Smaller Asymptotic Variance Research Articles

Accelerating reversible Markov chains

Reversibility is usually applied in most popular Markov chain Monte Carlo algorithms, such as the Metropolis–Hastings algorithm and the Gibbs sampler. However, several researchers have shown that non-reversible Markov chains are better than reversible ones. In this paper, we present a method for accelerating a reversible Markov chain. For any reversible Markov chain with a cycle on the corresponding graph, we construct a non-reversible Markov chain by adding some antisymmetric perturbations to the original chain. We prove that this non-reversible Markov chain is uniformly better than the original one in the sense of having a smaller asymptotic variance. Furthermore, we propose a conjecture that no uniformly better chain exists for the acyclic case.

Formula omitted]-statistic with side information

In this paper we study U-statistics with side information incorporated using the method of empirical likelihood. Some basic properties of the proposed statistics are investigated. We find that by implementing the side information properly, the proposed U-statistics can have smaller asymptotic variance than the existing U-statistics in the literature. The proposed U-statistics can achieve asymptotic efficiency in a formal sense and their weak limits admit a convolution result. We also find that the corresponding U-likelihood ratio procedure, as well as the U-empirical likelihood based confidence interval construction, do not benefit from incorporating side information, a result that is consistent with the result under the standard empirical likelihood ratio procedure. The impact of incorrect side information implementation in the proposed U-statistics is also explored. Simulation studies are conducted to assess the finite sample performance of the proposed method. The numerical results show that with side information implemented, the deduction of asymptotic variance can be substantial in some cases, and the coverage probability of the confidence interval using the U-empirical likelihood ratio based method outperforms that of the normal approximation based method, in particular in the cases when the underlying distribution is skewed.

Quantile Estimation in the Presence of Auxiliary Information under Negatively Associated Samples

In this article, we apply the empirical likelihood technique to propose a new class of quantile estimators in the presence of some auxiliary information under negatively associated samples. It is shown that the proposed quantile estimators are asymptotically normally distributed with smaller asymptotic variances than those of the usual quantile estimators. It is also shown that blocking technique is an useful tool in estimating asymptotic variance under negatively associated samples, which makes it possible to construct normal approximation based confidence intervals for quantiles.

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n − 1 / 2 , but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence log n / n .

Statistical inference on seemingly unrelated non-parametric regression models with serially correlated errors

This article is concerned with the inference on seemingly unrelated non-parametric regression models with serially correlated errors. Based on an initial estimator of the mean functions, we first construct an efficient estimator of the autoregressive parameters of the errors. Then, by applying an undersmoothing technique, and taking both of the contemporaneous correlation among equations and serial correlation into account, we propose an efficient two-stage local polynomial estimation for the unknown mean functions. It is shown that the resulting estimator has the same bias as those estimators which neglect the contemporaneous and/or serial correlation and smaller asymptotic variance. The asymptotic normality of the resulting estimator is also established. In addition, we develop a wild block bootstrap test for the goodness-of-fit of models. The finite sample performance of our procedures is investigated in a simulation study whose results come out very supportive, and a real data set is analysed to illustrate the usefulness of our procedures.

Superefficient Estimation of the Marginals by Exploiting Knowledge on the Copula

We consider the problem of estimating the marginals in case there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have rate of convergence n-1/2, but a smaller asymptotic variance. In this paper we show that smoothness assumptions on the copula are necessary: we construct both a (non-smooth) copula and, exploiting the information our copula provides, estimators of the marginals with rate of convergence log n/n.

Asymptotic representations for importance-sampling estimators of value-at-risk and conditional value-at-risk

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are important risk measures. They are often estimated by using importance-sampling (IS) techniques. In this paper, we derive the asymptotic representations for IS estimators of VaR and CVaR. Based on these representations, we are able to prove the consistency and asymptotic normality of the estimators and to provide simple conditions under which the IS estimators have smaller asymptotic variances than the ordinary Monte Carlo estimators.

Estimation of average treatment effects based on parametric propensity score model

In this paper, the estimation of average treatment effects is examined given that the propensity score is of a parametric form with some unknown parameters. Under the assumption that the treatment is ignorable given some observed characteristics, the MLEs for those unknown parameters in the probability assignment model have been achieved firstly and then three estimators have been defined by the inverse probability weighted, regression and imputation methods, respectively. All the estimators are shown asymptotically normal and more importantly, the substantial efficiency gains of the first two estimates have been obtained theoretically compared with the existing estimators in Hahn (1998) and Hirano et al. (2003), i.e., the inverse weighted probability estimator and the regression estimator have smaller asymptotic variances. Our simulation analysis verifies the theoretical results in terms of biases, SEs and MSEs.

