| 1. |
- Andersson, Patrik, et al.
(författare)
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Unbiased simulation of stochastic differential equations using parametrix expansions
- 2017
-
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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| 2. |
- Behme, Anita, et al.
(författare)
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A class of scale mixtures of Gamma(k)-distributions that are generalized gamma convolutions
- 2017
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 23:1, s. 773-787
-
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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| 3. |
- Chotard, Alexandre, et al.
(författare)
-
Verifiable conditions for the irreducibility and aperiodicity of Markov chains by analyzing underlying deterministic models
- 2019
-
Ingår i: Bernoulli. - : The International Statistical Institute. - 1350-7265 .- 1573-9759. ; 25:1, s. 112-147
-
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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| 4. |
- Djehiche, Boualem, 1962-, et al.
(författare)
-
Finite impulse response models : A non-asymptotic analysis of the least squares estimator
- 2021
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 27:2, s. 976-1000
-
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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| 5. |
- Erhardsson, Torkel
(författare)
-
Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions
- 2014
-
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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| 6. |
- Gantert, Nina, 1962, et al.
(författare)
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Strictly weak consensus in the uniform compass model on Z
- 2020
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 26:2, s. 1269-1293
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opinions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e., global agreement of all individuals) versus weak consensus (i.e., local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a “central” opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice Z with weak consensus but no strong consensus.
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| 7. |
- Gloaguen, Pierre, et al.
(författare)
-
A pseudo-marginal sequential Monte Carlo online smoothing algorithm
- 2022
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 28:4, s. 2606-2633
-
Tidskriftsartikel (refereegranskat)abstract
- We consider online computation of expectations of additive state functionals under general path probability measures proportional to products of unnormalised transition densities. These transition densities are assumed to be intractable but possible to estimate, with or without bias. Using pseudo-marginalisation techniques we are able to extend the particle-based, rapid incremental smoother (PaRIS) algorithm proposed in [Bernoulli 23(3) (2017) 1951-1996] to this setting. The resulting algorithm, which has a linear complexity in the number of particles and constant memory requirements, applies to a wide range of challenging path-space Monte Carlo problems, including smoothing in partially observed diffusion processes and models with intractable likelihood. The algorithm is furnished with several theoretical results, including a central limit theorem, establishing its convergence and numerical stability. Moreover, under strong mixing assumptions we establish a novel O(n epsilon) bound on the asymptotic bias of the algorithm, where n is the path length and epsilon controls the bias of the transition-density estimators.
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| 8. |
- Gut, Allan, et al.
(författare)
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Between the LIL and the LSL
- 2010
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Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 16:1, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- In two earlier papers, two of the present authors (A.G. and U.S.) extended Lai's [Ann. Probab. 2 (1974) 432-440] law of the single logarithm for delayed sums to a multiindex setting in which the edges of the nth window grow like |n|α, or with different α's, where the α's belong to (0,1). In this paper, the edge of the nth window typically grows like n/log n, thus at a higher rate than any power less than one, but not quite at the LIL-rate.
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| 9. |
- Gut, Allan, et al.
(författare)
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Laws of the single logarithm for delayed sums of random fields
- 2008
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 14:1, s. 249-276
-
Tidskriftsartikel (refereegranskat)abstract
- We extend a law of the single logarithm for delayed sums by Lai to delayed sums of random fields. A law for subsequences, which also includes the one-dimensional case, is obtained in passing.
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| 10. |
- Heiny, Johannes, et al.
(författare)
-
Maximum interpoint distance of high-dimensional random vectors
- 2024
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759.
-
Tidskriftsartikel (refereegranskat)abstract
- A limit theorem for the largest interpoint distance of p independent and identically distributed points in Rn to the Gumbel distribution is proved, where the number of points p=pn tends to infinity as the dimension of the points n→∞. The theorem holds under moment assumptions and corresponding conditions on the growth rate of p. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.
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| 11. |
- Hult, Henrik, et al.
