Nonparametric Kernel Estimation Research Articles

Dispersal Patterns, Dispersal Mechanisms, and Invasion Wave Speeds for Invasive Thistles

Understanding and predicting population spread rates is an important problem in basic and applied ecology. In this article, we link estimates of invasion wave speeds to species traits and environmental conditions. We present detailed field studies of wind dispersal and compare nonparametric (i.e., data-based) and mechanistic (fluid dynamics model-based) dispersal kernel and spread rate estimates for two important invasive weeds, Carduus nutans and Carduus acanthoides. A high-effort trapping design revealed highly leptokurtic dispersal distributions, with seeds caught up to 96 m from the source, far further than mean dispersal distances (approx. 2 m). Nonparametric wave speed estimates are highly sensitive to sampling effort. Mechanistic estimates are insensitive to sampling because they are obtained from independent data and more useful because they are based on the dispersal mechanism. Over a wide range of realistic conditions, mechanistic spread rate estimates were most sensitive to high winds and low seed settling velocities. The combination of integrodifference equations and mechanistic dispersal models is a powerful tool for estimating invasion spread rates and for linking these estimates to characteristics of the species and the environment.

Adaptive Estimation of Heteroscedastic Money Demand Model of Pakistan

For the problem of estimation of Money demand model of Pakistan, money supply (M1) shows heteroscedasticity of the unknown form. For estimation of such model we compare two adaptive estimators with ordinary least squares estimator and show the attractive performance of the adaptive estimators, namely, nonparametric kernel estimator and nearest neighbour regression estimator. These comparisons are made on the basis standard errors of the estimated coefficients, standard error of regression, Akaike Information Criteria (AIC) value, and the Durban-Watson statistic for autocorrelation. We further show that nearest neighbour regression estimator performs better when comparing with the other nonparametric kernel estimator.

Non-Parametric Confidence Bands in Deconvolution Density Estimation

SummaryUniform confidence bands for densities f via non-parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*ψ for an ordinary smooth error density ψ. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L∞-distance) between a deconvolution kernel estimator f^ and f. Further consistency of the simple non-parametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector that is based on heuristic arguments, which aims at minimizing the L∞-distance between f^ and f. Although not constructed explicitly to undersmooth the estimator, a simulation study reveals that the bandwidth selector suggested performs well in finite samples, in terms of both area and coverage probability of the resulting confidence bands. Finally the methodology is applied to measurements of the metallicity of local F and G dwarf stars. Our results confirm the ‘G dwarf problem’, i.e. the lack of metal poor G dwarfs relative to predictions from ‘closed box models’ of stellar formation.

Asymptotic properties of the kernel estimator of the conditional mode for the left truncated model

In this Note we propose a non-parametric kernel estimator of the conditional mode function, when the variable of interest is subject to random left-truncation. We establish the rate of the strong uniform consistency of the estimate as well as its asymptotic normality. To cite this article: E. Ould-Saïd, A. Tatachak, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Rates of strong uniform consistency for local least squares kernel regression estimators

We establish exact rates of strong uniform consistency for the multivariate Nadaraya–Watson kernel estimator of the regression function and its derivatives. As a special case, we treat the local linear estimator of the regression and the local polynomial smoothers of derivatives of the regression in the more convenient univariate setting. Our methods of proofs are based upon modern empirical process theory in the spirit of the results of Einmahl and Mason [2000. An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13.1, 1–37.] and Deheuvels and Mason [2004. General asymptotic confidence bands based on kernel-type function estimators. Statist. Infer. Stochastic Process. 7.3, pp. 225–277] relative to uniform deviations of nonparametric kernel estimators.

A NONPARAMETRIC REGRESSION ESTIMATOR THAT ADAPTS TO ERROR DISTRIBUTION OF UNKNOWN FORM

We propose a new kernel estimator for nonparametric regression with unknown error distribution. We show that the proposed estimator is adaptive in the sense that it is asymptotically equivalent to the infeasible local likelihood estimator (Staniswalis, 1989, Journal of the American Statistical Association 84, 276–283; Fan, Farmen, and Gijbels, 1998, Journal of the Royal Statistical Society, Series B 60, 591–608; and Fan and Chen, 1999, Journal of the Royal Statistical Society, Series B 61, 927–943), which requires knowledge of the error distribution. Hence, our estimator improves on standard nonparametric kernel estimators when the error distribution is not normal. A Monte Carlo experiment is conducted to investigate the finite-sample performance of our procedure.We thank Yuichi Kitamura, Yanqin Fan, Joel Horowitz, Roger Koenker, Jens Perch Nielsen, Peter Phillips, Peter Robinson, Tom Rothenberg, and two referees for helpful comments. Financial support from the NSF and the ESRC (UK) is gratefully acknowledged.

