Motivated by the problem of setting prediction intervals in time series analysis, we suggest two new methods for conditional distribution estimation. The first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation. It produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1. The second method involves an adjusted form of the Nadaraya–Watson estimator. It preserves the bias and variance properties of a class of second-order estimators introduced by Yu and Jones but has the added advantage of always being a distribution itself. Our methods also have application outside the time series setting; for example, to quantile estimation for independent data. This problem motivated the work of Yu and Jones.

/pdf/methods-for-estimating-a-conditional-distribution-function-6ud2i1s2wk.pdf

Methods for Estimating a Conditional Distribution Function

Motivated by the problem of setting prediction intervals in time series analysis, we suggest two new methods for conditional distribution estimation. The first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation. It produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1. The second method involves an adjusted form of the Nadaraya--Watson estimator. It preserves the bias and variance properties of a class of second-order estimators introduced by Yu and Jones but has the added advantage of always being a distribution itself. Our methods also have application outside the time series setting; for example, to quantile estimation for independent data. This problem motivated the work of Yu and Jones.

Methods for estimating a conditional distribution function

This paper develops a nonparametric density estimator with parametric overtones. Suppose f(x, θ) is some family of densities, indexed by a vector of parameters θ. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form f(x, o(x)), thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression.

/pdf/locally-parametric-nonparametric-density-estimation-1pcc26bnpc.pdf

Locally parametric nonparametric density estimation

Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.

/pdf/local-maximum-likelihood-estimation-and-inference-4jvtnh45ul.pdf

Local maximum likelihood estimation and inference

We suggest two improved methods for conditional density estimation. The first is based on locally fitting a log-linear model, and is in the spirit of recent work on locally parametric techniques in...

http://eprints.lse.ac.uk/6092/1/Nonparametric_estimation_and_symmetry_tests_for_conditional_density_functions(LSERO).pdf

Nonparametric estimation and symmetry tests for conditional density functions

By drawing an analogy with likelihood for censored data, a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space. Resulting methods can be used for assessing local departures from a parametric model, and for semiparametric density estimation. Some theory, and three examples, is given

J. B. Copas

Papers

Local Likelihood Based on Kernel Censoring