(författare)
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On Kesten's counterexample to the Cramer-Wold device for regular variation
- 2006
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 12:1, s. 133-142
-
Tidskriftsartikel (refereegranskat)abstract
- In 2002 Basrak, Davis and Mikosch showed that an analogue of the Cramer-Wold device holds for regular variation of random vectors if the index of regular variation is not an integer. This characterization is of importance when studying stationary solutions to stochastic recurrence equations. In this paper we construct counterexamples showing that for integer-valued indices, regular variation of all linear combinations does not imply that the vector is regularly varying. The construction is based on unpublished notes by Harry Kesten.
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| 12. |
- Hult, Henrik, 1975-, et al.
(författare)
-
Tail probabilities for infinite series of regularly varying random vectors
- 2008
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 14:3, s. 838-864
-
Tidskriftsartikel (refereegranskat)abstract
- A random vector X with representation X = Sigma(j >= 0)A(j)Z(j) is considered. Here, (Z(j)) is a sequence of independent and identically distributed random vectors and (A(j)) is a sequence of random matrices, 'predictable' with respect to the sequence (Z(j)). The distribution of Z(1) is assumed to be multivariate regular varying. Moment conditions on the matrices (A(j)) are determined under which the distribution of X is regularly varying and, in fact, 'inherits' its regular variation from that of the (Z(j))'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.
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| 13. |
- Janson, Svante, 1955-, et al.
(författare)
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Continuous-time digital search tree and a border aggregation model
- 2022
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 28:4, s. 2563-2577
-
Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We consider the continuous-time version of the random digital search tree, and construct a coupling with a border aggregation model as studied in Thacker and Volkov (Ann. Appl. Probab. 28 (2018) 1604???1633), showing a relation between the height of the tree and the time required for aggregation. This relation carries over to the corresponding discrete-time models. As a consequence we find a very precise asymptotic result for the time to aggregation, using recent results by Drmota et al. (Random Structures Algorithms 58 (2021) 430???467) for the digital search tree.
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| 14. |
- Janson, Svante
(författare)
-
Monotonicity, asymptotic normality and vertex degrees in random graphs
- 2007
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 13:4, s. 952-965
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Tidskriftsartikel (refereegranskat)abstract
- We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n, m) with a fixed number of edges from the corresponding result for the random graph G(n, p) with independent edges. We also give some simple applications to random allocations and to spacings. Finally, inspired by these results, but logically independent of them, we investigate whether a one-sided version of the Cramér-Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.
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| 15. |
- Jonsson, Fredrik
(författare)
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On the heavy-tailedness of Student's t-statistic
- 2011
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 17:1, s. 276-289
-
Tidskriftsartikel (refereegranskat)abstract
- Let {X-i}(i >= 1) be an i.i.d. sequence of random variables and define, for n >= 2, T-n={(n-1/2 (sigma) over capn-1Sn)(0), ((sigma) over capn > 0,)((sigma) over capn=0,) with S-n = Sigma(n)(i=1) X-i, (sigma) over cap (2)(n)=1/n-1 Sigma(n)(i=1)(X-i - n(-1)S(n))(2). We investigate the connection between the distribution of an observation X-i and finiteness of E vertical bar T-n vertical bar(r) for (n, r) is an element of N->= 2 x R+. Moreover, assuming T-n ->(d) T, we prove that for any r > 0, lim(n ->infinity) E vertical bar T-n vertical bar(r) = E vertical bar T vertical bar(r) < infinity, provided there is an integer n(0) such that E vertical bar T-n0 vertical bar(r) is finite.
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| 16. |
- Lagerås, Andreas N.
(författare)
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Copulas for Markovian dependence
- 2010
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 16:2, s. 331-342
-
Tidskriftsartikel (refereegranskat)abstract
- Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper demonstrates some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Frechet copulas) are shown to imply quite a restricted type of Markov process and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable or, equivalently, conditionally i.i.d.
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| 17. |
- Li, Yiting, et al.
