A small but practically useful modification to the Hodrick–Prescott filtering: A note

The Hodrick–Prescott (HP) filtering is widely applied to decompose macroeconomic time series, such as real Gross Domestic Product, into cyclical and trend components. This paper presents a small but practically useful modification to this approach. The reason why this modified filtering is of practical use is that it provides not only identical trend estimates as the HP filtering but also extrapolations of the trend. We provide a proof based on a ridge regression representation of the modified HP filtering. This is mainly because it enhances our understanding of the approach.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

Rate of convergence for asymptotic variance of the Horvitz–Thompson estimator

Drawing distinct units without replacement and with unequal probabilities from a population is a problem often considered in the literature (e.g. Hanif and Brewer, 1980, Int. Statist. Rev. 48, 317–355). In such a case, the sample mean is a biased estimator of the population mean. For this reason, we use the unbiased Horvitz–Thompson estimator (1951). In this work, we focus our interest on the variance of this estimator. The variance is cumbersome to compute because it requires the calculation of a large number of second-order inclusion probabilities. It would be helpful to use an approximation that does not need heavy calculations. The Hájek (1964) variance approximation provides this advantage as it is free of second-order inclusion probabilities. Hájek (1964) proved that this approximation is valid under restrictive conditions that are usually not fulfilled in practice. In this paper, we give more general conditions and we show that this approximation remains acceptable for most practical problems.     

A new estimator of Kullback–Leibler information and its application in goodness of fit tests*

We propose here a general statistic for the goodness of fit test of statistical distributions. The proposed statistic is constructed based on an estimate of Kullback–Leibler information. The proposed test is consistent and the limiting distribution of the test statistic is derived. Then, the established results are used to introduce goodness of fit tests for the normal, exponential, Laplace and Weibull distributions. A simulation study is carried out for examining the power of the proposed test and to compare it with those of some existing procedures. Finally, some illustrative examples are presented and analysed, and concluding comments are made.     

A class of one sample tests based on the Mann–Whitney–Wilcoxon functional

The main idea behind the proposed class of tests is rooted on an extension of the technique used in the derivation of the Mann–Whitney–Wilcoxon test. Just like the case of two-sample rank-based tests, the new class consists of tests defined through score functions. When properly selected, these score functions lead to consistent and often more powerful tests compared with classical goodness-of-fit tests. Theoretical results are supported by an extensive simulation study.     

A stochastic restricted k–d class estimator

In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakall?o?lu and S. Kaçiranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.     

Nonparametric estimation of the multivariate survivor function: the multivariate Kaplan–Meier estimator

The Dabrowska (Ann Stat 16:1475–1489, 1988) product integral representation of the multivariate survivor function is extended, leading to a nonparametric survivor function estimator for an arbitrary number of failure time variates that has a simple recursive formula for its calculation. Empirical process methods are used to sketch proofs for this estimator’s strong consistency and weak convergence properties. Summary measures of pairwise and higher-order dependencies are also defined and nonparametrically estimated. Simulation evaluation is given for the special case of three failure time variates.     

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y$y$ on x$x$ given by a pair of mean and variance functions with a parameter vector θ$\theta$ to be estimated that also appears in the distribution of the regressor variable x$x$. The estimation of θ$\theta$ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y$y$ unbiased estimating functions. Of special interest is the case where the distribution of x$x$ depends only on a subvector α$\alpha$ of θ$\theta$, which may be considered a nuisance parameter. In general, α$\alpha$ must be estimated simultaneously together with the rest of θ$\theta$, but there are cases where α$\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.     

A stable estimator of the information matrix under EM for dependent data

This article develops a new and stable estimator for information matrix when the EM algorithm is used in maximum likelihood estimation. This estimator is constructed using the smoothed individual complete-data scores that are readily available from running the EM algorithm. The method works for dependent data sets and when the expectation step is an irregular function of the conditioning parameters. In comparison to the approach of Louis (J. R. Stat. Soc., Ser. B 44:226–233, 1982), this new estimator is more stable and easier to implement. Both real and simulated data are used to demonstrate the use of this new estimator.     

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

An extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe

In this paper, an extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe is developed. Main new results are presented in theorems. The primary notions of discrete population are transferred to continuous population. First and second order inclusion probabilities for networks are delivered. Horvitz–Thompson estimator for adaptive cluster sampling in continuous universe is constructed. The unbiasedness of the estimator is proven. Variance and unbiased variance estimator are delivered. Finally, the theory is illustrated with an example.     

On the Kaplan–Meier estimator based on ranked set samples

When quantification of all sampling units is expensive but a set of units can be ranked, without formal measurement, ranked set sampling (RSS) is a cost-efficient alternate to simple random sampling (SRS). In this paper, we study the Kaplan–Meier estimator of survival probability based on RSS under random censoring time setup, and propose nonparametric estimators of the population mean. We present a simulation study to compare the performance of the suggested estimators. It turns out that RSS design can yield a substantial improvement in efficiency over the SRS design. Additionally, we apply the proposed methods to a real data set from an environmental study.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

A Property of Count Distributions in the Hinde–Demétrio Family

The Hinde–Demétrio (HD) family of distributions, which are discrete exponential dispersion models with an additional real index parameter p, have been recently characterized from the unit variance function μ + μ ^p . For p equals to 2, 3,…, the corresponding distributions are concentrated on non negative integers, overdispersed and zero-inflated with respect to a Poisson distribution having the same mean. The negative binomial (p = 2) and strict arcsine (p = 3) distributions are HD families; the limit case (p → ∞) is associated to a suitable Poisson distribution. Apart from these count distributions, none of the HD distributions has explicit probability mass functions p _k . This article shows that the ratios r _k  = k p _k /p _k?1, k = 1,…, p ? 1, are equal and different from r _p . This new property allows, for a given count data set, to determine the integer p by some tests. The extreme situation of p = 2 is of general interest for count data. Some examples are used for illustrations and discussions.     

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.