Efficient estimation for error component seemingly unrelated nonparametric regression models

Multivariate panel data provides a unique opportunity in studying the joint evolution of multiple response variables over time. In this paper, we propose an error component seemingly unrelated nonparametric regression model to fit the multivariate panel data, which is more flexible than the traditional error component seemingly unrelated parametric regression. By applying the undersmoothing technique and taking both of the correlations within and among responses into account, we propose an efficient two-stage local polynomial estimation for the unknown functions. It is shown that the resulting estimators are asymptotically normal, and have the same biases as the standard local polynomial estimators, which are only based on the individual response, and smaller asymptotic variances. The performance of the proposed procedure is evaluated through a simulation study and a real data set.

EFFICIENCY BOUNDS FOR SEMIPARAMETRIC ESTIMATION OF INVERSE CONDITIONAL-DENSITY-WEIGHTED FUNCTIONS

Consider the unconditional moment restriction E[m(y, υ, w; π0)/fV|w (υ|w) −s (w; π0)] = 0, where m(·) and s(·) are known vector-valued functions of data (y┬, υ, w┬)┬. The smallest asymptotic variance that $\root \of n $-consistent regular estimators of π0 can have is calculated when fV|w(·) is only known to be a bounded, continuous, nonzero conditional density function. Our results show that “plug-in” kernel-based estimators of π0 constructed from this type of moment restriction, such as Lewbel (1998, Econometrica 66, 105–121) and Lewbel (2007, Journal of Econometrics 141, 777–806), are semiparametric efficient.

Efficiency of the maximum partial likelihood estimator for nested case control sampling

In making inference on the relation between failure and exposure histories in the Cox semiparametric model, the maximum partial likelihood estimator (MPLE) of the finite dimensional odds parameter, and the Breslow estimator of the baseline survival function, are known to achieve full efficiency when data is available for all time on all cohort members, even when the covariates are time dependent. When cohort sizes become too large for the collection of complete data, sampling schemes such as nested case control sampling must be used and, under various models, there exist estimators based on the same information as the MPLE having smaller asymptotic variance. Though the MPLE is therefore not efficient under sampling in general, it approaches efficiency in highly stratified situations, or instances where the covariate values are increasingly less dependent upon the past, when the covariate distribution, not depending on the real parameter of interest, is unknown and there is no censoring. In particular, in such situations, when using the nested case control sampling design, both the MPLE and the Breslow estimator of the baseline survival function achieve the information lower bound both in the distributional and the minimax senses in the limit as the number of cohort members tends to infinity.

On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates

Varying-coefficient model Y = ∑ j = 1 p β j ( U ) X j + ɛ has been studied extensively when data are completely observed. When the covariates X are missing at random, we propose a locally weighted estimator based on the inverse selection probabilities. Distribution theory of β ^ ( · ) is derived when the selection probabilities are known, estimated parametrically or nonparametrically. We show that the resulting nonparametric estimator of β ^ ( · ) when the selection probabilities are estimated nonparametrically has a smaller asymptotic variance than that when the selection probabilities are known or estimated parametrically. Motivated by Robin et al. [1994. Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89, 846–866], we also consider simple locally augmented weighted estimator. However, we show that it does not improve the efficiency theoretically. We have constructed a bootstrap test for goodness of fit of models in the missing covariates case. The results of a simulation study are also given to illustrate our method. The proposed method is applied to analyze an AIDS dataset from a clinical study.

An Information-theoretic Approach to the Effective Usage of Auxiliary Information from Survey Data

In this paper, we propose an information-theoretic approach to the effective usage of auxiliary information from survey data, which is suitable for both simple and complex survey data. Our estimator under simple random sampling without replacement will be consistent and asymptotically normal. We show that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information. For more complex survey designs, the resulting estimator is in essence asymptotically equivalent to a pseudo empirical likelihood estimator. Results of a limited simulation study show that the proposed estimators perform well among a number of competitors.

