scholarly journals A Family of Non-Gaussian Martingales with Gaussian Marginals

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Kais Hamza ◽  
Fima C. Klebaner
Keyword(s):  
Poisson Distribution ◽  
Gamma Distribution ◽  
Quadratic Variation ◽  
Convolution Semigroup ◽  
Jump Processes ◽  
Marginal Distributions ◽  
Semigroup Property ◽  
Infinitesimal Generators ◽  
The Family ◽  
Non Gaussian

We construct a family of martingales with Gaussian marginal distributions. We give a weak construction as Markov, inhomogeneous in time processes, and compute their infinitesimal generators. We give the predictable quadratic variation and show that the paths are not continuous. The construction uses distributions Gσ having a log-convolution semigroup property. Further, we categorize these processes as belonging to one of two classes, one of which is made up of piecewise deterministic pure jump processes. This class includes the case where Gσ is an inverse log-Poisson distribution. The processes in the second class include the case where Gσ is an inverse log-gamma distribution. The richness of the family has the potential to allow for the imposition of specifications other than the marginal distributions.

Mass transportation problems with capacity constraints

1999 ◽  
Vol 36 (2) ◽  
pp. 433-445 ◽  
Author(s):  
S. T. Rachev ◽  
I. Olkin
Keyword(s):  
Joint Distribution ◽  
Bivariate Distribution ◽  
Capacity Constraints ◽  
Distribution Functions ◽  
Mass Transportation ◽  
Marginal Distributions ◽  
Transportation Problems ◽  
The Family ◽  
Additional Capacity ◽  
Additional Constraints

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

scholarly journals Exact inbreeding coefficient and effective size of finite populations under partial sib mating.

Genetics ◽  
1995 ◽  
Vol 140 (1) ◽  
pp. 357-363
Author(s):  
J Wang
Keyword(s):  
Poisson Distribution ◽  
Family Size ◽  
Inbreeding Coefficient ◽  
Stochastic Simulations ◽  
Effective Size ◽  
Finite Populations ◽  
A Value ◽  
The Family ◽  
Sib Mating ◽  
Special Case

Abstract An exact recurrence equation for inbreeding coefficient is derived for a partially sib-mated population of N individuals mated in N/2 pairs. From the equation, a formula for effective size (Ne) taking second order terms of 1/N into consideration is derived. When the family sizes are Poisson or equally distributed, the formula reduces to Ne = [(4 - 3 beta) N/(4 - 2 beta)] + 1 or Ne = [(4 - 3 beta) N/(2 - 2 beta)] - 8/(4 - 3 beta), approximately. For the special case of sib-mating exclusion and Poisson distribution of family size, the formula simplifies to Ne = N + 1, which differs from the previous results derived by many authors by a value of one. Stochastic simulations are run to check our results where disagreements with others are involved.

scholarly journals Estimation of marginal parameters of SUP-OU processes with long range dependence

2016 ◽  
Vol 4 (2) ◽  
pp. 102
Author(s):  
Emanuele Taufer
Keyword(s):  
Long Range ◽  
Asymptotic Theory ◽  
Marginal Distribution ◽  
Real Data ◽  
Function Technique ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Marginal Distributions ◽  
Non Gaussian ◽  
Basic Features

Superpositions of Ornstein Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non Gaussian marginal distributions. Estimation of the parameters of the marginal distribution is undertaken by means of a characteristic function technique. We provide the relevant asymptotic theory as well as results of simulations and real data applications.

scholarly journals The functional Itō formula under the family of continuous semimartingale measures

2016 ◽  
Vol 16 (04) ◽  
pp. 1650010 ◽  
Author(s):  
Harald Oberhauser
Keyword(s):  
Continuous Dependence ◽  
Quadratic Variation ◽  
Stochastic Integrals ◽  
Itô’S Formula ◽  
Itô Formula ◽  
Ito Formula ◽  
Underlying Process ◽  
Continuous Semimartingale ◽  
The Family ◽  
Nonlinear Functionals

