asymptotic method (ASYM) is developed for the EIV system. Firstly, an auto regressive with exogeneous (ARX) model estimation method is proposed, which is consistent for EIV systems. Then the asymptotic variance expression of the estimated high‐order ARX model is derived, which forms the basis of the ASYM method. In parameter estimation, the ASYM starts with a high‐order ARX model estimation followed by a frequency domain weighted model reduction. The obtained model is consistent, and its efficiency needs to be investigated. Besides parameter estimation, a criterion for model order selection is proposed, which is based on frequency domain considerations, and the frequency domain error bound is established that can be used for model validation. Simulations and comparisons with other methods are used to illustrate the performance of the method.

Estimating and testing the significance of correlation coefficient obtained from truncated bivariate normal distribution

In this paper we have given a methodology to estimate the correlation coefficient be- tween Combined Admission Test (CAT) score and Cumulative Grade Point Average (CGPA) of the students by truncating the CAT score at different cut-off points. An expression for asymptotic variance of estimated value of correlation coefficient has been derived for testing the hypothesis H0: ρ = 0 against the alternative H1: ρ > 0.

Tail index estimation with a fixed tuning parameter fraction

Semi-parametric tail index estimators, such as the Hill, Harmonic Moment, Pickands, and Dekkers, Einmahl and de Haan estimators, rely upon a tuning parameter that typically grows with sample size n. Proper selection of this tuning parameter k=k(n) is crucial for good practical performance, although asymptotic theory dictates that 1/k+k/n→0 as n→∞. A similar issue presents itself in the bandwidth literature in spectral density estimation and recent research shows that the use of asymptotic distributions when the bandwidth is a fixed ratio of sample size yields improved approximations to finite-sample distributions. Here, we study some semi-parametric tail index estimators utilizing the same perspective where k=bn and b∈(0,1) is a fixed constant. This allows us to derive asymptotic bias and variance expressions which are compatible with the small-b conventional theory. Our simulations corroborate that the finite-sample bias and variance are well described by the asymptotic bias and variance quantities arising from our fixed bandwidth ratio theory.

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

Asymptotic Variance Expression in Output Over-sampling Based Closed-loop Identification

In this work, asymptotic variance expression is derived for the transfer function estimates in output over-sampling based closed-loop identification. The result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance, respectively, is proportional to the "generalized" noise-to-signal ratio, where the spectrum is in the form of cyclo-stationary spectrum due to the cyclo-stationary properties of the observed signals. The factor of proportionality is the ratio of model order to number of data. Based on the result, it can be shown that only when the output disturbance contains some energy beyond the bandwidth of the plant, the closed-loop identifiability of the over-sampling scheme without external excitation is guaranteed. Numerical examples are used to illustrate the asymptotic variance expression and closed-loop identifiability. Finally, when test signal is used, both in open and closed-loop, the result points out that usingthe over-sampling scheme can increase model accuracy in real world applications.

Virtual matrix pencil method for 2-D DOA estimation with a two-nested-shape-array

A new two-nested-shape array geometry with three different inter-sensor spacings is designed to achieve enhanced degrees of freedom and doubled aperture length. The virtual matrix pencil method (VMPM) is proposed for this new array to estimate the two-dimensional direction-of-arrival (2-D DOA) of the signal sources, where the three different inter-sensor spacings are used to construct a virtual sparse two-parallel-shape array with much larger number sensors and achieve more accurate DOA estimations. Compared with the improved propagator method, the VMPM has better angle estimation accuracy, and is capable of resolving $$\hbox {O}(P^{2}/32)$$O(P2/32) sources with $$P$$P sensors. The statistical analysis of these two methods are studied, and the asymptotic variance expressions of their estimation errors are derived. Moreover, these asymptotic variance expressions are simplified for the case of one signal, and the quantitative comparisons are performed to demonstrate the performance advantage of the VMPM. The simulation results verify the validity of the VMPM.

Profile likelihood‐based confidence interval of the intraclass correlation for binary outcome data sampled from clusters

The intraclass correlation in binary outcome data sampled from clusters is an important and versatile measure in many biological and biomedical investigations. Properties of the different estimators of the intraclass correlation based on the parametric, semi-parametric, and nonparametric approaches have been studied extensively, mainly in terms of bias and efficiency [see, for example, Ridout et al., Biometrics 1999, 55:137-148; Paul et al., Journal of Statistical Computation and Simulation 2003, 73:507-523; and Lee, Statistical Modelling 2004, 4: 113-126], but little attention has been paid to extending these results to the problem of the confidence intervals. In this article, we generalize the results of the four point estimators by constructing asymptotic confidence intervals obtaining closed-form asymptotic and sandwich variance expressions of those four point estimators. It appears from simulation results that the asymptotic confidence intervals based on these four estimators have serious under-coverage. To remedy this, we introduce the Fisher's z-transformation approach on the intraclass correlation coefficient, the profile likelihood approach based on the beta-binomial model, and the hybrid profile variance approach based on the quadratic estimating equation for constructing the confidence intervals of the intraclass correlation for binary outcome data. As assessed by Monte Carlo simulations, these confidence interval approaches show significant improvement in the coverage probabilities. Moreover, the profile likelihood approach performs quite well by providing coverage levels close to nominal over a wide range of parameter combinations. We provide applications to biological data to illustrate the methods.