Estimators for Survival Function when Censoring Times are Known

Under the random censorship model with extra assumption that all censoring times are known, Chiu (1999) proposed a new estimator for the survival function of failure time. In this article, we explore the most likely uniformly consistent estimators of the survival function in such data settings and study large sample properties of these estimators. It turns out that the classical Kaplan-Meier estimator is of the smallest asymptotic variance among these possible estimators. This fact confirms the optimality of the Kaplan-Meier estimator in term of asymptotic variance and may suggest that the Kaplan-Meier estimator is the most recommendable even in such circumstances.

Estimators for Survival Function when Censoring Times are Known

Under the random censorship model with extra assumption that all censoring times are known, Chiu (1999) proposed a new estimator for the survival function of failure time. In this article, we explore the most likely uniformly consistent estimators of the survival function in such data settings and study large sample properties of these estimators. It turns out that the classical Kaplan-Meier estimator is of the smallest asymptotic variance among these possible estimators. This fact confirms the optimality of the Kaplan-Meier estimator in term of asymptotic variance and may suggest that the Kaplan-Meier estimator is the most recommendable even in such circumstances.

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.

Accelerated degradation-tests with tightened critical values

ALT (accelerated life tests) are widely used to provide quickly the information about life distributions of products. Life data at elevated stresses are extrapolated to estimate the life distribution at design stress. The existing estimation methods are efficient and easy to implement-given sufficient life data. However, ALT frequently results in few or no failures at low-level stress, making it difficult to estimate the life distribution. For products whose failures are defined in terms of performance characteristics exceeding their critical values, reliability assessment can be based on degradation measurements by using degradation models. The estimation, however, is usually mathematically complicated and computationally intensive. This paper presents a method for the estimation of life distribution by using life data from degradation measurements. Since the time-to-failure depends on the level of a critical value, more life data can be obtained by tightening the critical value. The relationship between life and critical value and stress is modeled and used to estimate the life distribution at a usual critical value and design stress. The model parameters are estimated by using maximum likelihood. The optimum test plans, which choose the critical values, stress levels, and proportions of sample size to each stress level, are devised by minimizing the asymptotic variance of the mean (log) life at a usual critical value and design stress. The comparison between the proposed and existing 2-level test plans shows that the proposed plans have smaller asymptotic variance and are less sensitive to the uncertainty of the pre-estimates of unknown parameters.

On the theory of ranked-set sampling and its ramifications

We consider in this article ranked-set sampling (RSS) and its ramifications including RSS with imperfect ranking, RSS by ranking a concomitant variable and RSS with multivariate samples, etc. We deal with a unified sampling scheme which is referred to as generalized ranked-set sampling (GRSS) and which includes RSS and its ramifications as special cases. We develop a general theory for GRSS in both parametric and nonparametric settings. In a parametric setting, it is shown that the Fisher information matrix about the unknown parameters of a GRSS sample minus that of an SRS sample of the same size is always positive definite. In a nonparametric setting, a particular model, the smooth-function-of-means model, is considered and it is proved that the method-of-moment estimates of parameters based on a GRSS sample will always have smaller asymptotic variances than those based on an SRS sample of the same size. An example for RSS with multivariate samples is treated in detail and a simulation study is reported. Some other issues and open problems such as those involving optimal designs for the GRSS are also discussed.

Efficiency and Convergence Properties of Slice Samplers

The slice sampler (SS) is a method of constructing a reversible Markov chain with a specified invariant distribution. Given an independence Metropolis–Hastings algorithm (IMHA) it is always possible to construct a SS that dominates it in the Peskun sense. This means that the resulting SS produces estimates with a smaller asymptotic variance than the IMHA. Furthermore the SS has a smaller second‐largest eigenvalue. This ensures faster convergence to the target distribution. A sufficient condition for uniform ergodicity of the SS is given and an upper bound for the rate of convergence to stationarity is provided.

Pooled Log Periodogram Regression

Estimation of the memory parameter in time series with long range dependence is considered. A pooled log periodogram regression estimator is proposed that utilizes a set of mL periodogram ordinates with L→∞ rather than m ordinates as in the conventional log periodogram estimator. Consistency and asymptotic normality of the pooled regression estimator are established. The pooled estimator is shown to have smaller asymptotic variance, but larger asymptotic bias, than the conventional log periodogram estimator. Finite sample performance is assessed in simulations and the methods are illustrated in an empirical application with inflation and stock returns.