2,248 research outputs found
Bootstrap-Based Inference for Cube Root Asymptotics
This paper proposes a valid bootstrap-based distributional approximation for
M-estimators exhibiting a Chernoff (1964)-type limiting distribution. For
estimators of this kind, the standard nonparametric bootstrap is inconsistent.
The method proposed herein is based on the nonparametric bootstrap, but
restores consistency by altering the shape of the criterion function defining
the estimator whose distribution we seek to approximate. This modification
leads to a generic and easy-to-implement resampling method for inference that
is conceptually distinct from other available distributional approximations. We
illustrate the applicability of our results with four examples in econometrics
and machine learning
Toward a Deterministic Model of Planetary Formation VII: Eccentricity Distribution of Gas Giants
The ubiquity of planets and diversity of planetary systems reveal planet
formation encompass many complex and competing processes. In this series of
papers, we develop and upgrade a population synthesis model as a tool to
identify the dominant physical effects and to calibrate the range of physical
conditions. Recent planet searches leads to the discovery of many
multiple-planet systems. Any theoretical models of their origins must take into
account dynamical interaction between emerging protoplanets. Here, we introduce
a prescription to approximate the close encounters between multiple planets. We
apply this method to simulate the growth, migration, and dynamical interaction
of planetary systems. Our models show that in relatively massive disks, several
gas giants and rocky/icy planets emerge, migrate, and undergo dynamical
instability. Secular perturbation between planets leads to orbital crossings,
eccentricity excitation, and planetary ejection. In disks with modest masses,
two or less gas giants form with multiple super-Earths. Orbital stability in
these systems is generally maintained and they retain the kinematic structure
after gas in their natal disks is depleted. These results reproduce the
observed planetary mass-eccentricity and semimajor axis-eccentricity
correlations. They also suggest that emerging gas giants can scatter residual
cores to the outer disk regions. Subsequent in situ gas accretion onto these
cores can lead to the formation of distant (> 30AU) gas giants with nearly
circular orbits.Comment: 54 pages, 14 Figures; accepted for publication in Astrophysical
Journa
Eccentricity Evolution of Extrasolar Multiple Planetary Systems due to the Depletion of Nascent Protostellar Disks
Most extrasolar planets are observed to have eccentricities much larger than
those in the solar system. Some of these planets have sibling planets, with
comparable masses, orbiting around the same host stars. In these multiple
planetary systems, eccentricity is modulated by the planets' mutual secular
interaction as a consequence of angular momentum exchange between them. For
mature planets, the eigenfrequencies of this modulation are determined by their
mass and semi-major axis ratios. But, prior to the disk depletion, self gravity
of the planets' nascent disks dominates the precession eigenfrequencies. We
examine here the initial evolution of young planets' eccentricity due to the
apsidal libration or circulation induced by both the secular interaction
between them and the self gravity of their nascent disks. We show that as the
latter effect declines adiabatically with disk depletion, the modulation
amplitude of the planets' relative phase of periapse is approximately invariant
despite the time-asymmetrical exchange of angular momentum between planets.
However, as the young planets' orbits pass through a state of secular
resonance, their mean eccentricities undergo systematic quantitative changes.
For applications, we analyze the eccentricity evolution of planets around
Upsilon Andromedae and HD168443 during the epoch of protostellar disk
depletion. We find that the disk depletion can change the planets' eccentricity
ratio. However, the relatively large amplitude of the planets' eccentricity
cannot be excited if all the planets had small initial eccentricities.Comment: 50 pages including 11 figures, submitted to Ap
Hierarchy of QM SUSYs on a Bounded Domain
We systematically formulate a hierarchy of isospectral Hamiltonians in
one-dimensional supersymmetric quantum mechanics on an interval and on a
circle, in which two successive Hamiltonians form N=2 supersymmetry. We find
that boundary conditions compatible with supersymmetry are severely restricted.
In the case of an interval, a hierarchy of, at most, three isospectral
Hamiltonians is possible with unique boundary conditions, while in the case of
a circle an infinite tower of isospectral Hamiltonians can be constructed with
two-parameter family of boundary conditions.Comment: 15 pages, 3 figure
Multiple-Planet Scattering and the Origin of Hot Jupiters
Exoplanets show a pile-up of Jupiter-size planets in orbits with a 3-day
period. A fraction of these hot Jupiters have retrograde orbits with respect to
the parent star's rotation. To explain these observations we performed a series
of numerical integrations of planet scattering followed by the tidal
circularization. We considered planetary systems having 3 and 4 planets
initially. We found that the standard Kozai migration is an inefficient
mechanism for the formation of hot Jupiters.
