: A density function f(dot), with 1/f (dot) both Lebesgue-integrable, has a representation as an autoregressive spectral density. This representation is used to obtain new density autoregressive estimators of different orders p based on the empirical characteristic function of a sample of size n. The consistency of these new estimators is shown under varying conditions on the smoothness of f(dot).

This article attempts to describe an approach to statistical data analysis which is simultaneously parametric and nonparametric. Given a random sample X 1, …, X n of a random variable X, one would like (1) to test the parametric goodness-of-fit hypothesis H 0 that the true distribution function F is of the form F(x) = F0[(x − μ)/σ)], where F 0 is specified, and (2) when H 0 is not accepted, to estimate nonparametrically the true density-quantile function fQ(u) and score function J(u) = − (fQ)'(u). The article also introduces density-quantile functions, autoregressive density estimation, estimation of location and scale parameters by regression analysis of the sample quantile function, and quantile-box plots.

Consistency of the Autoregressive Method of Density Estimation.

Nonparametric Statistical Data Modeling

Publisher Summary The problem of spectral analysis of time series is clearly of great interest to the many applied scientists who use spectral analysis in their scientific research. It should be of great interest to statisticians, because it embodies the prototypes of two of the great problems of modern statistics: functional inference and modeling. The estimation of a function often has similar features to model identification, because a function can be parameterized exactly by a countable infinity of parameters. It should be noted that the inverse-correlation function is positive definite. However, the cepstral-correlation function is not. These new types of correlation functions are introduced, because they may provide more parsimonious parameterizations in the sense that they decay to 0 faster than does the correlation function. The use of autoregressive spectral densities as exact models and as approximating models for true spectral densities is often questioned by skeptical statisticians on the ground that their use in general is ad hoc and without theoretical justification.

Autoregressive Spectral Estimation.

: Functions used to describe the probability distributions of time series (both Gaussian and non-Gaussian) are introduced. The concept of type of a time series is defined. Autoregressive spectral densities are defined. Order determining criteria are motivated. through the concept of model identification by estimating information. An approach to empirical spectral analysis is suggested. (Author)

Consistency of the Autoregressive Method of Density Estimation.

Citations

Nonparametric Statistical Data Modeling

Autoregressive Spectral Estimation.

Autoregressive Spectral Estimation and Functional Inference.

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