(författare)
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On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices
- 2021
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 27:2, s. 1057-1076
-
Tidskriftsartikel (refereegranskat)abstract
- We consider an N by N real or complex generalized Wigner matrix H-N, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, s(ij) := E vertical bar H-ij vertical bar(2), satisfies Sigma(N)(i=1) s(ij) = 1, for all 1 <= j <= N and c(-1) <= Ns(ij) <= c for all 1 <= i, j <= N with some constant c >= 1. We establish Gaussian fluctuations for the linear eigenvalue statistics of HN on global scales, as well as on all mesoscopic scales up to the spectral edges, with the expectation and variance formulated in terms of the variance profile. We subsequently obtain the universal mesoscopic central limit theorems for the linear eigenvalue statistics inside the bulk and at the edges, respectively.
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| 18. |
- Lindskog, Filip, et al.
(författare)
-
Exact long time behavior of some regime switching stochastic processes
- 2020
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 26:4, s. 2572-2604
-
Tidskriftsartikel (refereegranskat)abstract
- Regime switching processes have proved to be indispensable in the modeling of various phenomena, allowing model parameters that traditionally were considered to be constant to fluctuate in a Markovian manner in line with empirical findings. We study diffusion processes of Ornstein–Uhlenbeck type where the drift and diffusion coefficients a and b are functions of a Markov process with a stationary distribution π on acountable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: Eπa(·)>0,=0,<0, respectively. Alongside we provide exact time limit results for integrals of form ∫t0b2(Xs)e−2∫tsa(Xr)drds for the three different regimes. Finally, we demonstrate natural applications of the findings in terms of Cox–Ingersoll–Ross diffusion and deterministic SIS epidemic models inMarkovian environments. The time asymptotic behaviors are naturally expressed in terms of solutions tothe well-studied fixed-point equation in law Xd=AX+B with X⊥⊥(A, B).
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| 19. |
- Mazur, Stepan, 1988-, et al.
(författare)
-
Estimation of the linear fractional stable motion
- 2020
-
Ingår i: Bernoulli. - : The International Statistical Institute. - 1350-7265 .- 1573-9759. ; 26:1, s. 226-252
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter σ>0, the self-similarity parameter H∈(0,1) and the stability index α∈(0,2) of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in (Ann. Probab. 45 (2017) 4477–4528). More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of (σ,α,H). We present the law of large numbers and some fully feasible weak limit theorems.
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| 20. |
- Muratov, Anton, et al.
(författare)
-
Neighbour-dependent point shifts and random exchange models : Invariance and attractors
- 2017
-
Ingår i: Bernoulli. - : International statistical institute. - 1350-7265 .- 1573-9759. ; 23:1, s. 539-551
-
Tidskriftsartikel (refereegranskat)abstract
- Consider a partition of the real line into intervals by the points of a stationary renewal point process. Subdivide the intervals in proportions given by i.i.d. random variables with distribution G supported by [0, 1]. We ask ourselves for what interval length distribution F and what division distribution G, the subdivision points themselves form a renewal process with the same F? An evident case is that of degenerate F and G. As we show, the only other possibility is when F is Gamma and G is Beta with related parameters. In particular, the process of division points of a Poisson process is again Poisson, if the division distribution is Beta: B(r, 1 - r) for some 0 < r < 1. We show a similar behaviour of random exchange models when a countable number of "agents" exchange randomly distributed parts of their "masses" with neighbours. More generally, a Dirichlet distribution arises in these models as a fixed point distribution preserving independence of the masses at each step. We also show that for each G there is a unique attractor, a distribution of the infinite sequence of masses, which is a fixed point of the random exchange and to which iterations of a non-equilibrium configuration of masses converge weakly. In particular, iteratively applying B(r, 1 - r)-divisions to a realisation of any renewal process with finite second moment of F yields a Poisson process of the same intensity in the limit.
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| 21. |
- Ojeda, Gabriel Berzunza, et al.