The asymptotic variance of the PBSIDopt algorithm

It is well-known that system identification is a valuable technique to obtain compact models for controller design and prediction. Subspace identification methods are of interest since they solely rely on tools from linear algebra and their ability to directly work with data generated by systems with multiple inputs and outputs. With respect to other methods (e.g. Prediction Error (PE)), subspace methods do not have straightforward asymptotic variance expressions. However, recently some papers appeared with asymptotic variance expressions for a certain class of identification algorithms that merge ideas from PE and subspace methods. In this paper we derive the asymptotic variance expression for the PBSIDopt algorithm and derive manageable expressions under some reasonable assumptions. We conclude the paper with two simulation examples where we show the strength of the proposed method.

Asymptotics for spectral regularization estimators in statistical inverse problems

While optimal rates of convergence in L 2 for spectral regularization estimators in statistical inverse problems have been much studied, the pointwise asymptotics for these estimators have received very little consideration. Here, we briefly discuss asymptotic expressions for bias and variance for some such estimators, mainly in deconvolution-type problems, and also show their asymptotic normality. The main part of the paper consists of a simulation study in which we investigate in detail the pointwise finite sample properties, both for deconvolution and the backward heat equation as well as for a regression model involving the Radon transform. In particular we explore the practical use of undersmoothing in order to achieve the nominal coverage probabilities of the confidence intervals.

LOCAL INSTRUMENTAL VARIABLE METHOD FOR THE GENERALIZED ADDITIVE-INTERACTIVE NONLINEAR VOLATILITY MODEL ESTIMATION

In this article we consider a new separable nonparametric volatility model that includes second-order interaction terms in both mean and conditional variance functions. This is a very flexible nonparametric ARCH model that can potentially explain the behavior of the wide variety of financial assets. The model is estimated using the generalized version of the local instrumental variable estimation method first introduced in Kim and Linton (2004, Econometric Theory 20, 1094–1139). This method is computationally more effective than most other nonparametric estimation methods that can potentially be used to estimate components of such a model. Asymptotic behavior of the resulting estimators is investigated and their asymptotic normality is established. Explicit expressions for asymptotic means and variances of these estimators are also obtained.

Conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller

It is well known that if we intend to use a minimum variance control strategy, which is designed based on a model obtained from an identification experiment, the best experiment which can be performed on the system to determine such a model (subject to output power constraints, or for some specific model structures) is to use the true minimum variance controller. This result has been derived under several circumstances, first using asymptotic (in model order) variance expressions but also more recently for ARMAX models of finite order. In this paper we re-approach this problem using a recently developed expression for the variance of parametric frequency function estimates. This allows a geometric analysis of the problem and the generalization of the aforementioned finite model order ARMAX results to general linear model structures.

Estimation Over Fading Channels With Limited Feedback Using Distributed Sensing

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We consider a wireless sensor network for distributed estimation over fading channels. The sensors transmit their observations over a multiple access fading channel to a fusion center (FC), where a source parameter is estimated. The sensor transmissions add incoherently over a multiple access channel, which motivates the need for channel knowledge at the sensors to improve performance. We consider the effects of different fading channel models on the performance of the system, and characterize the effect of different amounts of channel information at the sensors. We calculate the variance of the estimate for cases when both perfect, and differing amounts of partial channel information are available at the sensors. Asymptotic variance expressions for large number of sensors are derived for different channel statistics and feedback scenarios. We show that the degradation in variance due to using only channel phase information is at most a factor of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$4/\pi$</tex></formula></emphasis> over Rayleigh fading channels. We consider continuous feedback error and evaluate the degradation in performance due to differing degrees of error. The loss in performance due to feedback quantization, and effect of error in feedback are also quantified. We also consider speed of convergence, and compare the rate of convergence under different conditions. Further, we characterize the effect of correlated channels between sensors and the FC, and provide the different values for the speed of convergence for this case. Simulation results are provided to show that only a few tens of sensors are required for the asymptotic results to hold. Numerical results corroborate our analytical results. </para>