Nonparametric density estimation with grouped observations

A smooth and detailed distribution is fitted to coarsely grouped frequency data by a nonparametric approach, based on penalized maximum likelihood. The estimated distribution conserves mean and variance of the data. The numerical solution is described and a compact and simplified algorithm is given. The procedure is applied to two empirical datasets.     

Nonparametric estimation and inference on conditional quantile processes

This paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process. Two estimators based on local linear regressions are proposed. The first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity. The bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity. For inference, the paper first establishes a uniform Bahadur representation and then shows that the two estimators converge weakly to the same limiting Gaussian process. As an empirical illustration, the paper considers a dataset from Project STAR and delivers two new findings.     

Nonparametric density estimation for symmetric distributions by contaminated data

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. We consider three estimates for the pdf of primary component: a naive one, a symmetrized naive estimate and a symmetrized estimate with adaptive weights. Asymptotic behavior and small sample performance of the estimates are investigated. Some rules of thumb for bandwidth selection are discussed.     

Nonparametric covariance estimation in multivariate distributions

The estimation problem of the unknown covariance matrix of a multivariate distribution with the known mean is studied under a matrix-valued quadratic loss function. The conditions on the sample sizes for the best unbiased estimator to have a smaller risk than the sample covariance matrix is established. The former estimator is completely (without exceptional sets of Lebesgue measure zero) characterized by its expectation in the class of all multivariate distributions with zero mean and finite fourth moments. Received: November 1998     

Nonparametric estimation in case of endogenous selection

This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are obtained. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration.     

Nonparametric least squares estimation in derivative families

Cost function estimation often involves data on a function and a family of its derivatives. Such data can substantially improve convergence rates of nonparametric estimators. We propose series-type estimators which incorporate the various derivative data into a single nonparametric least-squares procedure. Convergence rates are obtained and it is shown that for low-dimensional cases, much of the beneficial impact is realized even if only data on ordinary first-order partials are available. In instances where root-n$n$ consistency is attained, smoothing parameters can often be chosen very easily, without resort to cross-validation. Simulations and an illustration of cost function estimation are included.     

Nonparametric estimation of a discontinuity in regression

We propose and study a new method to nonparametrically estimate a discontinuity of a regression function. The optimal rate of convergence n ⁻¹ is obtained under minimal assumptions. No smoothing is required.     

Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Let (X, Y) be a pair of random variables withsupp(X)⊆[0,1] ^l andEY ²<∞. Letm ^* be the best approximation of the regression function of (X, Y) by sums of functions of at mostd variables (1≤d≤l). Estimation ofm ^* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at mostd variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X, Y) the weak and strongL ₂-consistency of the estimate is shown. Furthermore, for everyp≥1 and every distribution of (X, Y) withsupp(X)⊆[0,1] ^l ,y bounded andm ^* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate      

Regression density estimation using smooth adaptive Gaussian mixtures

We model a regression density flexibly so that at each value of the covariates the density is a mixture of normals with the means, variances and mixture probabilities of the components changing smoothly as a function of the covariates. The model extends the existing models in two important ways. First, the components are allowed to be heteroscedastic regressions as the standard model with homoscedastic regressions can give a poor fit to heteroscedastic data, especially when the number of covariates is large. Furthermore, we typically need fewer components, which makes it easier to interpret the model and speeds up the computation. The second main extension is to introduce a novel variable selection prior into all the components of the model. The variable selection prior acts as a self-adjusting mechanism that prevents overfitting and makes it feasible to fit flexible high-dimensional surfaces. We use Bayesian inference and Markov Chain Monte Carlo methods to estimate the model. Simulated and real examples are used to show that the full generality of our model is required to fit a large class of densities, but also that special cases of the general model are interesting models for economic data.     

Nonparametric maximum likelihood estimation in a non locally compact setting

Maximum likelihood estimation is considered in the context of infinite dimensional parameter spaces. It is shown that in some locally convex parameter spaces sequential compactness of the bounded sets ensures the existence of minimizers of objective functions and the consistency of maximum likelihood estimators in an appropriate topology. The theory is applied to revisit some classical problems of nonparametric maximum likelihood estimation, to study location parameters in Banach spaces, and finally to obtain Varadarajan’s theorem on the convergence of empirical measures in the form of a consistency result for a sequence of maximum likelihood estimators. Several parameter spaces sharing the crucial compactness property are identified. This research was supported by grants from the National Sciences and Engineering Research Council of Canada and the Fonds FCAR de la Province de Québec.     

Mean estimation bias in least squares estimation of autoregressive processes

Estimation of the parameters of an autoregressive process with a mean that is a function of time is considered. Approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived. For the case of a mean function that is a polynomial in time, a reparameterization that isolates the bias is given. Using the approximate expressions, a method of modifying the least squares estimator is proposed. A Monte Carlo study of the second-order autoregressive process is presented. The Monte Carlo results agree well with the approximate theory and, generally speaking, the modified least squares estimators performed better than the least squares estimator. For the second-order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one, two and three periods in the future.     