(författare)
-
Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees
- 2023
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265 .- 1573-9759. ; 29:4, s. 2745-2770
-
Tidskriftsartikel (refereegranskat)abstract
- Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on n labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as n -> infinity to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner. In this work, we continue the above investigation and study the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree t(n) conditioned on having n vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index alpha is an element of(1,2]. Our main results establish that, after rescaling, the fragmentation process of t(n) converges as n -> infinity to the fragmentation process obtained by cutting-down at a rate proportional to the length measure on the skeleton of an alpha-stable Levy tree. We further show that the latter can be constructed by considering the partitions of the unit interval induced by the normalized alpha-stable Levy excursion with a deterministic drift studied by Miermont (2001). This extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.
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| 22. |
- Olsson, Jimmy, et al.
(författare)
-
Efficient particle-based online smoothing in general hidden Markov models : The PaRIS algorithm
- 2017
-
Ingår i: Bernoulli. - : INT STATISTICAL INST. - 1350-7265 .- 1573-9759. ; 23:3, s. 1951-1996
-
Tidskriftsartikel (refereegranskat)abstract
- This paper presents a novel algorithm, the particle-based, rapid incremental smoother (PaRIS), for efficient online approximation of smoothed expectations of additive state functionals in general hidden Markov models. The algorithm, which has a linear computational complexity under weak assumptions and very limited memory requirements, is furnished with a number of convergence results, including a central limit theorem. An interesting feature of PaRIS, which samples on-the-fly from the retrospective dynamics induced by the particle filter, is that it requires two or more backward draws per particle in order to cope with degeneracy of the sampled trajectories and to stay numerically stable in the long run with an asymptotic variance that grows only linearly with time.
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| 23. |
- Olsson, Jimmy, et al.
(författare)
-
Numerically stable online estimation of variance in particle filters
- 2019
-
Ingår i: Bernoulli. - : INT STATISTICAL INST. - 1350-7265 .- 1573-9759. ; 25:2, s. 1504-1535
-
Tidskriftsartikel (refereegranskat)abstract
- This paper discusses variance estimation in sequential Monte Carlo methods, alternatively termed particle filters. The variance estimator that we propose is a natural modification of that suggested by H.P. Chan and T.L. Lai [Ann. Statist. 41 (2013) 2877-2904], which allows the variance to be estimated in a single run of the particle filter by tracing the genealogical history of the particles. However, due particle lineage degeneracy, the estimator of the mentioned work becomes numerically unstable as the number of sequential particle updates increases. Thus, by tracing only a part of the particles' genealogy rather than the full one, our estimator gains long-term numerical stability at the cost of a bias. The scope of the genealogical tracing is regulated by a lag, and under mild, easily checked model assumptions, we prove that the bias tends to zero geometrically fast as the lag increases. As confirmed by our numerical results, this allows the bias to be tightly controlled also for moderate particle sample sizes.
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| 24. |
- Parolya, Nestor, et al.
(författare)
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Logarithmic law of large random correlation matrices
- 2024
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 30:1, s. 346-370
-
Tidskriftsartikel (refereegranskat)abstract
- Consider a random vector y=Σ1∕2x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ1∕2 is a deterministic p×p matrix such that the eigenvalues of the population correlation matrix R of y are uniformly bounded away from zero and infinity. In this paper, we find that the log determinant of the sample correlation matrix based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p∕n→γ∈(0,1] and p≤n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after re-centering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. Finally, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R=I, the test statistic becomes distribution-free and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.
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| 25. |
- Thulin, Måns, 1986-, et al.
(författare)
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Exact confidence intervals and hypothesis tests for parameters of discrete distributions
- 2017
-
Ingår i: Bernoulli. - 1350-7265 .- 1573-9759. ; 23:1, s. 479-502
-
Tidskriftsartikel (refereegranskat)abstract
- We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing p-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, including the Clopper-Pearson interval for a binomial proportion and the Garwood interval for a Poisson mean, are optimal.
