| 1. |
- Alm, Sven Erick
(författare)
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Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence
- 2003
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Ingår i: Bernoulli. - 1350-7265. ; 9:2, s. 351-371
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Tidskriftsartikel (refereegranskat)abstract
- For $n\ge0$, let $\lambda_n$ be the median of the $\Gamma(n+1,1)$ distribution. We prove that the sequence $\{\alpha_n=\lambda_n-n\}$ decreases from $\log 2$ to $2/3$ as $n$ increases from 0 to $\infty$. The difference, $1-\alpha_n$, between the mean and the median thus increases from $1-\log 2$ to $1/3$.This result also proves the following conjecture by Chen \& Rubin about the Poisson distributions: Let $Y_{\mu}\sim\text{Poisson}(\mu)$, and \lambda_n$ be the largest $\mu$ such that $P(Y_{\mu}\le n)=1/2$, then $\lambda_n-n$ is decreasing in $n$.The sequence $\{\alpha_n\}$ is related to a sequence $\{\theta_n\}$, introduced by Ramanujan, which is known to be decreasing and of the form$\theta_n=\frac13+\frac4{135(n+k_n)}$, where $\frac2{21}
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| 2. |
- Andersson, Patrik, et al.
(författare)
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Unbiased simulation of stochastic differential equations using parametrix expansions
- 2017
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Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 23:3, s. 2028-2057
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we consider an unbiased simulation method for multidimensional diffusions based on the parametrix method for solving partial differential equations with Holder continuous coefficients. This Monte Carlo method which is based on an Euler scheme with random time steps, can be considered as an infinite dimensional extension of the Multilevel Monte Carlo method for solutions of stochastic differential equations with Holder continuous coefficients. In particular, we study the properties of the variance of the proposed method. In most cases, the method has infinite variance and therefore we propose an importance sampling method to resolve this issue.
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| 3. |
- Anevski, Dragi, et al.
(författare)
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Limit properties of the monotone rearrangement for density and regression function estimation
- 2019
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Ingår i: Bernoulli. - 1350-7265. ; 25:1, s. 549-583
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Tidskriftsartikel (refereegranskat)abstract
- The monotone rearrrangement algorithm was introduced by Hardy, Littlewood and Po ́lya as a sorting device for functions. As- suming that x is a monotone function and that an estimate xn of x is given, consider the monotone rearrangement xˆn of xn. This new estimator is shown to be uniformly consistent. Under suitable as- sumptions, pointwise limit distribution results for xˆn are obtained. The framework is general and allows for weakly dependent and long range dependent stationary data. Applications in monotone density and regression function estimation are detailed.
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| 4. |
- Behme, Anita, et al.
(författare)
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A class of scale mixtures of Gamma(k)-distributions that are generalized gamma convolutions
- 2017
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Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 23:1, s. 773-787
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Tidskriftsartikel (refereegranskat)abstract
- Let k > 0 be an integer and Y a standard Gamma(k) distributed random variable. Let X be an independent positive random variable with a density that is hyperbolically monotone (HIM) of order k. Then Y . X and Y/X both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case k = 1 but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Levy processes are mentioned.
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| 5. |
- Belomestny, Denis, et al.
(författare)
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Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
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Ingår i: Bernoulli. - 1350-7265. ; 28:4, s. 2151-2180
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Tidskriftsartikel (refereegranskat)abstract
- We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.
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| 6. |
- Belomestny, D., et al.
(författare)
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Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
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Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 28:4, s. 2151-2180
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Tidskriftsartikel (refereegranskat)abstract
- We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.
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| 7. |
- Bolin, David, et al.
(författare)
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Gaussian Whittle–Matérn fields on metric graphs
- 2024
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Ingår i: Bernoulli. - 1350-7265. ; 30:2, s. 1611-1639
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Tidskriftsartikel (refereegranskat)abstract
- We define a new class of Gaussian processes on compact metric graphs such as street or river networks. The proposed models, the Whittle–Matérn fields, are defined via a fractional stochastic differential equation on the compact metric graph and are a natural extension of Gaussian fields with Matérn covariance functions on Euclidean domains to the non-Euclidean metric graph setting. Existence of the processes, as well as some of their main properties, such as sample path regularity are derived. The model class in particular contains differentiable processes. To the best of our knowledge, this is the first construction of a differentiable Gaussian process on general compact metric graphs. Further, we prove an intrinsic property of these processes: that they do not change upon addition or removal of vertices with degree two. Finally, we obtain Karhunen–Loève expansions of the processes, provide numerical experiments, and compare them to Gaussian processes with isotropic covariance functions.
