Incompatibility of trends in multi-year estimates from the American Community Survey

The American Community Survey (ACS) provides one-year (1y), three-year (3y) and five-year (5y) multi-year estimates (MYEs) of various demographic and economic variables for each "community", although the 1y and 3y may not be available for communities with a small population. These survey estimates are not truly measuring the same quantities, since they each cover different time spans. Using some simplistic models, we demonstrate that comparing different period-length MYEs results in spurious conclusions about trend movements. A simple method utilizing weighted averages is presented that reduces the bias inherent in comparing trends of different MYEs. These weighted averages are nonparametric, require only a short span of data, and are designed to preserve polynomial characteristics of the time series that are relevant for trends. The basic method, which only requires polynomial algebra, is outlined and applied to ACS data. In some cases there is an improvement to comparability, although a final verdict must await additional ACS data. We draw the conclusion that MYE data is not comparable across different periods.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS259 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Computer-intensive rate estimation, diverging statistics and scanning

A general rate estimation method is proposed that is based on studying the in-sample evolution of appropriately chosen diverging/converging statistics. The proposed rate estimators are based on simple least squares arguments, and are shown to be accurate in a very general setting without requiring the choice of a tuning parameter. The notion of scanning is introduced with the purpose of extracting useful subsamples of the data series; the proposed rate estimation method is applied to different scans, and the resulting estimators are then combined to improve accuracy. Applications to heavy tail index estimation as well as to the problem of estimating the long memory parameter are discussed; a small simulation study complements our theoretical results.Comment: Published in at http://dx.doi.org/10.1214/009053607000000064 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stable marked point processes

In many contexts such as queuing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation involves considering the observation points as generated by a Poisson process. Under this assumption, we study the limit behavior of the partial sums of the marked point process $\{(t_i,X(t_i))\}$, where X(t) is a stationary random field and the points t_i are generated from an independent Poisson random measure $\mathbb{N}$ on $\mathbb{R}^d$. We define the sample mean and sample variance statistics and determine their joint asymptotic behavior in a heavy-tailed setting, thus extending some finite variance results of Karr [Adv. in Appl. Probab. 18 (1986) 406--422]. New results on subsampling in the context of a marked point process are also presented, with the application of forming a confidence interval for the unknown mean under an unknown degree of heavy tails.Comment: Published at http://dx.doi.org/10.1214/009053606000001163 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Signal extraction revision variances as a goodness-of-fit measure

erworben im Rahmen der Schweizer Nationallizenzen (www.nationallizenzen.ch)Typically, model misspecification is addressed by statistics relying on model-residuals, i.e., on one-step ahead forecasting errors. In practice, however, users are often also interested in problems involving multi-step ahead forecasting performances, which are not explicitly addressed by traditional diagnostics. In this article, we consider the topic of misspecification from the perspective of signal extraction. More precisely, we emphasize the connection between models and real-time (concurrent) filter performances by analyzing revision errors instead of one-step ahead forecasting errors. In applications, real-time filters are important for computing trends, for performing seasonal adjustment or for inferring turning-points towards the current boundary of time series. Since revision errors of real-time filters generally rely on particular linear combinations of one- and multi-step ahead forecasts, we here address a generalization of traditional diagnostics. Formally, a hypothesis testing paradigm for the empirical revision measure is developed through theoretical calculations of the asymptotic distribution under the null hypothesis, and the method is assessed through real data studies as well as simulations. In particular, we analyze the effect of model misspecification with respect to unit roots, which are likely to determine multi-step ahead forecasting performances. We also show that this framework can be extended to general forecasting problems by defining suitable artificial signals

Optimal real-time filters for linear prediction problems

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)The classic model-based paradigm in time series analysis is rooted in the Wold decomposition of the data-generating process into an uncorrelated white noise process. By design, this universal decomposition is indifferent to particular features of a specific prediction problem (e. g., forecasting or signal extraction) – or features driven by the priorities of the data-users. A single optimization principle (one-step ahead forecast error minimization) is proposed by this classical paradigm to address a plethora of prediction problems. In contrast, this paper proposes to reconcile prediction problem structures, user priorities, and optimization principles into a general framework whose scope encompasses the classic approach. We introduce the linear prediction problem (LPP), which in turn yields an LPP objective function. Then one can fit models via LPP minimization, or one can directly optimize the linear filter corresponding to the LPP, yielding the Direct Filter Approach. We provide theoretical results and practical algorithms for both applications of the LPP, and discuss the merits and limitations of each. Our empirical illustrations focus on trend estimation (low-pass filtering) and seasonal adjustment in real-time, i. e., constructing filters that depend only on present and past data

Estimating the Spectral Density at Frequencies Near Zero

Estimating the spectral density function $f(w)$ for some $w\in [-\pi, \pi]$ has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, i.e., approximating $f(w)$ by a constant over a window of small width. Although $f(w)$ is uniformly continuous and periodic with period $2\pi$, in this paper we recognize the fact that $w=0$ effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for $w=\pm \pi$. It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when $w$ is at (or near) the points 0 or $\pm \pi$. The case $w=0$ is of particular importance since $f(0)$ is the large-sample variance of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of inference on the mean