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| 26. |
- Alm, Sven Erick
(författare)
-
Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence
- 2003
-
Ingår i: Bernoulli. - 1350-7265. ; 9:2, s. 351-371
-
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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| 27. |
- Anevski, Dragi, et al.
(författare)
-
Limit properties of the monotone rearrangement for density and regression function estimation
- 2019
-
Ingår i: Bernoulli. - 1350-7265. ; 25:1, s. 549-583
-
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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| 28. |
- Belomestny, Denis, et al.
(författare)
-
Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
-
Ingår i: Bernoulli. - 1350-7265. ; 28:4, s. 2151-2180
-
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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| 29. |
- Belomestny, D., et al.
(författare)
-
Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 28:4, s. 2151-2180
-
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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| 30. |
- Bolin, David, et al.
(författare)
-
Gaussian Whittle–Matérn fields on metric graphs
- 2024
-
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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| 31. |
- Fan, J. Y., et al.
(författare)
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Convergence of the age structure of general schemes of population processes
- 2020
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 26:2, s. 893-926
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter K, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper (Proc. Steklov Inst. Math. 282 (2013) 90-105), the Law of Large Numbers as K -> infinity was derived. Here we prove the central limit theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.
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| 32. |
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| 33. |
- Gut, A
(författare)
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Mixed shock models
- 2001
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Ingår i: BERNOULLI. - : INT STATISTICAL INST. - 1350-7265. ; 7:3, s. 541-555
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Tidskriftsartikel (refereegranskat)abstract
- Traditionally, shock models are of two kinds. The failure (of a system) is related either to the cumulative effect of a (large) number of shocks or it is caused by a shock which is larger than some critical level. The present paper is devoted to a mixed m
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| 34. |
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| 35. |
- Hall, Peter, et al.
(författare)
-
Locating lines among scattered points
- 2006
-
Ingår i: Bernoulli. - 1350-7265. ; 12:5, s. 821-839
-
Tidskriftsartikel (refereegranskat)abstract
- Consider a process of events on a line L, where, for the most part, the events occur randomly in both time and location. A scatterplot of the pair that represents position on the line, and occurrence time, will resemble a bivariate stochastic point process in a plane, P say. If, however, some of the points on L arise through a more regular phenomenon which travels along the line at an approximately constant speed, creating new points as it goes, then the corresponding points in P will occur roughly in a straight line. It is of interest to locate such lines, and thereby identify, as nearly as possible, the points on L which are associated with the (approximately) constant-velocity process. Such a problem arises in connection with the study of seismic data, where L represents a fault-line and the constant-velocity process there results from the steady diffusion of stress. We suggest methodology for solving this needle-in-a-haystack problem, and discuss its properties. The technique is applied to both simulated and real data. In the latter case it draws particular attention to events occurring along the San Andreas fault, in the vicinity of Parkville, California, on 5 April 1995.
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| 36. |
- Jagers, Peter, 1941, et al.
(författare)
-
Convergence of the age structure of general schemes of population processes
- 2020
-
Ingår i: Bernoulli. - 1350-7265. ; 26:2, s. 893-926
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter K, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper [8] the Law of Large Numbers as K Ñ 8 was derived. Here we prove the Central Limit Theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.
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| 37. |
- Jagers, Peter, 1941, et al.
(författare)
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General branching processes in discrete time as random trees.
- 2008
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Ingår i: Bernoulli. - 1350-7265. ; 14:4, s. 949-962
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Tidskriftsartikel (refereegranskat)abstract
- The simple Galton-Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyse the xlog x condition for exponential growth of supercritical general processes, and also the relation between simple Galton-Watson and more general branching processes.
