Numerical Analysis for Statisticians (Statistics and by Kenneth Lange

By Kenneth Lange

Numerical research is the research of computation and its accuracy, balance and infrequently its implementation on a working laptop or computer. This booklet specializes in the foundations of numerical research and is meant to equip these readers who use statistics to craft their very own software program and to appreciate the benefits and drawbacks of alternative numerical tools.

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Obviously, the relation f(x) = o(g(x)) implies the weaker relation f(x) = O(g(x)). Finally, K. 1007/978-1-4419-5945-4_4, © Springer Science+Business Media, LLC 2010 39 40 4. Asymptotic Expansions if limx→x0 f(x)/g(x) = 1, then f(x) is said to be asymptotic to g(x). This is usually written f(x) g(x). In many problems, the functions f(x) and g(x) are defined on the integers {1, 2, . } instead of on an interval I, and x0 is taken as ∞. For example, on I = (1, ∞) one has ex = O(sinh x) as x → ∞ because ex ex − e−x 2 = 2 1 − e−2x ≤ 2 .

7) as Pr(X ≤ k − 1) = 1 − P (k, λ). The most illuminating proof of this result relies on constructing a Poisson process of unit intensity on [0, ∞). In this framework Pr(X ≤ k − 1) is the probability of k − 1 or fewer random points on [0, λ]. Since the waiting time until the kth random point in the process follows a gamma distribution with parameters a = k and b = 1, the probability of k − 1 or fewer random points on [0, λ] coincides with the probability 1 − P (k, λ) that the kth random point falls beyond λ.

Continued Fraction Expansions 2 F 1 (a = + 1, b + 1, c + 2; x) F (a, b + 1, c + 1; x) 2 1 1 . 7). 8). 9) with d2n+1 = d2n+2 = (a + n)(c − b + n) (c + 2n)(c + 2n + 1) (b + n + 1)(c − a + n + 1) − (c + 2n + 1)(c + 2n + 2) − for n ≥ 0. 9) is most useful when b = 0, for then 2 F 1 (a, b, c; x) = 1. For instance, the hypergeometric expansion of (1 − x)−a has coefficients d1 = −a d2n+1 = − d2n+2 = (a + n) , 2(2n + 1) (n + 1 − a) , − 2(2n + 1) n≥1 n ≥ 0. 7) continues to hold for b = c = 0, provided 2 F 1 (a, 0, 0; x) and the ratio (c − b)/c are both interpreted as 1.

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