By Tormod Naes

ISBN-10: 0952866625

ISBN-13: 9780952866626

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Example text

If g is a Borelmeasurable function on [a, b], its integral with respect to the signed meab b sure µf , denoted by a g(x)df (x) or a g(x)dµf (x), is defined as the difference of the integrals with respect to the measures µF1 and µF2 , b b a b g(x)dµF1 (x) − g(x)df (x) = a g(x)dµF2 (x). a It is called the Lebesgue-Stieltjes integral of g with respect to f . Example. For an interval I, let In = I ∩ [−n, n]. Define mn (I) as the length of In . As in the first example, mn is a σ-additive function. Thus mn gives rise to a measure on the Borel sets of the real line, which will be denoted by λn and referred to as the Lebesgue measure on the segment [−n, n].

F ∼ g implies that g ∼ f . 3. f ∼ g and g ∼ h imply that f ∼ h. It follows from general set theory that the set of measurable functions can be viewed as a union of non-intersecting subsets, the elements of the same subset being all equivalent, and the elements which belong to different subsets not being equivalent. We next introduce the Lp (Ω, F, µ) spaces, whose elements are some of the equivalence classes of measurable functions. We shall not distinguish between a measurable function and the equivalence class it represents.

The distribution function Fωn (t) is called the empirical distribution function. There are many problems in mathematical statistics where it is needed to estimate F (t) by means of the observed empirical distribution function. Such estimates are based on the following theorem. 9. (Glivenko-Cantelli Theorem) If F (t) is continuous, then for any ε > 0 P(sup |F n (t) − F (t)| < ε) → 1 as n → ∞. t∈R Proof. For each t the value F n (t) is a random variable and Fωn (t) = {i : ωi ≤ t} = k. Therefore, P(F n (t) = k n if n!

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A user-friendly guide to multivariate calibration and classification by Tormod Naes


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