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| 38. |
- Lindberg, Carl, 1978
(författare)
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Portfolio optimization when expected stock returns are determined by exposure to risk
- 2009
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Ingår i: Bernoulli. - 1350-7265. ; 15:2, s. 464-474
-
Tidskriftsartikel (refereegranskat)abstract
- It is widely recognized that when classical optimal strategies are applied with parameters estimated from data, the resulting portfolio weights are remarkably volatile and unstable over time. The predominant explanation for this is the difficulty of estimating expected returns accurately. In this paper, we modify the n stock Black-Scholes model by introducing a new parametrization of the drift rates. We solve Markowitz' continuous time portfolio problem in this framework. The optimal portfolio weights correspond to keeping 1/n of the wealth invested in stocks in each of the n Brownian motions. The strategy is applied out-of-sample to a large data set. The portfolio weights are stable over time and obtain a significantly higher Sharpe ratio than the classical 1/n strategy.
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| 39. |
- Lindgren, Georg, et al.
(författare)
-
How reliable are contour curves?
- 1995
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Ingår i: Bernoulli. - 1350-7265. ; 1:4, s. 301-319
-
Tidskriftsartikel (refereegranskat)abstract
- We present a method to add confidence limits to level curves, drawn on a map of a Gaussian random surface, measured with or without error on a regular or irregular grid. The method is based on the conditional level crossing intensity measure for non-stationary Gaussian fields.
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| 40. |
- Mogensen, Søren Wengel, et al.
(författare)
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Graphical modeling of stochastic processes driven by correlated noise
- 2022
-
Ingår i: Bernoulli. - 1350-7265. ; 28:4, s. 3028-3050
-
Tidskriftsartikel (refereegranskat)abstract
- We study a class of graphs that represent local independence structures in stochastic processes allowing for correlated noise processes. Several graphs may encode the same local independencies and we characterize such equivalence classes of graphs. In the worst case, the number of conditions in our characterizations grows superpolyno-mially as a function of the size of the node set in the graph. We show that deciding Markov equivalence of graphs from this class is coNP-complete which suggests that our characterizations cannot be improved upon substantially. We prove a global Markov property in the case of a multivariate Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.
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| 41. |
- Muratov, Anton, 1988, et al.
(författare)
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Neighbour-dependent point shifts and random exchange models: Invariance and attractors
- 2017
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 23:1, s. 539-551
-
Tidskriftsartikel (refereegranskat)abstract
- Consider a partition of the real line into intervals by the points of a stationary renewal point process. Subdivide the intervals in proportions given by i.i.d. random variables with distribution G supported by [0, 1]. We ask ourselves for what interval length distribution F and what division distribution G, the subdivision points themselves form a renewal process with the same F? An evident case is that of degenerate F and G. As we show, the only other possibility is when F is Gamma and G is Beta with related parameters. In particular, the process of division points of a Poisson process is again Poisson, if the division distribution is Beta: B(r, 1 - r) for some 0
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| 42. |
- Olsson, Jimmy, et al.
(författare)
-
Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models
- 2008
-
Ingår i: Bernoulli. - 1350-7265. ; 14:1, s. 155-179
-
Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations.
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| 43. |
- Rootzén, Holger, et al.
(författare)
-
Multivariate generalized Pareto distributions
- 2006
-
Ingår i: Bernoulli. - 1350-7265. ; 12:5, s. 917-930
-
Tidskriftsartikel (refereegranskat)abstract
- Statistical inference for extremes has been a subject of intensive research over the past couple of decades. One approach is based on modelling exceedances of a random variable over a high threshold with the generalized Pareto (GP) distribution. This has proved to be an important way to apply extreme value theory in practice and is widely used. We introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: first, exceedances asymptotically have a multivariate GP distribution if and only if maxima asymptotically are extreme value distributed; and second, the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also discuss a bivariate example and lower-dimensional marginal distributions.
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| 44. |
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| 45. |
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| 46. |
- Wermuth, Nanny, 1943, et al.
