Conditional Distribution and Expectation

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

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NONLINEAR PANEL DATA MODELS WITH DISTRIBUTION-FREE CORRELATED RANDOM EFFECTS

Econometric Theory ◽

10.1017/s0266466620000481 ◽

2021 ◽

pp. 1-25

Author(s):

Yu-Chin Hsu ◽

Ji-Liang Shiu

Keyword(s):

Panel Data ◽

Data Model ