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Every house-monotone method can be defined as a recursive function of the house size h. [1]: Thm.7.2 Formally, an apportionment method M ( t , h ) {\displaystyle M(\mathbf {t} ,h)} is house-monotone and satisfies both quotas if-and-only-if it is constructed recursively as follows (see mathematics of apportionment for the definitions and notation):
The issue of sample size for Mokken scaling is largely unresolved. Work using simulated samples and varying the item quality in the scales (Loevinger's coefficient and the correlation between scales) suggests that, where the quality of the items is high that lower samples sizes in the region of 250–500 are required compared with sample sizes ...
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 ]
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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 ...
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 .