scholarly journals On theq-Konhauser biorthogonal polynomials

1987 ◽  
Vol 10 (2) ◽  
pp. 413-415
Author(s):  
H. C. Madhekar ◽  
V. T. Chamle
Keyword(s):  
Distribution Function ◽  
Discrete Distribution ◽  
Operational Formula ◽  
Biorthogonal Polynomials

Recently Al-Salam and Verma discussed two polynomial sets{Zn(α)(x,k|q)}and{Yn(α)(x,k|q)}which are biorthogonal on(0,∞)with respect to a continuous or discrete distribution function. For the polynomialsYn(α)(x,k|q)the operational formula is derived.

Binary decompositions of probability densities and random-bit simulation

2020 ◽  
Vol 26 (2) ◽  
pp. 163-169
Author(s):  
Vladimir Nekrutkin
Keyword(s):  
New York ◽  
Distribution Function ◽  
Random Number ◽  
Discrete Distribution ◽  
Random Number Generation ◽  
Academic Press ◽  
Bernoulli Trials ◽  
Algorithms And Complexity ◽  
Binary Decomposition ◽  
Probability Densities

AbstractThis paper is devoted to random-bit simulation of probability densities, supported on {[0,1]}. The term “random-bit” means that the source of randomness for simulation is a sequence of symmetrical Bernoulli trials. In contrast to the pioneer paper [D. E. Knuth and A. C. Yao, The complexity of nonuniform random number generation, Algorithms and Complexity, Academic Press, New York 1976, 357–428], the proposed method demands the knowledge of the probability density under simulation, and not the values of the corresponding distribution function. The method is based on the so-called binary decomposition of the density and comes down to simulation of a special discrete distribution to get several principal bits of output, while further bits of output are produced by “flipping a coin”. The complexity of the method is studied and several examples are presented.

Optimal selection based on relative ranks of a sequence by ties

1984 ◽  
Vol 16 (01) ◽  
pp. 131-146
Author(s):  
Gregory Campbell
Keyword(s):  
Distribution Function ◽  
Dirichlet Process ◽  
Random Distribution ◽  
Discrete Distribution ◽  
Optimal Selection ◽  
Dirichlet Process Prior ◽  
Limiting Behavior ◽  
Relative Rank ◽  
Atomic Parameter ◽  
Selection Of

The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.

scholarly journals A nonparametric approach toward upper bounds to transit time distribution functions

2017 ◽  
Author(s):  
Earl Bardsley
Keyword(s):  
Distribution Function ◽  
Transit Time ◽  
Time Distribution ◽  
Discrete Distribution ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Linear Measure ◽  
Time Varying ◽  
Transit Time Distribution ◽  
The Individual

Abstract. A nonparametric method is proposed as a possible approach to obtaining upper bounds to distribution functions of time-varying transit times for catchment environmental tracers. A discretization is employed for the tracer throughput process, with tracer input represented as a sequence of K discrete pulses over a given time period. Each input pulse is associated with a different and unknown upper-bounded nonparametric discrete transit time distribution. The model transit time distribution function is therefore a K-component finite mixture of different and unknown discrete distribution functions, weighted by the relative magnitudes of the respective tracer pulses. Upper bounds to this distribution function can be obtained by linear programming to achieve a sequence of K discrete optimised transit time distributions which yield the maximum possible value of tracer fraction less than a given age, subject to a constraint of matching the catchment tracer output time series to some specified linear measure of accuracy. The individual optimised distributions do not estimate actual transit time distributions and the optimisation procedure is not hydrological modelling. This is actually a strength of the methodology in that the true transit time distributions are permitted to be created as a consequence of any time-varying nonlinear catchment process with complete or partial mixing. However, a negative aspect is that the extreme flexibility of K different nonparametric distributions is likely to give transit time distribution functions upper bounds near 1, unless sufficient constraints can be imposed on the form of the individual optimised distributions. There is a possibility, however, that optimising just a single nonparametric L-shaped discrete distribution could yield useful distribution function upper bounds for time-varying situations.

Optimal selection based on relative ranks of a sequence by ties

1984 ◽  
Vol 16 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Gregory Campbell
Keyword(s):  
Distribution Function ◽  
Dirichlet Process ◽  
Random Distribution ◽  
Discrete Distribution ◽  
Optimal Selection ◽  
Dirichlet Process Prior ◽  
Limiting Behavior ◽  
Relative Rank ◽  
Atomic Parameter ◽  
Selection Of

The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.

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