## A Bayesian model for local smoothing in kernel density by Brewer M. J. PDF

By Brewer M. J.

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In the above-mentioned article of 1774, Laplace treated approximation problems in an analytical style closely related to that of Euler. Laplace discussed the behavior of the peak with an “infinitely large” parameter, carefully considering “infinitely” large or small quantities. In his later work, however, he abandoned the “Eulerian” style of calculating with infinite quantities of different gradations and, influenced by Lagrange’s algebraic analysis, developed a special algebraic-algorithmic style dealing primarily with formal series expansions, as we have just seen in connection with the Gamma function.

3 The Role of the Central Limit Theorem in Poisson’s Work As we will see in the following, Poisson’s work on the CLT was based on Laplace’s ideas on the one hand; on the other hand, however, Poisson’s discussion of new analytical aspects paved the way toward a more rigorous treatment of the CLT. 1 Poisson’s Version of the Central Limit Theorem Poisson’s results concerning the CLT can be summarized in modern terminology essentially as follows: Let X1 ; : : : ; Xs be a great number of independent random variables with density functions which decrease sufficiently fast (Poisson did not specify exactly how fast) 34 2 The Central Limit Theorem from Laplace to Cauchy as their arguments tend to ˙1.

In 1856 Anton Meyer8 submitted a proof of the CLT in the special case of twovalued random variables to the Academy in Brussels. Meyer’s proof was not based 8 Meyer was the author of a rather influential treatise of probability and error theory [Meyer 1874], which was also translated into German [Meyer 1874/79] and constitutes an important source for the state of the art at the beginning of the last quarter of the 19th century. 1 Laplace’s Central “Limit” Theorem 25 on the usual procedure which can be traced back to de Moivre, and which had also been elaborated in Laplace’s Théorie analytique.

### A Bayesian model for local smoothing in kernel density estimation by Brewer M. J.

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