scholarly journals A Bi-Invariant Statistical Model Parametrized by Mean and Covariance on Rigid Motions

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 432
Author(s):  
Emmanuel Chevallier ◽  
Nicolas Guigui
Keyword(s):  
Statistical Model ◽  
Convergence Rates ◽  
Probability Distributions ◽  
Exponential Map ◽  
Jacobian Determinant ◽  
Numerical Comparison ◽  
Rigid Motions ◽  
Invariant Statistics ◽  
The Moment ◽  
Explicit Expressions

This paper aims to describe a statistical model of wrapped densities for bi-invariant statistics on the group of rigid motions of a Euclidean space. Probability distributions on the group are constructed from distributions on tangent spaces and pushed to the group by the exponential map. We provide an expression of the Jacobian determinant of the exponential map of S E ( n ) which enables the obtaining of explicit expressions of the densities on the group. Besides having explicit expressions, the strengths of this statistical model are that densities are parametrized by their moments and are easy to sample from. Unfortunately, we are not able to provide convergence rates for density estimation. We provide instead a numerical comparison between the moment-matching estimators on S E ( 2 ) and R 3 , which shows similar behaviors.

A comparison of convergence rates for three models in the theory of dams

1997 ◽  
Vol 34 (01) ◽  
pp. 74-83
Author(s):  
Robert Lund ◽  
Walter Smith
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Sample Path ◽  
Jump Process ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Input Process ◽  
The Moment ◽  
Finite Dam ◽  
General Conditions

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

scholarly journals Some Moment Relations for the Hipp approximation

1996 ◽  
Vol 26 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Jan Dhaene ◽  
Bjørn Sundt ◽  
Nelson De Pril
Keyword(s):  
Finite Number ◽  
Present Note ◽  
Order Approximation ◽  
Probability Distributions ◽  
Higher Order ◽  
Exact Distribution ◽  
Higher Order Moments ◽  
The Moment

AbstractIn the present note we consider the Hipp approximation to the convolution of a finite number of probability distributions on the non-negative integers. It is shown that the moment up to and including order r of the rth order approximation are equal to the corresponding moments of the exact distribution. We also give a relation between the exact and approximated (r + 1)th order moments and indicate how similar relations can be obtained for higher order moments.

A comparison of convergence rates for three models in the theory of dams

1997 ◽  
Vol 34 (1) ◽  
pp. 74-83
Author(s):  
Robert Lund ◽  
Walter Smith
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Sample Path ◽  
Jump Process ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Input Process ◽  
The Moment ◽  
Finite Dam ◽  
General Conditions

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

A Dam with seasonal input

1994 ◽  
Vol 31 (2) ◽  
pp. 526-541 ◽  
Author(s):  
Robert B. Lund
Keyword(s):  
Laplace Transform ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Stochastic Monotonicity ◽  
The Laplace Transform ◽  
Monotonicity Result ◽  
The Mean ◽  
The Moment ◽  
Necessary And Sufficient

This paper examines the infinitely high dam with seasonal (periodic) Lévy input under the unit release rule. We show that a periodic limiting distribution of dam content exists whenever the mean input over a seasonal cycle is less than 1. The Laplace transform of dam content at a finite time and the Laplace transform of the periodic limiting distribution are derived in terms of the probability of an empty dam. Necessary and sufficient conditions for the periodic limiting distribution to have finite moments are given. Convergence rates to the periodic limiting distribution are obtained from the moment results. Our methods of analysis lean heavily on the coupling method and a stochastic monotonicity result.

UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH

2016 ◽  
Vol 33 (4) ◽  
pp. 874-914 ◽  
Author(s):  
Shin Kanaya
Keyword(s):  
Uniform Convergence ◽  
Continuous Time ◽  
Convergence Rates ◽  
Diffusion Processes ◽  
Kernel Functions ◽  
Damping Function ◽  
Nonparametric Estimators ◽  
Empirical Process Theory ◽  
Convergence Results ◽  
The Moment

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called adamping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

scholarly journals Stieltjes classes for discrete distributions of logarithmic type

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2533-2540
Author(s):  
Sofiya Ostrovska ◽  
Mehmet Turan
Keyword(s):  
Significant Role ◽  
Moment Problem ◽  
Probability Distributions ◽  
Infinite Family ◽  
Discrete Distributions ◽  
Discrete Case ◽  
Continuous Distributions ◽  
Absolutely Continuous ◽  
The Moment ◽  
Logarithmic Type

Stieltjes classes play a significant role in the moment problem since they permit to expose explicitly an infinite family of probability distributions all having equal moments of all orders. Mostly, the Stieltjes classes have been considered for absolutely continuous distributions. In this work, they have been considered for discrete distributions. New results on their existence in the discrete case are presented.

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