Our results show the formation of two distinct populations of hot Jupiters.
The inner population of hot Jupiters with semimajor axis a < 0.03 AU formed in
the systems where no planetary ejections occurred. This group contained a
significant fraction of highly inclined and retrograde orbits, with
distributions largely independent of the initial setup. However, our follow-up
integrations showed that this populations was transient with most planets
falling inside the Roche radius of the star in <1 Gyr. The outer population of
hot Jupiters formed in systems where at least one planet was ejected. This
population survived the effects of tides over >1 Gyr. The semimajor axis
distribution of Population II fits nicely the observed 3-day pile-up.
The inclination distribution of the outer hot planets depends on the number
of planets in the initial systems and the 4-planet case showed a larger
proportion (up to 10%), and a wider spread in inclination values. As the later
results roughly agrees with observations, this may suggest that the planetary
systems with observed hot Jupiters were originally rich in the number of
planets, some of which were ejected. In a broad perspective, our work therefore
hints on an unexpected link between the hot Jupiters and recently discovered
free floating planets.Comment: submitted to Ap
Bootstrap-Assisted Inference for Generalized Grenander-type Estimators
Westling and Carone (2020) proposed a framework for studying the large sample
distributional properties of generalized Grenander-type estimators, a versatile
class of nonparametric estimators of monotone functions. The limiting
distribution of those estimators is representable as the left derivative of the
greatest convex minorant of a Gaussian process whose covariance kernel can be
complicated and whose monomial mean can be of unknown order (when the degree of
flatness of the function of interest is unknown). The standard nonparametric
bootstrap is unable to consistently approximate the large sample distribution
of the generalized Grenander-type estimators even if the monomial order of the
mean is known, making statistical inference a challenging endeavour in
applications. To address this inferential problem, we present a
bootstrap-assisted inference procedure for generalized Grenander-type
estimators. The procedure relies on a carefully crafted, yet automatic,
transformation of the estimator. Moreover, our proposed method can be made
``flatness robust" in the sense that it can be made adaptive to the (possibly
unknown) degree of flatness of the function of interest. The method requires
only the consistent estimation of a single scalar quantity, for which we
propose an automatic procedure based on numerical derivative estimation and the
generalized jackknife. Under random sampling, our inference method can be
implemented using a computationally attractive exchangeable bootstrap
procedure. We illustrate our methods with examples and we also provide a small
simulation study. The development of formal results is made possible by some
technical results that may be of independent interest
Entropy, time irreversibility and Schroedinger equation in a primarily discrete space-time
In this paper we show that the existence of a primarily discrete space-time
may be a fruitful assumption from which we may develop a new approach of
statistical thermodynamics in pre-relativistic conditions. The discreetness of
space-time structure is determined by a condition that mimics the Heisenberg
uncertainty relations and the motion in this space-time model is chosen as
simple as possible. From these two assumptions we define a path-entropy that
measures the number of closed paths associated with a given energy of the
system preparation. This entropy has a dynamical character and depends on the
time interval on which we count the paths. We show that it exists an
like-equilibrium condition for which the path-entropy corresponds exactly to
the usual thermodynamic entropy and, more generally, the usual statistical
thermodynamics is reobtained. This result derived without using the Gibbs
ensemble method shows that the standard thermodynamics is consistent with a
motion that is time-irreversible at a microscopic level. From this change of
paradigm it becomes easy to derive a . A comparison with the
traditional Boltzmann approach is presented. We also show how our approach can
be implemented in order to describe reversible processes. By considering a
process defined simultaneously by initial and final conditions a well defined
stochastic process is introduced and we are able to derive a Schroedinger
equation, an example of time reversible equation.Comment: latex versio
On the stochastic mechanics of the free relativistic particle
Given a positive energy solution of the Klein-Gordon equation, the motion of
the free, spinless, relativistic particle is described in a fixed Lorentz frame
by a Markov diffusion process with non-constant diffusion coefficient. Proper
time is an increasing stochastic process and we derive a probabilistic
generalization of the equation . A
random time-change transformation provides the bridge between the and the
domain. In the domain, we obtain an \M^4-valued Markov process
with singular and constant diffusion coefficient. The square modulus of the
Klein-Gordon solution is an invariant, non integrable density for this Markov
process. It satisfies a relativistically covariant continuity equation
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