Structural estimation of jump-diffusion processes in macroeconomics

This paper shows how to solve and estimate a continuous-time dynamic stochastic general equilibrium (DSGE) model with jumps. It also shows that a continuous-time formulation can make it simpler (relative to its discrete-time version) to compute and estimate the deep parameters using the likelihood function when non-linearities and/or non-normalities are considered. We illustrate our approach by solving and estimating the stochastic AK and the neoclassical growth models. Our Monte Carlo experiments demonstrate that non-normalities can be detected for this class of models. Moreover, we provide strong empirical evidence for jumps in aggregate US data.     

Nonparametric methods in sample surveys. Application to the estimation of cancer prevalence

We address the problem of the estimation of the population mean and the distribution function using nonparametric regression. These methods are being used in a wide range of settings and areas of research. In particular, they are a good alternative to other classical methods in the survey sampling context, since they work under the assumption that the underlying regression function is smooth. Some relevant nonparametric regression methods in survey sampling are presented. Data on breast cancer prevalence derived from 40 European countries are used to study the application of the nonparametric estimators to the estimation of cancer prevalence. Result derived from an empirical study show that nonparametric estimators have a good empirical performance in this study on cancer prevalence.     

Gains in efficiency from joint estimation of systems of autoregressive-moving average processes

The paper examines gains in efficiency from joint estimation of systems of ARMA processes where cross-correlation is due to contemporaneous correlation among disturbances. The asymptotic variance of joint estimates is derived and it involves only variances and covariances among purely AR processes corresponding to the AR and MA parts of the constituent processes. Small sample gains are evaluated by Monte Carlo methods. Application of joint estimation to two short-term interest rates is shown to result in more accurate post-sample predictions relative to both univariate models and the FMP econometric model.     

On the relative efficiency of estimators which include the initial observations in the estimation of seemingly unrelated regressions with first-order autoregressive disturbances

The generalized least squares estimator for a seemingly unrelated regressions model with first-order vector autoregressive disturbances is outlined, and its efficiency is compared with that of an approximate generalized least squares estimator which ignores the first observation. A scalar index for the loss of efficiency is developed and applied to a special case where the matrix of autoregressive parameters is diagonal and the regressors are smooth. Also, for a more general model, a Monte Carlo study is used to investigate the relative efficiencies of various estimators. The results suggest that Maeshiro (1980) has overstated the case for the exact generalized least squares estimator, because, in many circumstances, it is only marginally better than the approximate generalized least squares estimator.     

Quantile interval estimation in finite population using a multivariate ratio estimator

A new method to derive confidence intervals for quantiles in a finite populations is presented. This method uses multi-auxiliary information through a multi-variate ratio type estimator of the population distribution function.     

Duration dependence in US business cycles: An analysis using the modulated power law process

The modulated power law process is used to analyze the duration dependence in US business cycles. The model makes less restricting assumptions than traditional models do and measures both the local and global performance of business cycles. The results indicate evidence of positive duration dependence in the U.S. business cycles. Structural change after WWII in both expansion and contraction phases of business cycles is also documented. Hypothesis tests confirm that the model fits US business cycles.      

Minimisation of uncertainty in decision-making processes using optimised probabilistic Fuzzy Cognitive Maps: A case study for a rural sector

Several studies have focused on methods of increasing system and uncertainty knowledge for socio-economic and environmental policies; however, the nonlinearity and dynamism of real world increase the gap between uncertainty depiction and its evaluation in policy strategies. This work attempts to implement a methodology that is able to minimise uncertainty in decision support tools related to rural planning and management. Fuzzy Cognitive Maps, the Dempster–Shafer theory and nonlinear optimisation were applied to achieve the above-mentioned goal. The method was tested to describe suitable policies and intervention strategies to address the effects of the recent economic crisis in the agricultural sector.     

Studying the association between air pollution and lung cancer incidence in a large metropolitan area using a kernel density function

In the absence of patient-specific data, composite level data are often used in epidemiological studies. However, since individual exposure levels cannot accurately be inferred from aggregate data, such an approach may lead to erroneous estimates of health effects of potential environmental risk factors. In the present study, we attempt to address this information-loss problem by using the “kernel density function”, which estimates the intensity of events across a surface, by calculating the overall number of cases situated within a given search radius from a target point. The present paper illustrates the use of this analytical technique for a study of association between the geographical distributions of lung cancer cases and SO₂ air pollution estimates in the Greater Haifa Metropolitan Area (GHMA). In the analysis, the results obtained by kernel smoothing are contrasted with those obtained by areal aggregation techniques more commonly used in empirical studies.     

The fundraising efficiency in U.S. non-profit art organizations: an application of a Bayesian estimation approach using the stochastic frontier production model

This article examines how efficient art organizations are in raising funds from private giving. We measure fundraising efficiency using a Bayesian estimation approach using the stochastic frontier production model. We show that fundraising efficiencies are generally quite low for art organizations in the U.S. when private giving is only considered as a fundraising output; however, when the effect of fundraising on ticket sales is considered, fundraising efficiencies improve substantially. We also show that government grants have a negative impact on fundraising efficiency and therefore partially crowd out private giving.