Estimating the intensity of a cyclic Poisson process in the presence of linear trend

Helmers, Roelof; Mangku, I Wayan

doi:10.1007/s10463-007-0160-2

Cited by 15 publications

(18 citation statements)

References 11 publications

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“…Therefore, all terms on the r.h.s. of (13) are indeed of order o(1) as n → ∞, which imply (2). This completes the proof of Lemma 3.…”

Section: Proof Of Lemmasupporting

confidence: 66%

“…The estimator given in (3) is a generalization of the estimator discussed and investigated in Helmers and Mangku [2] for the case that the period τ is known.…”

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

“…as n → ∞, which directly yields the estimator given in (2). Note also that, if N were a Poisson proses with intensity function λ(s) = as, thenâ n would be the maximum likelihood estimator of a (see [8]).…”

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

“…(9) Now note that the estimatorλ c,n (s) given in (9) is a special case of the estimator λ c,n,K (s) in (3), that is in (9) we use the uniform kernelK = In Helmers and Mangku [2] has been proved the following lemma.…”

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

“…The present paper (cf. also [2]) aims at extending previous work for the purely cyclic case, i.e. a = 0, (cf.…”

Section: Introductionmentioning

confidence: 85%

See 4 more Smart Citations

Consistency of a Kernel-Type Estimator of the Intensity of the Cyclic Poisson Process With the Linear Trend

Mangku

Siswadi

Budiarti

2012

JIMS

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“…Therefore, all terms on the r.h.s. of (13) are indeed of order o(1) as n → ∞, which imply (2). This completes the proof of Lemma 3.…”

Section: Proof Of Lemmasupporting

confidence: 66%

“…The estimator given in (3) is a generalization of the estimator discussed and investigated in Helmers and Mangku [2] for the case that the period τ is known.…”

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

Section: Construction Of the Estimator And Resultsmentioning

confidence: 99%

“…The present paper (cf. also [2]) aims at extending previous work for the purely cyclic case, i.e. a = 0, (cf.…”

Section: Introductionmentioning

confidence: 85%

See 3 more Smart Citations

Consistency of a Kernel-Type Estimator of the Intensity of the Cyclic Poisson Process With the Linear Trend

Mangku

Siswadi

Budiarti

2012

JIMS

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Predicting a cyclic Poisson process

Helmers

Mangku

2012

Ann Inst Stat Math

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Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

et al. 2011

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Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.

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Estimating the intensity of a cyclic Poisson process in the presence of linear trend

Abstract: Cyclic Poisson process, Intensity function, Linear trend, Nonparametric estimation, Consistency, Bias, Variance, Mean-squared error,

Cited by 15 publications

References 11 publications

Consistency of a Kernel-Type Estimator of the Intensity of the Cyclic Poisson Process With the Linear Trend

Consistency of a Kernel-Type Estimator of the Intensity of the Cyclic Poisson Process With the Linear Trend

Predicting a cyclic Poisson process

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

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