Discretely Observed Diffusions: Classes of Estimating Functions and Small Δ-optimality

Ergodic diffusions in several dimensions, depending on an unknown multivariate parameter are considered. For estimation, when the diffusion is observed only at finitely many equidistant time points, unbiased estimating functions leading to consistent and asymptotically Gaussian estimators are used. Different types of estimating functions are discussed and the concept of small Δ-optimality is introduced to help select good estimating functions. Explicit criteria for small Δ-optimality are given. Also some exact optimality conditions are presented as well as, for one-dimensional diffusions, methods for improving estimators using time reversibility.     

On the nonparametric smooth estimation of the reversed Hazard Rate function

The Reversed Hazard Rate (RHR) function is an important measure as a tool in the analysis of the reliability of both natural and man-made systems. In this paper, we present several new estimators of the RHR function using nonparametric techniques. These estimators are obtained by incorporating different binning techniques with fixed design local polynomial regression. We show that these estimators are asymptotically unbiased and consistent and, to determine the bandwidth, we propose two simple yet efficient plug-in bandwidth selection methods for even and odd order local polynomial estimators. Simulated and real life data are subsequently used to evaluate the performances of these estimators.     

Asymptotic bayesian inference in some nonstandard cases: Bernstein–von Mises Results and regular bayes' estimators

Asymptotically with probability close to one, the convergence in variation (also in distribution) to the multivariate normal, of the aposteriori density function of a parameter agains an apriori density, viz. the BERNSTEIN–VON MISES results are established when observations are not necessarily indenpendent or identically distributed but satisfy weak regularity assumptions on their joint density function. Regular BAYES' estimators are defined with respect to regular loss functions and a positive apriori density and proved consistent, asymptotically efficient and asymptotically normal. Examples and applications to conjugate families of densities, to inference in MARKOV Chains and other nonstandard cases illustrate results     

On the efficiency of regression analysis with AR(p) errors

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

Sequential point estimation of the means of u-statistics in finite population sampling

In simple random sampling without replacement from a finite population, sequential point estimators of the means of U-statistics are proposed. The proposed procedure is shown to be asymptotically risk efficient in the sense of Starr (Ann. Math. Statist. (1966), 1173-1185)     

Estimators based on ranks for arma models

In this paper we introduce a new family of robust estimators for ARMA models. These estimators are defined by replacing the residual sample autocovariances in the least squares equations by autocovariances based on ranks. The asymptotic normality of the proposed estimators is provided. The efficiency and robustness properties of these estimators are studied. An adequate choice of the score functions gives estimators which have high efficiency under normality and robustness in the presence of outliers. The score functions can also be chosen so that the resulting estimators are asymptotically as efficient as the maximum likelihood estimators for a given distribution.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

Asymptotic behavior of iterative m-estimators for the linear model

In this paper, we study the M-estimators for the linear model when they are computed by a class of numerical iterative procedures. This class includes the usual method of Newton-Raphson, iteratively reweighted least squares and iterative winsorization. We show that under mild conditions, the numerical iterative procedures converge and the resulting estimators are consistent and asymptotically normal.     

Location and scale parameter estimation from randomly censored data

The problem of location and scale parameter estimation from randomly censored data is analyzed through use of a regression model for the Kaplan-Meier quantlle process. Continuous time regression techniques are employed to construct estimators that are both asymptotically normal and efficient. Estimators with a particularly simple form are obtained for the Koziol-Green model for random censorship. In the event of no censoring the regression model, and resulting estimators, reduce to those proposed by Parzen (1979 a, b).     

Estimating fault hitting rates by recapture sampling

For the recapture debugging design introduced by Nayak (1988) we consider the problem of estimating the hitting rates of the faults remaining in a system. In the context of a conditional likelihood, moment estimators are derived and are shown to be asymptotically normal and fully efficient. Fixed sample properties of the moment estimators are compared, through simulation, with those of the conditional maximum likelihood estimators. Also considered is a procedure for testing the assumption that faults have identical hitting rates; this provides a test of fit of the Jelinski-Moranda (1972) model. It is assumed that the residual hitting rates follow a log linear rate model and that the testing process is truncated when the gaps between the detection of new errors exceed a fixed amount of time.     