Robust kernel estimator for densities of unknown smoothness

Results on non-parametric kernel estimators of density differ according to the assumed degree of density smoothness. A kernel/bandwidth pair that was optimal for a twice differentiable function may not be suitable when the density is piecewise linear. If there is uncertainty about the degree of smoothness, an inappropriate choice may lead to under- or oversmoothing. To examine various possible outcomes we provide asymptotic results on kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair and derive the limit joint distribution of kernel density estimators corresponding to different bandwidths and kernel functions. Using these results, we propose a combined estimator constructed as an optimal linear combination of several estimators with different bandwidth/kernel pairs. Its theoretical properties [Kotlyarova, Y. and Zinde-Walsh, V., 2006, Non- and semi-parametric estimation in models with unknown smoothness. Economics Letters, 93, 379–386] are such that it automatically attains the best possible rate without a priori knowledge of the degree of smoothness. Our Monte-Carlo results confirm the advantages of the combined estimator of density.

On Nonparametric Identification of Wiener Systems

In this paper, a new method for the identification of the Wiener nonlinear system is proposed. The system, being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element, is identified by a two-step semiparametric approach. The impulse response function of the linear part is identified via the nonlinear least-squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate. The obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the Wiener system. The proposed method permits recovery of a wide class of nonlinearities which need not be invertible. As a result, the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate. Furthermore, our approach allows non-Gaussian input signals and the presence of additive measurement noise. However, only linear systems with a finite memory are admissible. The conditions for the convergence of the proposed estimates are given. Computer simulations are included to verify the basic theory

Analysis of Shallow and Deep Earthquake Doublets in the Fiji-Tonga-Kermadec Region

For the Fiji-Tonga-Kermadec area and for the period from January 1977 to July 2003, the Harvard CMT catalogue lists 1022 shallow, 410 intermediate and 633 deep earthquakes of moment magnitude from 4.9 to 8.0. The magnitude threshold, above which the catalogue is complete, is 5.3–5.4, and the number of earthquakes of magnitude above this value is 691 for shallow, 329 for intermediate and 476 for deep events, respectively. The proportion of earthquakes associated with doublets and multiplets against the total number of earthquakes is approximately the same in both data sets and therefore all earthquake pairs were considered regardless of their magnitude. We investigated all the pairs of earthquakes that occurred at a centroid distance of less than 40, 60 or 90 km from each other and within a time interval of 200, 300 or 450 days, depending on their magnitude. We found 208 pairs of shallow, 31 of intermediate and 92 of deep events. To ascertain whether these earthquakes in pairs are not connected by chance, the possibility of their occurrence in an uncorrelated Poissonian catalogue was considered. It was assumed that in such a catalogue the inter-event time is exponentially distributed, the earthquake magnitude follows the Gutenberg-Richter relation, and the distribution of centroid distances between the events in pairs is controlled by its non-parametric kernel estimate. The probability of the appearance of the observed proportion of doublets of shallow earthquakes in the Poissonian catalogue was found to be very low. The low probability of occurrence in a semi-random catalogue, created by randomising centroid locations in the actual data set, also indicates major importance of the distance criterion used for a doublet specification. In general, shallow earthquakes tend to form pairs at shorter distances and within shorter time intervals than deep earthquakes. Both the distance and the time intervals do not depend on the magnitude of involved events. The largest number of pairs of deep earthquakes is observed at a depth of about 600 km, and the proportion of deep events associated with doublets against the number of all events increases with depth. From comparison of the focal mechanism of earthquakes in pairs, measured by the 3-D rotation angle, it follows that deep earthquakes forming pairs have a more diverse focal mechanism than shallow events; the rotation angle for three quarters of shallow pairs and only for about one third of deep pairs is reasonably small. The azimuth between two events forming a doublet is in about 60–65% of cases close to the strike of one of nodal planes of the first or the second event.

Parametric versus nonparametric tolerance regions in detection problems

A major problem in statistical quality control is to detect a change in the distribution of independent sequentially observed random vectors. The case of a Gaussian pre-change distribution has been extensively analyzed. Here we are concerned with the non-normal multivariate case. In this setup it is natural to use tolerance regions as detection tools. These regions are defined in terms of density level sets, which can be estimated in a plug-in fashion. Under a normal mixture model we compare, through a simulation study, the performance of such a detection scheme for two density estimators: a (parametric) normal mixture and a (nonparametric) kernel estimator. The problem of the bandwidth choice for the latter is addressed. We also obtain a result concerning the convergence rates of the error probabilities under a general parametric model. Finally, a real data example is discussed.

Demographic change and economic growth: An inverted-U shape relationship

The cross-country regression and non-parametric kernel estimation using the panel data from OECD countries over the 1960–2000 periods show the inverted-U shape relationship between demographic changes and economic growth; growth rates initially increase and then decrease with population aging.

Time series clustering based on forecast densities

A new clustering method for time series is proposed, based on the full probability density of the forecasts. First, a resampling method combined with a nonparametric kernel estimator provides estimates of the forecast densities. A measure of discrepancy is then defined between these estimates and the resulting dissimilarity matrix is used to carry out the required cluster analysis. Applications of this method to both simulated and real life data sets are discussed.

