Collection Of Probability Measures Research Articles

On a robust risk measurement approach for capital determination errors minimization

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. An empirical illustration is carried out to demonstrate the use of our approach in capital determination.

Transience and recurrence of state-dependent branching processes with an immigration component

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P(i)k} of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μi, the variances σ2i, and the (2 + γ)th central absolute moments β2+γ,i of the {P(i)k}'s which ensure transience of recurrence of {Zn}.

Transience and recurrence of state-dependent branching processes with an immigration component

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P (i) k } of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μ i , the variances σ2 i , and the (2 + γ)th central absolute moments β2+γ,i of the {P (i) k }'s which ensure transience of recurrence of {Zn }.

On the Existence of a Minimal Sufficient Subfield

Let $(X, S)$ be a measurable space and $M$ a collection of probability measures on S. Pitcher (Pacific J. Math. (1965), pages 597-611) introduced a condition called compactness on the statistical structure $(X, S, M)$, more general than domination by a fixed $\sigma$-finite measure. Under this condition he gave a construction of a minimal sufficient subfield. In this paper a counterexample invalidating this construction is presented. We give a nonconstructive proof of the existence of a minimal sufficient subfield under a condition slightly weaker than compactness. The proof proceeds by considering the intersection of an uncountable collection of sufficient subfields, and relies on a martingale convergence theorem with directed index set, due to Krickeberg.