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**11 - 13**of**13**### Splines in Statistics*

"... ABSTRACT. Spline functions are particularly appropriate in fitting a smooth non-parametric model to noisy data. The usc of spline functions in non-parametric density estimation and spectral estimation is surveyed. The requisite spline theory background is also developed. Key Words and Phrases. ..."

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ABSTRACT. Spline functions are particularly appropriate in fitting a smooth non-parametric model to noisy data. The usc of spline functions in non-parametric density estimation and spectral estimation is surveyed. The requisite spline theory background is also developed. Key Words and Phrases.

### Assume that Y 1

"... are i.i.d. observations from a distribu-tion with a continuous density function g. Let yE(-0. A density n function estimator which can be written in the form 1 K(y,Yj) is called a kernel estimator with kernel K. Whittle (1958) proposed selecting a kernel estimator of a density, using as criterion ex ..."

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are i.i.d. observations from a distribu-tion with a continuous density function g. Let yE(-0. A density n function estimator which can be written in the form 1 K(y,Yj) is called a kernel estimator with kernel K. Whittle (1958) proposed selecting a kernel estimator of a density, using as criterion expected square error, and observed that implementation of this approach requires the specification of only first and second moments of the joint distri-bution of the values of the density function g(.) at the various values of its argument. Hartigan-(1969) described Whittle's approach as a "linear Bayes " approach. Brunk (1980) preferred to refer to it as "Bayesian Least Squares " because, as for ordinary least squares, both input and output involve only first and second moments. In this thesis, Brunk (1980)'s Bayesian Least Squares method has been slightly modified and applied to the estimation of univariate and mixing densities. For the univariate density estimation, we begin with a prescribed prior mean probability density g of g, and let {P r (y)} be a pre--scribed sequence of functions orthonormal w.r.t. g o, with P

### Nonparametric Maximum Likelihood Estimation of Probability Measures: Existence

, 2004

"... This paper formulates the nonparametric maximum likelihood es-timation of probability measures and generalizes the consistency result on the maximum likelihood estimator (MLE). We drop the indepen-dence assumption on the underlying stochastic process and replace it with the assumption that the stoch ..."

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This paper formulates the nonparametric maximum likelihood es-timation of probability measures and generalizes the consistency result on the maximum likelihood estimator (MLE). We drop the indepen-dence assumption on the underlying stochastic process and replace it with the assumption that the stochastic process is stationary and ergodic. The present proof employs Birkhoff’s ergodic theorem and the martingale convergence theorem. The main result is applied to the parametric and nonparametric maximum likelihood estimation of density functions.