Frequentist model averaging for envelope models

The envelope method produces efficient estimation in multivariate linear regression, and is widely applied in biology, psychology, and economics. This paper estimates parameters through a model averaging methodology and promotes the predicting abilities of the envelope models. We propose a frequentist model averaging method by minimizing a cross-validation criterion. When all the candidate models are misspecified, the proposed model averaging estimator is proved to be asymptotically optimal. When correct candidate models exist, the coefficient estimator is proved to be consistent, and the sum of the weights assigned to the correct models, in probability, converges to one. Simulations and an empirical application demonstrate the effectiveness of the proposed method.     

Rank-based estimation for semiparametric accelerated failure time model under length-biased sampling

Length-biased sampling appears in many observational studies, including epidemiological studies, labor economics and cancer screening trials. To accommodate sampling bias, which can lead to substantial estimation bias if ignored, we propose a class of doubly-weighted rank-based estimating equations under the accelerated failure time model. The general weighting structures considered in our estimating equations allow great flexibility and include many existing methods as special cases. Different approaches for constructing estimating equations are investigated, and the estimators are shown to be consistent and asymptotically normal. Moreover, we propose efficient computational procedures to solve the estimating equations and to estimate the variances of the estimators. Simulation studies show that the proposed estimators outperform the existing estimators. Moreover, real data from a dementia study and a Spanish unemployment duration study are analyzed to illustrate the proposed method.     

Minimum Variance Quadratic Unbiased Estimation for the Variance Components in Simple Linear Regression with Onefold Nested Error

The explicit forms of the minimum variance quadratic unbiased estimators (MIVQUEs) of the variance components are given for simple linear regression with onefold nested error. The resulting estimators are more efficient as the ratio of the initial variance components estimates increases and are asymptotically efficient as the ratio tends to infinity.     

Analysis of categorical data obtained by stratified random sampling

Analysis of categorical data by linear models is extended to data obtained by stratified random sampling. It is shown that, asymptotically, proportional allocation reduces the variances of estimators from those obtained hy simple random sampling. The difference between the asymptotic covariance matrices of estimated parameters obtained by simple random sampling and stratified random sampling with proportional allocation is shown to be positive definite vinder fairly non-restrictive conditions, when an asymptotically efficient method of estimation is used. Data from a major community study of mental health are used to illustrate application of the technique.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

Flat-Top Realized Kernel Estimation of Quadratic Covariation With Nonsynchronous and Noisy Asset Prices

This article develops a general multivariate additive noise model for synchronized asset prices and provides a multivariate extension of the generalized flat-top realized kernel estimators, analyzed earlier by Varneskov (2014), to estimate its quadratic covariation. The additive noise model allows for α-mixing dependent exogenous noise, random sampling, and an endogenous noise component that encompasses synchronization errors, lead-lag relations, and diurnal heteroscedasticity. The various components may exhibit polynomially decaying autocovariances. In this setting, the class of estimators considered is consistent, asymptotically unbiased, and mixed Gaussian at the optimal rate of convergence, n^1/4. A simple finite sample correction based on projections of symmetric matrices ensures positive definiteness without altering the asymptotic properties of the estimators. It, thereby, guarantees the existence of nonlinear transformations of the estimated covariance matrix such as correlations and realized betas, which inherit the asymptotic properties from the flat-top realized kernel estimators. An empirically motivated simulation study assesses the choice of sampling scheme and projection rule, and it shows that flat-top realized kernels have a desirable combination of robustness and efficiency relative to competing estimators. Last, an empirical analysis of signal detection and out-of-sample predictions for a portfolio of six stocks of varying size and liquidity illustrates the use and properties of the new estimators.     

Additive transformation models for clustered failure time data

We propose a class of additive transformation risk models for clustered failure time data. Our models are motivated by the usual additive risk model for independent failure times incorporating a frailty with mean one and constant variability which is a natural generalization of the additive risk model from univariate failure time to multivariate failure time. An estimating equation approach based on the marginal hazards function is proposed. Under the assumption that cluster sizes are completely random, we show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also provide goodness-of-fit test statistics for choosing the transformation. Simulation studies and real data analysis are conducted to examine the finite-sample performance of our estimators.     

Hypothesis Testing in Functional Comparative Calibration Models

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.