Paul Shaman

The Bias of Autoregressive Coefficient Estimators

Abstract This article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary, but known, finite order. The results include models both with and without a constant term. The effects of overspecification of the model order on the bias are described. The emphasis is on least-squares and Yule-Walker estimators, but the methods extend to other estimators of similar design. Although only the order T -1 component of the bias is captured, where T is the series length, this asymptotic approximation is shown to be very accurate for least-squares estimators through some numerical simulations. The simulations examine fourth-order autoregressions chosen to resemble some data series from the literature. The order T -1 bias approximations for Yule-Walker estimators need not be accurate, especially if the zeros of the associated polynomial have moduli near 1. Examples are given where the approximation is accurate and where it is useless. The bias expressions are...

Complications of Ethmoidectomy: A Survey of Fellows of the American Academy of Otolaryngology—Head and Neck Surgery

A survey regarding complications of sinus surgery was mailed to 6969 otolaryngologists; 3933 responses (56.44%) were obtained, and 3043 of these physicians (77.37%) reported that they performed ethmoidectomy. Completed questionnaires were available for review from 42.21% of all Academy fellows (2942 physicians). Responses were tabulated and summarized question by question. Physicians generally did not rate their residency training in ethmoidectomy highly. The survey confirmed that there has been a marked rise in the frequency of ethmoidectomy and in the amount of training in ethmoidectomy since 1985. Empirical complication rates were calculated for different procedures and time periods. Poisson regression models were then constructed to describe the rate of complications under varying conditions such as the type of surgery performed, time period, experience and training of the surgeon, and type of complication encountered. The models permitted determination of the statistical significance of variables in relation to Incidence of complications. The study did not demonstrate a clear and consistent statistical relationship between the incidence of complications, the type of surgery performed, and the quality of training. Moreover, physicians who provided data from record review tended to report higher rates than those who estimated responses. The majority of physicians discussed specific potential complications with their patients before surgery and routinely performed preoperative computed tomography. The study demonstrated that physicians who experienced complications at higher rates were more likely to discuss these complications with patients before surgery.

Topographic coding of odorant quality is maintained at different concentrations in the salamander olfactory epithelium.

In a recent study in the tiger salamander, Ambystoma tigrinum, were demonstrated topographic patterns of responsivity across the olfactory epithelium which were characteristic for each odorant. The present study was initiated to investigate whether these patterns remain constant when odorant concentration is varied. Odorant-induced electro- olfactograms were recorded from at least 12 sites on each epithelium. The odorants used were pinene, amyl acetate and propanol. Each epithelium was tested with one odorant, delivered at 3 concentrations. For comparison between animals, the epithelia were divided into 3 regions with at least 4 recording sites per region. An analysis of variance model was used to study odorants, concentrations, regions and animals. Odorant-induced regional patterns in responsivity were similar across all concentrations. In particular, the region of highest responsivity at one concentration was the region of highest responsivity at all concentrations. It is concluded that topographic patterns of receptor cell responses may reflect an underlying genetic component in the distribution of receptor cells. This distribution is related to two aspects of receptor cell responses: responsivity to particular odorants (Fig. 4) and general responsivity to all odorants (Fig. 5).

The inverted complex Wishart distribution and its application to spectral estimation

The inverted complex Wishart distribution and its use for the construction of spectral estimates are studied. The density, some marginals of the distribution, and the first- and second-order moments are given. For a vector-valued time series, estimation of the spectral density at a collection of frequencies and estimation of the increments of the spectral distribution function in each of a set of frequency bands are considered. A formal procedure applies Bayes theorem, where the complex Wishart is used to represent the distribution of an average of adjacent periodogram values. A conjugate prior distribution for each parameter is an inverted complex Wishart distribution. Use of the procedure for estimation of a 2 - 2 spectral density matrix is discussed.

Bilateral and Unilateral Olfactory Sensitivity: Relationship to Handedness and Gender

Unlike most major sensory systems, the majority of olfactory projections are ipsilateral. Both hemispheres can process olfactory information in a manner analogous to what is seen in other sensory systems, although, as noted below, they may do so differently. Gordon and Sperry (1969) found that patients whose corpus callosum and other forebrain commissures were surgically sectioned could only names odors presented to the left nostril; odors presented to the right nostril could be identified by pointing to an object associated with the smell.

PROPERTIES OF ESTIMATES OF THE MEAN SQUARE ERROR OF PREDICTION IN AUTOREGRESSIVE MODELS

Publisher Summary This chapter discusses the properties of estimates of the mean square error of prediction in autoregressive models. A selection criterion is used to choose a suitable finite order approximation to the infinite order autoregression. There are several criteria available: final prediction error method of Akaike; AIC, an information criterion; and the criterion autoregressive transfer function method of Parzen. All of these methods require an estimate of the mean error of one-step-ahead prediction when an optimal predictor of finite memory is used. The chapter also discusses the bias to terms of order n-1 of several estimates of the mean square error of one-step-ahead prediction for an optimal predictor of finite memory. In the estimates, the autoregressive coefficients, forming the optimal predictor of finite memory, are estimated by regression methods. The chapter describes estimates of the autoregressive coefficients, constructed from both biased and unbiased estimates of the population covariances.

Photochemical effects of laser irradiation on neuritic outgrowth of olfactory neuroepithelial explants.

The photochemical effect of low-intensity laser irradiation (LILI) on the maturation and regeneration of olfactory-Immature estrus day 15 (E15) and olfactory-mature estrus day 22 (E22) rat fetuses was studied. Neuritic outgrowths of olfactory bipolar receptor cells were quantified in olfactory neuroepithelial explants. Explants in the experimental groups were irradiated with a helium-neon laser using different incident energy densities (IEDs). Explants in another group were exposed to fluorescent light. Control explants did not receive laser or fluorescent light irradiation. Neuritic outgrowths were analyzed on a regular basis for 12 days. Analysis of variance was used to evaluate the data. The parameters of neuritic outgrowth in E15 fetuses showed a significant increase of 30% to 50% vs. the control with a single laser irradiation of 0.5 J/cm2 IED. The rate of neuritic outgrowth observed in the E22 fetuses was less than in the E15 fetuses. The parameters of neuritic outgrowth in E22 fetuses showed a significant and substantially greater percentage increase than in the E15 fetuses with daily laser irradiations of 0.05 and 0.5 J/cm2 IED when compared to the control. The magnitude of these increases appears to be of biological significance as well as statistical significance.

On the calculation of cumulants of estimators arising from a linear time series regression model

Bounds for higher-order cumulants of statistics arising from a linear time series regression model are investigated. A result given in Brillinger is proved and extended. The bounds permit derivation of asymptotic moments and asymptotic normality for estimators of parameters in the model. Two examples are given as illustrations.

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.

Bias of autoregressive spectral estimators

Abstract Bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator. Under the assumption of an autoregressive generating process of known finite order, we obtain an expression for the order 1/T bias, where T is the sample length of the observed series. This approximation is a sum of several simple functions of the unknown coefficients. When the spectral density has sharp peaks, one of these functions dominates the bias. The attenuation from this dominant component can be substantial when the spectral peak is well defined, and several examples illustrate this effect. Since the integral of the order 1/T bias components that are frequency dependent is 0, unbiased estimation of entropy to this order is possible for autoregressive processes. These bias expressions extend to autoregressive models in which the mean is a polynomial function of time. Similar results obtain for the log of the Yule-Walker spectral estimator, for which the order ...