Subsampling,symmetrization, and robust interpolation

The recently developed subsampling methodology has been shown to be valid for the construction of large-sample confidence regions for a general unknown parameter 9 under very minimal conditions. Nevertheless, in some specific cases—e.g. in the case of the sample mean of i.i.d. data—it has been noted that the subsampling distribution estimator underperforms as compared to alternative estimators such as the bootstrap or the asymptotic normal distribution (with estimated variance). In the present report we introduce a (partially) symmetrized.     

Subsampling quantile estimators and uniformity criteria

Several asymptotically equivalent quantile estimators recently have been proposed as alternative to the conventional sample quantile. A variety of weight functions have been obtained either by subsampling considerations or by a kernel approach, analogous to density estimation techniques. Focusing on the former approach, a unified treatment of quantile estimators derived by subsampling is developed. Closely related to the generalized Harrell-Davis (HD) and Kaigh-Lachenbruch (KL) estimators, a new statistic performed well in Monte Carlo effiency comparisons presented here. Moreover, the new estimator shares certain desirable computational and finite-sample theeoretical properties with the KL estimator to yield convenient components representations for tests of uniformity and goodness-of-fit criteria. Similar analytic treatment for the HD statistics and kernel quantile estimators, however, is precluded by intractable eigenvalue problems.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

Estimation of tail parameters under type i censoring

This paper is a continuation of previous work concerning the estimation of tail-parameters under Type II censoring (Weissman 1978). The same estimation problem is considered here, this truip under Type I censoring. A sample of size n is censored below aE a given level x₀it is assumed that che underlying distriibution .function (df)belogs to the domain of attraction of a known extreme-value distribution and that K - K(x_o) , the number of observed values, remains finite as on - ∞ . We offer here estimators, which are asymptotically maximum likelihood estimators (MLE's), for quantiles associated with the tail of F such as location and scale parameters, quantiles and F(x) itself (for x in the tail). The results are applied to two illustrative examples.     

Moment estimation for statistics from marked point processes

In spatial statistics the data typically consist of measurements of some quantity at irregularly scattered locations; in other words, the data form a realization of a marked point process. In this paper, we formulate subsampling estimators of the moments of general statistics computed from marked point process data, and we establish their L ₂-consistency. The variance estimator in particular can be used for the construction of confidence intervals for estimated parameters. A practical data-based method for choosing a subsampling parameter is given and illustrated on a data set. Finite sample simulation examples are also presented.     

Ranked set sampling with random subsamples

A ranked sampling procedure with random subsamples is proposed to estimate the population mean. Four methods of obtaining random subsamples are described. Several estimators of the mean of the population based on random subsamples in ranked set sampling are proposed. These estimators are compared with the mean of a simple random sample for estimating the mean of symmetric and skew distributions. Extensive simulation under several subsampling distributions, sample sizes, and symmetric and skew distributions shows that the estimators of the mean based on random subsamples are more accurate than existing methods.     

Subsampling Variance Estimation for Non-stationary Spatial Lattice Data

ABSTRACT.  Most proposed subsampling and resampling methods in the literature assume stationary data. In many empirical applications, however, the hypothesis of stationarity can easily be rejected. In this paper, we demonstrate that moment and variance estimators based on the subsampling methodology can also be employed for different types of non-stationarity data. Consistency of estimators are demonstrated under mild moment and mixing conditions. Rates of convergence are provided, giving guidance for the appropriate choice of subshape size. Results from a small simulation study on finite-sample properties are also reported.     

On the maximal inequalities for the average of pairwise i.i.d. random variables

We obtain two sided inequalities for the tail of the maximal function of the averages of a multiple sequence of pairwise i.i.d. random variables taking values in a separable Banach space. We then use the results to establish a necessary and sufficient con¬dition, in terms of the common distribution of the norm of the random variables, for the maximal function to be in L , 1< p << &z.rdang;     

Estimation of the sum of differences distribution

Consider the distribution of Z_i d_iwhere the d.d_i?s are 1=1 lldifferences independently, identically and symmetrically distributed with mean zero. The problem is to determine properties of the sdd given the distribution of the d._i?fs and the sample size n. The standardized moments as a function of the moments of the d._i!s are developed. A variance reduction technique for estimating the quantiles of the sdd using Monte Carlo methods is developed based on using the randomization sample consisting of the 2ⁿ values of Z _i+d. rather than the single observation i=l lZ d. corresponding to each sample d_id_n. The randomization sample is shown to produce unbiased and consistent estimators.     

