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The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [ 6 ]
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.
A monotone matrix is nonsingular. [1]Proof: Let be a monotone matrix and assume there exists with =.Then, by monotonicity, and , and hence =. Let be a real square matrix. is monotone if and only if .
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ] [ 2 ] [ 3 ] This concept first arose in calculus , and was later generalized to the more abstract setting of order theory .
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
Monotone likelihood-functions are used to construct median-unbiased estimators, using methods specified by Johann Pfanzagl and others. [ 2 ] [ 3 ] One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators : The procedure holds for a smaller class of probability distributions than does the Rao–Blackwell ...