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| 8. |
- Chotard, Alexandre, et al.
(författare)
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Verifiable conditions for the irreducibility and aperiodicity of Markov chains by analyzing underlying deterministic models
- 2019
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Ingår i: Bernoulli. - : The International Statistical Institute. - 1350-7265 .- 1573-9759. ; 25:1, s. 112-147
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Tidskriftsartikel (refereegranskat)abstract
- We consider Markov chains that obey the following general non-linear state space model: Phi(k)(+)(1) = F (Phi(k), alpha (Phi(k), U-k+(1))) where the function F is C-1 while alpha is typically discontinuous and {U-k : k is an element of Z(>0)} is an independent and identically distributed process. We assume that for all x, the random variable a (x, U-1) admits a density p(x) such that (x, w) bar right arrow p(x) (w) is lower semi-continuous. We generalize and extend previous results that connect properties of the underlying deterministic control model to provide conditions for the chain to be phi-irreducible and aperiodic. By building on those results, we show that if a rank condition on the controllability matrix is satisfied for all x, there is equivalence between the existence of a globally attracting state for the control model and phi-irreducibility of the Markov chain. Additionally, under the same rank condition on the controllability matrix, we prove that there is equivalence between the existence of a steadily attracting state and the phi-irreducibility and aperiodicity of the chain. The notion of steadily attracting state is new. We additionally derive practical conditions by showing that the rank condition on the controllability matrix needs to be verified only at a globally attracting state (resp. steadily attracting state) for the chain to be a phi-irreducible T-chain (resp. phi-irreducible aperiodic T-chain). Those results hold under considerably weaker assumptions on the model than previous ones that would require (x, u) bar right arrow F (x, alpha (x, u)) to be C-infinity (while it can be discontinuous here). Additionally the establishment of a necessary and sufficient condition on the control model for the phi-irreducibility and aperiodicity without a structural assumption on the control set is novel - even for Markov chains where (x, u) bar right arrow F (x, alpha (x, u)) is C-infinity. We illustrate that the conditions are easy to verify on a non-trivial and non-artificial example of Markov chain arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario.
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| 9. |
- Djehiche, Boualem, 1962-, et al.
(författare)
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Finite impulse response models : A non-asymptotic analysis of the least squares estimator
- 2021
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Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 27:2, s. 976-1000
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Tidskriftsartikel (refereegranskat)abstract
- We consider a finite impulse response system with centered independent sub-Gaussian design covariates and noise components that are not necessarily identically distributed. We derive non-asymptotic near-optimal estimation and prediction bounds for the least squares estimator of the parameters. Our results are based on two concentration inequalities on the norm of sums of dependent covariate vectors and on the singular values of their covariance operator that are of independent value on their own and where the dependence arises from the time shift structure of the time series. These results generalize the known bounds for the independent case.
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| 10. |
- Erhardsson, Torkel
(författare)
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Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions
- 2014
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Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 20:2, s. 990-1005
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Tidskriftsartikel (refereegranskat)abstract
- A d-dimensional RCA(1) process is a generalization of the d-dimensional AR(1) process, such that the coefficients {M-t; t =1, 2, ...} are i.i.d. random matrices. In the case d =1, under a nondegeneracy condition, Goldie and Mailer gave necessary and sufficient conditions for the convergence in distribution of an RCA(1) process, and for the almost sure convergence of a closely related sum of random variables called a perpetuity. We here prove that under the condition parallel to Pi(n)(t=1) M-t parallel to -greater than(a.s.) 0 as n -greater than infinity, most of the results of Goldie and Mailer can be extended to the case d greater than 1. If this condition does not hold, some of their results cannot be extended.
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