Bounding Standard Gaussian Tail Probabilities

We review various inequalities for Mills' ratio (1 - Φ)= Ø, where Ø and Φ denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.

Regularity conditions in the realisability problem in applications to point processes and random closed sets

We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive ex-tension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extens ion can be chosen to possess some invariance properties. The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order q generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for q = 2 and q = 4. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and U-statistics on the Poisson space.

Higher order elicitability and Osband’s principle

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance, this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the “correct” quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.

Statistical analysis of the goodness of classical factor analysis regression (CFAR) /

Cover title.

On the problem of estimating the number of schools of fish.

"A paper presented at the joint meeting of the Bometric Society and the institute of Mathematical Statistics, November 27, 1948, at Seattle, Washington."

Statistical theory of extreme values and some practical applications; a series of lectures.

Bibliography: p. 48-49.

Statistical decision functions.

Mode of access: Internet.

Notes on the theory of statistical estimation and of testing hypotheses : lectures /

"Notes prepared by Ralph J. Brookner."

Notes on decision theory.

"These notes were prepared at Cornell University from lectures given in Mathematics 742 in 1953 ... and were recorded by J. Sacks. The notes are intended only for the convenience of the student and are not intended to be considered as a text on the subject matter. These notes were mimeographed under a contract with the Office of Naval Research."

Estimation after preliminary tests of significance.

Bibliography: p. 185-186.

Statistical techniques based on probablistic models /

Includes list of the author's publications to December, 1980: p. 119-122-III (inserted leaves 122-I-III, in typescript, cover 1962 to 1980).

Theory of probability with applications /

At the head of title page: Princeton University Station, Division 2, National Defense Research Committee.

The moments of the non-central t-distribution,

"This work was supported by the Office of Naval Reseaarch under Contract Nonr. 404(16)."