Predictive Inference for Integrated Volatility

In recent years, numerous volatility-based derivative products have been engineered. This has led to interest in constructing conditional predictive densities and confidence intervals for integrated volatility. In this paper, we propose nonparametric kernel estimators of the aforementioned quantities. The kernel functions used in our analysis are based on different realized volatility measures, which are constructed using the ex post variation of asset prices. A set of sufficient conditions under which the estimators are asymptotically equivalent to their unfeasible counterparts, based on the unobservable volatility process, is provided. Asymptotic normality is also established. The efficacy of the estimators is examined via Monte Carlo experimentation, and an empirical illustration based upon data from the New York Stock Exchange is provided.

Nonparametric Bayesian flood frequency estimation

A novel nonparametric Bayesian Monte-Carlo method is presented to estimate flood frequency. This method accommodates complex flood behaviors such as event clustering (repeated instances of similar magnitude floods) and can use varied data, such as gage and historical peak discharges, and paleohydrologic upper and lower bounds on peak discharge, while rigorously accounting for a wide variety of measurement uncertainties. In contrast to nonparametric kernel estimation approaches, the stochastic assumption is used to generate flood frequency models that span the data and provide about twice the number of degrees of freedom of the data. Each generated flood frequency model is scored using likelihoods that account for data measurement uncertainties. A parametric estimation approach ensures high precision because posterior sampling is known. However, parametric approaches can produce substantial biases because the classes of allowed flood frequency models are restricted. These biases are completely undetectable within a parametric paradigm. The nonparametric approach used here surrenders some precision in the pursuit of reduced bias and greater overall accuracy and assurance; it reveals the annual probabilities where discharge becomes unconstrained by the data, thereby eliminating unsubstantiated extrapolation. Parametric flood frequency estimation introduces strong extrapolation priors that make it difficult, if not impossible, to determine when flood frequency is not longer constrained by the data. Nonparametric and parametric flood frequency estimation using a demonstration data set shows that while parametric functions may sometimes provide reasonable fits to subsets of paleohydrologic data, parametric flood frequency estimates are likely to produce substantial biases over entire log cycles of annual exceedance probability, when using paleohydrologic data spanning thousands of years.

Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.

Kernel estimates of the mean and the volatility functions in a nonlinear autoregressive model with ARCH errors

In this paper, we deal with the nonparametric kernel estimation of the regression and volatility functions pertaining to nonlinear autoregressive model with ARCH errors. Under stationarity and ergodicity, we establish the strong uniform consistency and asymptotic normality of the estimators. Our results hold without any mixing condition and do not require the existence of marginal densities. Furthermore, rates of convergence are obtained.

Twicing Kernels and a Small Bias Property of Semiparametric Estimators

The purpose of this note is to show how semiparametric estimators with a small bias property can be constructed. The small bias property (SBP) of a semiparametric estimator is that its bias converges to zero faster than the pointwise and integrated bias of the nonparametric estimator on which it is based. We show that semiparametric estimators based on twicing kernels have the SBP. We also show that semiparametric estimators where nonparametric kernel estimation does not affect the asymptotic variance have the SBP. In addition we discuss an interpretation of series and sieve estimators as idempotent transformations of the empirical distribution that helps explain the known result that they lead to the SBP. In Monte Carlo experiments we find that estimators with the SBP have mean-square error that is smaller and less sensitive to bandwidth than those that do not have the SBP.

Testing Marshall-Lerner condition: a non-parametric approach

In this study, non-parametric kernel estimation technique has been employed to estimate import and export price elasticities for six developed countries. Based on the estimates of these elasticities Marshall-Lerner condition has been examined. In general the condition is only partially satisfied in the sub-sample periods. The results also suggest that the condition is more likely to be satisfied under fixed exchange rate regime.

Kernel Estimation of the Spectral Density of Stationary Random Closed Sets

SummaryA non‐parametric kernel estimator of the spectral density of stationary random closed sets is studied. Conditions are derived under which this estimator is asymptotically unbiased and mean‐square consistent. For the planar Boolean model with isotropic compact and convex grains, an averaged version of the kernel estimator is compared with the theoretical spectral density.

Bayesian marginal inference via candidate's formula

Computing marginal probabilities is an important and fundamental issue in Bayesian inference. We present a simple method which arises from a likelihood identity for computation. The likelihood identity, called Candidate's formula, sets the marginal probability as a ratio of the prior likelihood to the posterior density. Based on Markov chain Monte Carlo output simulated from the posterior distribution, a nonparametric kernel estimate is used to estimate the posterior density contained in that ratio. This derived nonparametric Candidate's estimate requires only one evaluation of the posterior density estimate at a point. The optimal point for such evaluation can be chosen to minimize the expected mean square relative error. The results show that the best point is not necessarily the posterior mode, but rather a point compromising between high density and low Hessian. For high dimensional problems, we introduce a variance reduction approach to ease the tension caused by data sparseness. A simulation study is presented.