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A random walk that carries information starts at 0 and moves inside the set of integers ℤ. When the random walk reaches a new point this point obtains the information. A given set Vf containing 0 is considered to be the set of friendly stations and its complement Vh=ℤ\Vf is considered to be the set of hostile stations. An observer who knows the partition of ℤ into Vf and Vh obtains a signal only if a new station is informed. There are two kinds of signal: from friendly stations and from hostile ones. This is the whole knowledge about the random walk the observer possesses. Thus, the observer's time is measured by the number of informed stations. The observer can stop the walk at a newly informed point and wins n if there have been n friendly stations informed and no hostile one or wins 0 if at least one hostile station has been informed. We consider two models one with Vf={−m+1,−m+2,…, m−2,m−1} and one with Vf={0,1,2,…} for which we find the observer's optimal stopping time maximizing the expected value of the win. [ABSTRACT FROM PUBLISHER]

We investigate the optimal control of neuronal spiking activity for classical Stein's model, Stein's model with reversal potentials with continuous random inputs, characterized by a positive parameter α and Stein's model with Poisson inputs. We solve the optimal control problems and obtain optimal rates λ(t) for different kinds of models. The numerical simulations on variable parameters show that it is possible to make the interval of spikes the same as our expected time in the range of the values of parameters. [ABSTRACT FROM AUTHOR]

Analysis times for clinical trials are often based on the time when a specified total number of events occur. A modification is proposed in which this time is based on when either arm reaches a prespecified number of events. The implications in terms of the expected number of events required to achieve the same size and power at a specified alternative are explored. Savings are achieved when alternatives for which there is reasonable power hold, but more events are required at the null hypothesis. [ABSTRACT FROM PUBLISHER]