148 research outputs found
Local proper scoring rules of order two
Scoring rules assess the quality of probabilistic forecasts, by assigning a
numerical score based on the predictive distribution and on the event or value
that materializes. A scoring rule is proper if it encourages truthful
reporting. It is local of order if the score depends on the predictive
density only through its value and the values of its derivatives of order up to
at the realizing event. Complementing fundamental recent work by Parry,
Dawid and Lauritzen, we characterize the local proper scoring rules of order 2
relative to a broad class of Lebesgue densities on the real line, using a
different approach. In a data example, we use local and nonlocal proper scoring
rules to assess statistically postprocessed ensemble weather forecasts.Comment: Published in at http://dx.doi.org/10.1214/12-AOS973 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Strictly and non-strictly positive definite functions on spheres
Isotropic positive definite functions on spheres play important roles in
spatial statistics, where they occur as the correlation functions of
homogeneous random fields and star-shaped random particles. In approximation
theory, strictly positive definite functions serve as radial basis functions
for interpolating scattered data on spherical domains. We review
characterizations of positive definite functions on spheres in terms of
Gegenbauer expansions and apply them to dimension walks, where monotonicity
properties of the Gegenbauer coefficients guarantee positive definiteness in
higher dimensions. Subject to a natural support condition, isotropic positive
definite functions on the Euclidean space , such as Askey's and
Wendland's functions, allow for the direct substitution of the Euclidean
distance by the great circle distance on a one-, two- or three-dimensional
sphere, as opposed to the traditional approach, where the distances are
transformed into each other. Completely monotone functions are positive
definite on spheres of any dimension and provide rich parametric classes of
such functions, including members of the powered exponential, Mat\'{e}rn,
generalized Cauchy and Dagum families. The sine power family permits a
continuous parameterization of the roughness of the sample paths of a Gaussian
process. A collection of research problems provides challenges for future work
in mathematical analysis, probability theory and spatial statistics.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP06 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Copula Calibration
We propose notions of calibration for probabilistic forecasts of general
multivariate quantities. Probabilistic copula calibration is a natural analogue
of probabilistic calibration in the univariate setting. It can be assessed
empirically by checking for the uniformity of the copula probability integral
transform (CopPIT), which is invariant under coordinate permutations and
coordinatewise strictly monotone transformations of the predictive distribution
and the outcome. The CopPIT histogram can be interpreted as a generalization
and variant of the multivariate rank histogram, which has been used to check
the calibration of ensemble forecasts. Climatological copula calibration is an
analogue of marginal calibration in the univariate setting. Methods and tools
are illustrated in a simulation study and applied to compare raw numerical
model and statistically postprocessed ensemble forecasts of bivariate wind
vectors
Onα-Symmetric Multivariate Characteristic Functions
AbstractAnn-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the formϕ((|u1|α+…+|un|α)1/α). We study the classesΦn(α) of all admissible functionsϕ:[0, ∞)→R. It is known that members ofΦn(2) andΦn(1) are scale mixtures of certain primitivesΩnandωn, respectively, and we show thatωnis obtained fromΩ2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: Ifϕ(0)=1,ϕis continuous, limt→∞ϕ(t)=0, andϕ(2n−2)(t) is convex, thenϕ∈Φn(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution
Predicting Inflation: Professional Experts Versus No-Change Forecasts
We compare forecasts of United States inflation from the Survey of
Professional Forecasters (SPF) to predictions made by simple statistical
techniques. In nowcasting, economic expertise is persuasive. When projecting
beyond the current quarter, novel yet simplistic probabilistic no-change
forecasts are equally competitive. We further interpret surveys as ensembles of
forecasts, and show that they can be used similarly to the ways in which
ensemble prediction systems have transformed weather forecasting. Then we
borrow another idea from weather forecasting, in that we apply statistical
techniques to postprocess the SPF forecast, based on experience from the recent
past. The foregoing conclusions remain unchanged after survey postprocessing
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