ESTIMATION OF PARAMETERS IN THE MARGINAL FISHER DISTRIBUTION1

The Fisher distribution is frequently used as a model for the probability distribution of directional data, which may be specified either in terms of unit vectors or angular co-ordinates (co-latitude and azimuth). If, in practical situations, only the co-latitudes can be observed, the available data must be regarded as a sample from the corresponding marginal distribution. This paper discusses the estimation by Maximum Likelihood (ML) and the Method of Moments of the two parameters of this marginal Fisher distribution. The moment estimators are generally simpler to compute than the ML estimators, and have high asymptotic efficiency.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

Asymptotics of cross-validated risk estimation in estimator selection and performance assessment

Risk estimation is an important statistical question for the purposes of selecting a good estimator (i.e., model selection) and assessing its performance (i.e., estimating generalization error). This article introduces a general framework for cross-validation and derives distributional properties of cross-validated risk estimators in the context of estimator selection and performance assessment. Arbitrary classes of estimators are considered, including density estimators and predictors for both continuous and polychotomous outcomes. Results are provided for general full data loss functions (e.g., absolute and squared error, indicator, negative log density). A broad definition of cross-validation is used in order to cover leave-one-out cross-validation, V-fold cross-validation, Monte Carlo cross-validation, and bootstrap procedures. For estimator selection, finite sample risk bounds are derived and applied to establish the asymptotic optimality of cross-validation, in the sense that a selector based on a cross-validated risk estimator performs asymptotically as well as an optimal oracle selector based on the risk under the true, unknown data generating distribution. The asymptotic results are derived under the assumption that the size of the validation sets converges to infinity and hence do not cover leave-one-out cross-validation. For performance assessment, cross-validated risk estimators are shown to be consistent and asymptotically linear for the risk under the true data generating distribution and confidence intervals are derived for this unknown risk. Unlike previously published results, the theorems derived in this and our related articles apply to general data generating distributions, loss functions (i.e., parameters), estimators, and cross-validation procedures.     

Estimating and testing group effects from type i censored normal samples in experimental design

The maximum likelihood (ML) equations calculated from censored normal samples do not admit explicit solutions. A principle of modification is given and modified maximum likelihood (MML) equations, which admit explicit solutions, are defined. This approach makes it possible to tackle the hitherto unresolved problem of estimating and testing hypotheses about group-effects in one-way classification experimental designs based on Type I censored normal samples. The MML estimators of group-effects are obtained as explicit functions of sample observations and shown to be asymptotically identical with the ML estimators and hence BAN (best asymptotic normal) estimators. A statistic t is defined to test a linear contrast of group-effects and shown to be asymptotically normally distributed. A numerical example is presented which illustrates the procedure.     

The Finite Sample Performance of Inference Methods for Propensity Score Matching and Weighting Estimators

ABSTRACT

This article investigates the finite sample properties of a range of inference methods for propensity score-based matching and weighting estimators frequently applied to evaluate the average treatment effect on the treated. We analyze both asymptotic approximations and bootstrap methods for computing variances and confidence intervals in our simulation designs, which are based on German register data and U.S. survey data. We vary the design w.r.t. treatment selectivity, effect heterogeneity, share of treated, and sample size. The results suggest that in general, theoretically justified bootstrap procedures (i.e., wild bootstrapping for pair matching and standard bootstrapping for “smoother” treatment effect estimators) dominate the asymptotic approximations in terms of coverage rates for both matching and weighting estimators. Most findings are robust across simulation designs and estimators.     

Contributions to adaptive estimation

There are many statistics which can be used to characterize data sets and provide valuable information regarding the data distribution, even for large samples. Traditional measures, such as skewness and kurtosis, mentioned in introductory statistics courses, are rarely applied. A variety of other measures of tail length, skewness and tail weight have been proposed, which can be used to describe the underlying population distribution. Adaptive statistical procedures change the estimator of location, depending on sample characteristics. The success of these estimators depends on correctly classifying the underlying distribution model. Advocates of adaptive distribution testing propose to proceed by assuming (1) that an appropriate model, say Omega , is such that Omega { Omega , Omega , i i 1 2 … , Omega }, and (2) that the character of the model selection process is statistically k independent of the hypothesis testing. We review the development of adaptive linear estimators and adaptive maximum-likelihood estimators.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

Summary.  We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates t  ∈  R ^d ,  d =1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for d =1 and d =2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension d this becomes more drastic. If d 4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when d 4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.     

Estimation of the Jump Size Density in a Mixed Compound Poisson Process

In this paper, we consider a mixed compound Poisson process, that is, a random sum of independent and identically distributed (i.i.d.) random variables where the number of terms is a Poisson process with random intensity. We study nonparametric estimators of the jump density by specific deconvolution methods. Firstly, assuming that the random intensity has exponential distribution with unknown expectation, we propose two types of estimators based on the observation of an i.i.d. sample. Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of the random intensity, we propose two non‐parametric estimators of the jump density based on the joint observation of the number of jumps and the random sum of jumps. Risks bounds are provided, leading to unusual rates for one of the two estimators. The methods are implemented and compared via simulations.     

A Note on Robustness of Normal Variance Estimators Under t-Distributions

ABSTRACT

In many real life problems one assumes a normal model because the sample histogram looks unimodal, symmetric, and/or the standard tests like the Shapiro-Wilk test favor such a model. However, in reality, the assumption of normality may be misplaced since the normality tests often fail to detect departure from normality (especially for small sample sizes) when the data actually comes from slightly heavier tail symmetric unimodal distributions. For this reason it is important to see how the existing normal variance estimators perform when the actual distribution is a t-distribution with k degrees of freedom (d.f.) (t _k -distribution). This note deals with the performance of standard normal variance estimators under the t _k -distributions. It is shown that the relative ordering of the estimators is preserved for both the quadratic loss as well as the entropy loss irrespective of the d.f. and the sample size (provided the risks exist).