Mathematische Methoden für Ökonomen

Abstract

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Some Technical Remarks on Negations of Discrete Probability Distributions and Their Information Loss

“…. , 1/n) gives a local maximum of (24). For α = 0.2, we get p * = 0.0235 such that 1/n < 0.0235 for n > 42.…”

“…φ is strictly concave for 1/2 ≤ α ≤ 1. Now, we want to show that (24), depending on n, has a local maximum or a local minimum for (1/n, . .…”

“…For α < 1, φ (p) < 0 for p > p * and φ (p) > 0 for p < p * . By using this property, a similar argument can be applied to show that the uniform distribution can be a point at which (24) has either a local maximum or a local minimum.…”

“…The discussion so far seems to give the impression that negation leads to an information loss for all φ-entropies. To show that this impression is misleading, let us again consider Leik entropy ( 23) and the entropy (24).…”

“…, n + 1.Proof. The statement is true for m = 3: Laplace expansion[24] (p. 292) along the third row givesΛ 3 = 1(−x 1 ) + x 2 (−1) = −x 1 − x 2 = ∑ {i 1 ∈ {1, 2} 1 } = {1, 2}.Assume that the statement is true for m. Laplace expansion of the upper left m + 1 × m + 1-matrix along the m + 1-th row givesΛ m+1 = −i=1 x i + x m Λ m Inserting Λ m = −x 1 x 2 . .…”