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August 22, 2011
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This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 ≤ m < n be fixed positive integers, some open domain, and a function yielding a full partitioning of D into a family, denoted M ( h ), of lower-dimensional surfaces/manifolds via inverse mapping h –1 as D = ⋃ M ( h ), where M ( h ) = d { h –1 ( t ) : t in range( h )}, noting each h –1 ( t ) can also be considered the solution set of all X in D of the simultaneous equations h ( X ) = t . Let X be a random vector (rv) over D having a probability density function (pdf) ƒ. Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh ( X )/ dX over D , etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M ( h ) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r , we can pose a similar question concerning rv ψ ( X ), when is some bounded a.e. continuous function, not necessarily admitting a pdf.
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June 18, 2011
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When count data are observed, one might be interested in making inference concerning damage proportions for a certain product. The goal of this article is to construct approximate confidence intervals for a linear combination of the proportions (which results in the confidence intervals for the damage total) for a certain product based on the ‘bivariate Poisson model’ for paired count data. The simulated coverage is evaluated using computer simulation, and recommendations are provided for selecting reasonable confidence intervals (their coverage close to the nominal value or the true coverage).
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August 28, 2011
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In this paper, we consider the conditional least squares (CLS) estimators for periodic ARCH models (P-ARCH). The CLS estimators applied to the square-transformed P-ARCH model have an explicit form which does not depend on the distribution of the innovation. Since the CLS are not asymptotically efficient in general, we give a necessary and sufficient condition ensuring the asymptotic efficiency of the CLS based on the local asymptotic normality (LAN) approach.
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June 18, 2011
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This note is provided to correct the mistakes in our previous work ‘BSDEs driven by infinite dimensional martingales and their applications to stochastic optimal control’, Random Oper. Stoch. Equ. 19 (2011), 45–61. The reader is advised to consider changes here when reading that paper.