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A characterization of mixed Poisson processes in terms of disintegrations is proven. As a consequence some further characterizations of such processes via claim interarrival processes, martingales and claim measures are obtained.
We consider the classic problem of estimating T, the total number of species in a population, from repeated counts in a simple random sample. We first show that the frequently used Chao-Lee estimator can in fact be obtained by Bayesian methods with a Dirichlet prior, and then use such clarification to develop a new estimator; numerical tests and some real experiments show that the new estimator is...
Specific features of the regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing are discussed. It is shown that the classical Bayesian statement of the problem of statistical hypotheses testing in the form of an unconditional optimizing problem is a special case of conditional Bayesian problems of hypotheses testing set in the form of conditional optimizing...
We obtain an inequality complementary to the Cauchy-Schwarz inequality in Hilbert space. The inequalities involving first three powers of a self-adjoint operator are derived. The inequalities include the bounds for the third central moment, as a special case. It is shown that an upper bound for the spectral radius of a matrix is a root of a particular cubic equation, provided all eigenvalues are positive.
Spatial linear models and the corresponding likelihood-based statistical inference are important tools for the analysis of spatial lattice data and have been applied in a wide range of disciplines. However, understanding of the asymptotic properties of maximum likelihood estimates is limited. Here we consider a unified asymptotic framework that encompasses increasing domain, infill, and a combination...
The asymptotic distribution of the Lasso estimator for regression models with independent errors has been investigated by Knight and Fu (2000). In this note we extend these results to regression models with a general weak dependence structure. We determine the asymptotic distribution of the Lasso estimator when the number of parameters M is fixed and the number of observations, n, converges to infinity...
It is known that bootstrapping maximum for estimating the endpoint of a distribution function is inconsistent and subsample bootstrap method is needed. Under an extreme value condition, some other estimators for the endpoint have been studied in the literature, which are preferrable to the maximum in regular cases. In this paper, we show that the full sample bootstrap method is consistent for the...
For an invariant statistical model we consider the induced right Haar prior distribution and the resulting right Haar posterior. Using this posterior distribution, HPD (highest posterior density) regions of constant posterior probability are constructed and shown to exhibit probability matching. Further, HPD regions for equivariant functions of the parameter are also shown to exhibit probability matching...
Results of Jain and Khan (1979) and Khan and Jain (1978) on the time to first emptiness of a reservoir are generalized to include the case of defective random variables where the mass at ∞ can be positive. The assumption of an underlying exponential family is not needed — the general condition is infinite divisibility and closure under convolutions. The support of the distributions can be nonnegative...
Brown and Zhao (2012) (Sankhyā, Series A, Volume 64, pp 611–625) developed a new test for the Poisson distribution and compared it with the likelihood ratio test (LRT) and some other tests. They claimed that under the null hypothesis, the asymptotic distribution of the LRT statistic was . In this paper we derive the asymptotic distribution of the LRT statistic reported in Brown...
A semiparametric model is considered where the functional of interest is a shift parameter between two curves. A surprising example is provided where two at first sight indistinguishable Gaussian priors lead to quite different behaviours of the posterior distribution of the functional of interest. This phenomenon also illustrates that a condition introduced in Castillo (2012) of the approximation...
A classical characterization result, which can be traced back to Gauss, states that the maximum likelihood estimator (MLE) of the location parameter equals the sample mean for any possible univariate samples of any possible sizes n if and only if the samples are drawn from a Gaussian population. A similar result, in the two-dimensional case, is given in von Mises (1918) for the Fisher-von Mises-Langevin...
We show that in a Polish space if {Pn} is a sequence of probability measures then the existence of for every bounded continuous function guarantees the existence of a probability P such that Pn converges weakly to P.
A betting game establishes a sense in which confidence measures, confidence distributions in the form of probability measures, are the only reliable inferential probability distributions. In addition, because confidence measures are Kolmogorov probability distributions, they are as coherent as Bayesian posterior distributions in their avoidance of sure loss under the usual Dutch-book betting game...
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