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Local regression and likelihood. (English) Zbl 0929.62046
Statistics and Computing. New York, NY: Springer. xiii, 290 p. DM 129.00; öS 942.00; sFr 117.50; £49.50; \$ 64.95 (1999).

Suppose a dataset consists of $n$ pairs of observations $\left({x}_{1},{Y}_{1}\right),\phantom{\rule{4pt}{0ex}}\left({x}_{2},{Y}_{2}\right),...,\left({x}_{n},{Y}_{n}\right)$, where $x$ is a (nonrandom) predictor variable (real or vector valued) and $Y$ is a (random) response variable related to the predictor variable. The problem is to predict $Y$ for a fixed value $x$ which may not belong to the set $\left\{{x}_{1},{x}_{2},...,{x}_{n}\right\}$. It is obvious that a kind of interpolation (or perhaps extrapolation) between the points ${x}_{1},{x}_{2},...,{x}_{n}$ is needed, and that those of the points which are nearer to $x$ should play a greater role. If a model of the form $Y=\mu \left(x\right)+\epsilon$ with a deterministic function $\mu$ and random error $\epsilon$ is assumed, the problem is known as that of local regression.

Ch. 2 of the book gives details. For a model of the form “$Y$ has a probability density function $f\left(y,\mu \left(x,\theta \right)\right)$” with given form of $f$ and $\mu \left(x,\theta \right)$, a local maximum likelihood estimate for the parameter $\theta$ may be conctructed. The Local Likelihood Model is discussed in Ch. 4. Similarly one can formulate the problem of local density estimation (Ch. 5), local Survival and Failure Time Analysis (Ch. 6), and so on.

The book gives us a complete and uptodate review of the state of the art. Practical applications and computer software (LOCFIT) are presented in detail.

##### MSC:
 62G08 Nonparametric regression 62-02 Research monographs (statistics) 62F10 Point estimation 62J99 Linear statistical inference 62J02 General nonlinear regression 65C60 Computational problems in statistics
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