On the Sequential Detection Problem

1989 ◽  
Vol 38 (3-4) ◽  
pp. 129-146
Author(s):  
Uttam Bandyopadhyay
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Stopping Rule ◽  
Independent Random Variables ◽  
Sequential Detection ◽  
Detection Problem ◽  
Asymptotic Results ◽  
Rule Based ◽  
Unknown Point ◽  
Cumulative Sums

In this paper, for an infinite sequence of independent random variables, we have considered the problem of estimation of an unknown point ( q) where a change in the distribution of the random variables occurs. Attaching suitable scores for the observed values. of the random variables, a stopping rule based on the cumulative sums of these scores has been proposed. Some asymptotic results useful for studying the performance of the proposed procedure havo beon obtained.

scholarly journals On the asymptotic behaviour of the extinction time of the simple branching process

1989 ◽  
Vol 21 (2) ◽  
pp. 470-472 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Asymptotic Behaviour ◽  
Branching Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Independent Random Variables ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Geometric Type ◽  
Made In

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

scholarly journals An optimal sequential procedure for a buying-selling problem with independent observations

2006 ◽  
Vol 43 (02) ◽  
pp. 454-462 ◽  
Author(s):  
G. Sofronov ◽  
Jonathan M. Keith ◽  
Dirk P. Kroese
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Stopping Rule ◽  
Sequential Procedure ◽  
Independent Random Variables ◽  
Value Of A Game ◽  
Optimal Stopping Rule ◽  
Independent Observations

We consider a buying-selling problem when two stops of a sequence of independent random variables are required. An optimal stopping rule and the value of a game are obtained.

scholarly journals An optimal sequential procedure for a buying-selling problem with independent observations

2006 ◽  
Vol 43 (2) ◽  
pp. 454-462 ◽  
Author(s):  
G. Sofronov ◽  
Jonathan M. Keith ◽  
Dirk P. Kroese
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Stopping Rule ◽  
Sequential Procedure ◽  
Independent Random Variables ◽  
Value Of A Game ◽  
Optimal Stopping Rule ◽  
Independent Observations

We consider a buying-selling problem when two stops of a sequence of independent random variables are required. An optimal stopping rule and the value of a game are obtained.

scholarly journals On the asymptotic behaviour of the extinction time of the simple branching process

1989 ◽  
Vol 21 (02) ◽  
pp. 470-472 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Asymptotic Behaviour ◽  
Branching Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Independent Random Variables ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Geometric Type ◽  
Made In

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

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