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The probability function and binomial moments of the number Nn of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q being the inverse of the proportion λ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables Nn, n=1,2,…, are deduced. As a corollary the probability function of the time Tk of the kth record is also expressed in terms of the signless q -Stirling numbers of the first kind. The mean of Tk is obtained as a q -series with terms of alternating sign. Finally, the probability function of the inter-record time Wk=Tk-Tk-1 is obtained as a sum of a finite number of terms of q -numbers. The mean of Wk is expressed by a q-series. As k increases to infinity the distribution of Wk converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived. 相似文献
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In this paper, we study a random field U?(t,x) governed by some type of stochastic partial differential equations with an unknown parameter θ and a small noise ?. We construct an estimator of θ based on the continuous observation of N Fourier coefficients of U?(t,x), and prove the strong convergence and asymptotic normality of the estimator when the noise ? tends to zero. 相似文献
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In this paper, we consider the following simple linear Errors-in-Variables (EV) regression model ηi=θ+βxi+?i, ξi=xi+δi, 1?i?n. The moderate deviation principle for the least squares (LS) estimators of the unknown parameters θ, β in the model are obtained. 相似文献
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Consider the model where there are I independent multivariate normal treatment populations with p×1 mean vectors μi, i=1,…,I, and covariance matrix Σ. Independently the (I+1)st population corresponds to a control and it too is multivariate normal with mean vector μI+1 and covariance matrix Σ. Now consider the following two multiple testing problems. 相似文献