(författare)
-
Covariance chains
- 2006
-
Ingår i: BERNOULLI. - 1350-7265. ; 12:5, s. 841-862
-
Tidskriftsartikel (refereegranskat)abstract
- Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, orthogonal decompositions for covariance chains are derived first in explicit form. Covariance chains are also contrasted to concentration chains, for which estimation is explicit and simple. For this purpose, maximum-likelihood equations are derived first for exponential families when some parameters satisfy zero value constraints. From these equations explicit estimates are obtained, which are asymptotically efficient, and they are applied to covariance chains. Simulation results confirm the satisfactory behaviour of the explicit covariance chain estimates also in moderate-size samples
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| 47. |
- Wermuth, Nanny, 1943
(författare)
-
Probability distributions with summary graph structure
- 2011
-
Ingår i: BERNOULLI. - 1350-7265. ; 17:3, s. 845-879
-
Tidskriftsartikel (refereegranskat)abstract
- A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class contains regression graphs. Regression graphs are a type of so-called chain graph and describe stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that result after possible marginalising or conditioning, we introduce summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences that remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of paths in graphs.
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| 48. |
- Zholud, Dmitrii, 1984
(författare)
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Tail approximations for the Student t-, F-, and Welch statistics for non-normal and not necessarily i.i.d. random variables
- 2014
-
Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 20:4, s. 2102-2130
-
Tidskriftsartikel (refereegranskat)abstract
- Let T be the Student one- or two-sample t-, F-, or Welch statistic. Now release the underlying assumptions of normality, independence and identical distribution and consider a more general case where one only assumes that the vector of data has a continuous joint density. We determine asymptotic expressions for P(T > u) as u -> infinity for this case. The approximations are particularly accurate for small sample sizes and may be used, for example, in the analysis of High-Throughput Screening experiments, where the number of replicates can be as low as two to five and often extreme significance levels are used. We give numerous examples and complement our results by an investigation of the convergence speed - both theoretically, by deriving exact bounds for absolute and relative errors, and by means of a simulation study.
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| 49. |
- Zuyev, Sergei, 1962, et al.
(författare)
-
Stability for random measures, point processes and discrete semigroups.
- 2011
-
Ingår i: BERNOULLI. - 1350-7265. ; 17:3, s. 1015-1043
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Tidskriftsartikel (refereegranskat)abstract
- Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.
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| 50. |
- Pontarp, Mikael, et al.
(författare)
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Adaptation of timing of life history traits and population dynamic responses to climate change in spatially structured populations
- 2015
-
Ingår i: Evolutionary Ecology. - : Springer Science and Business Media LLC. - 1573-8477 .- 0269-7653. ; 29:4, s. 565-579
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Tidskriftsartikel (refereegranskat)abstract
- Changes in the seasonal timing of life history events are documented effects of climate change. We used a general model to study how dispersal and competitive interactions affect eco-evolutionary responses to changes in the temporal distribution of resources over the season. Specifically, we modeled adaptation of the timing of reproduction and population dynamic responses in two competing populations that disperse between two habitats characterized by an early and late resource peak. We investigated three scenarios of environmental change: (1) food peaks advance in both habitats, (2) in the late habitat only and (3) in the early habitat only. At low dispersal rates the evolutionarily stable timing of reproduction closely matched the local resource peak and the environmental change typically caused population decline. Larger dispersal rates rendered less intuitive eco-evolutionary population responses. First, dispersal caused mismatch between evolutionarily stable timing of reproduction and local resource peaks and as a result, reproductive output for subpopulations could increase as well as decrease when resource availability underwent temporal shifts. Second, population responses were contingent on competition between populations. This could accelerate population declines and cause extinctions or even reverse population trends from negative to positive compared to the low dispersal case. When dispersal rate was large and the early resource peak was advanced available niche space was reduced. Hence, even when a population survived the environmental change and obtained positive equilibrium population density, subsequent adaptation of competing populations could drive it to extinction due to convergent evolution and competitive exclusion. These results shed new light on the role of competition and dispersal for the evolution of timing of life history events and provide guidelines for understanding short and long-term population response to climate change.
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