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

INFORMATION CONTENT IN THRESHOLD DATA WITH NON-GAUSSIAN NOISE

2007 ◽  
Vol 07 (01) ◽  
pp. L79-L89 ◽  
Author(s):  
PRISCILLA E. GREENWOOD ◽  
PETR LANSKY
Keyword(s):  
Information Content ◽  
Fisher Information ◽  
Binary Data ◽  
Interval Data ◽  
Threshold Data ◽  
The Family ◽  
Fixed Proportion ◽  
Non Gaussian ◽  
Rich Data ◽  
Signal Threshold

If data is binary, it is probable that the combination of a signal of interest plus a noise has been simplified by a thresholding mechanism, as in, e.g., a neuron firing mechanism. For identifying optimal signal or coding range of binary data, Fisher information is an attractive measure. A general formula allows the signal level or signal range producing the most information-rich data to be identified if the noise distribution is known. In this paper we study the information content of binary data resulting from threshold exceedance of a signal plus an arbitrary type of noise. For a specified parametric family of distributions a fixed proportion of exceedances is optimal for any combination of signal, threshold, and noise amplitude. If the ratio of noise to signal level is constant, Fisher information is unimodal for many noise distributions. The results extend to the case of a random signal and to inter-exceedance-interval data. The family of gamma noise distributions is used for illustration.

scholarly journals On the Discretization of Continuous Probability Distributions Using a Probabilistic Rounding Mechanism

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 555
Author(s):  
Chénangnon Frédéric Tovissodé ◽  
Sèwanou Hermann Honfo ◽  
Jonas Têlé Doumatè ◽  
Romain Glèlè Kakaï
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Probability Distributions ◽  
Continuous Distribution ◽  
Simple Expression ◽  
Distribution Model ◽  
Stochastic Representation ◽  
Gamma Distributions ◽  
The Family ◽  
Jensen Shannon Divergence

Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). The new method, referred to as “balanced discretization”, consists of discretizing continuous probability distributions while preserving expectations. It is easy to generate pseudo random variates from the resulting balanced discrete distribution since it has a simple stochastic representation (probabilistic rounding) in terms of the continuous distribution. For illustrative purposes, we develop the family of balanced discrete gamma distributions that can model equi-, under-, and over-dispersed count data. This family of count distributions is appropriate for building flexible count regression models because the expectation of the distribution has a simple expression in terms of the parameters of the distribution. Using the Jensen–Shannon divergence measure, we show that under the equidispersion restriction, the family of balanced discrete gamma distributions is similar to the Poisson distribution. Based on this, we conjecture that while covering all types of dispersions, a count regression model based on the balanced discrete gamma distribution will allow recovering a near Poisson distribution model fit when the data are Poisson distributed.

scholarly journals Some estimates of the normal approximation for mixture of Poisson and gamma random variables

2010 ◽  
Vol 51 ◽  
Author(s):  
Jonas Kazys Sunklodas
Keyword(s):  
Poisson Distribution ◽  
Gamma Distribution ◽  
Upper Bound ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Lp Norms

In the paper, we present the upper bound of Lp norms ∆p of the order (a1 + a2)/(DZ)-1/2 for all 1 < p< ∞, of the normal approximation for a standardized random variable (Z - EZ)/√DZ, where the random variable Z = a1X + a2Y , a1 + a2 = 1, ai > 0, i = 1, 2, the random variable X is distributed by the Poisson distribution with the parameter λ > 0, and the random variable Y by the standard gamma distribution Γ (α, 0, 1) with the parameter α > 0.

Mass transportation problems with capacity constraints

1999 ◽  
Vol 36 (02) ◽  
pp. 433-445 ◽  
Author(s):  
S. T. Rachev ◽  
I. Olkin
Keyword(s):  
Joint Distribution ◽  
Bivariate Distribution ◽  
Capacity Constraints ◽  
Distribution Functions ◽  
Mass Transportation ◽  
Marginal Distributions ◽  
Transportation Problems ◽  
The Family ◽  
Additional Capacity ◽  
Additional Constraints

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g.Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

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