By William A. Veech

Author William A. Veech, the Edgar Odell Lovett Professor of arithmetic at Rice collage, provides the Riemann mapping theorem as a unique case of an life theorem for common masking surfaces. His specialize in the geometry of advanced mappings makes common use of Schwarz's lemma. He constructs the common overlaying floor of an arbitrary planar zone and employs the modular functionality to boost the theorems of Landau, Schottky, Montel, and Picard as results of the life of sure coverings. Concluding chapters discover Hadamard product theorem and major quantity theorem.

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Let t! = (El', 2£, P B, e} be a statistical experiment and let P B be absolutely continuous with respect to measure Jl. on 2£ and dPB/dJl. = p(x; 0). Let X be the observation generating 8. The function p(X; 0) is called the likelihood function corresponding to 8 and X; thus P(X; 0) is a random function of 0 defined on e c: Rk. 5) Beec is called the maximum (or maximal) likelihood estimator for the parameter 0 (based on the observation X). 5) has no solution*; however, below we shall consider the case when the solution does exist (without specifically mentioning this fact).

Jn, k = X. 2. I ~ 2(1 + In) = an· Thus the standard estimator X is better than the minimax in the region Ip - tl > an and is inferior in the region I p - tl < an. For small n on the larger part of the interval [0, 1] the estimator T is superior over X, but as n increases, X is preferable over T everywhere except for the interval whose length approaches zero as n- 1/4. 29 3 Some Examples 4 Estimation of the Mean of a Uniform Distribution Let Xj possess the uniform distribution on the interval [0 - t,o means that X j possesses t e density on the real line given by + tJ.

An estimator for 0 will be sought among the estimators of the form Bn = X + n- 1g(X), where the function g(x) = (gh ... , gk): Rk - Rk. : = -2n- 2Eo (X)) - n-2Eolg(XW. Assume now that the function 9 can be represented in the form g(x) = grad (In q>(x)), where q> is a twice differentiable function from Rk into RI. Then ° ° ) k og. k (1---q>(x) = -Igl + -Llq>, 1 L-' (x) = LOXi OXi q>(X) OXi q> 1 1 2 where Ll = L~ 02/oxf is the Laplace operator. Consequently, for the above choice of g, EolX - 01 2 - Eolt1i'In - {I -} 01 2 = n- 2 Eoigi 2 - n- 2 Eo q>(X) Llq>(X) .

### A second course in complex analysis by William A. Veech

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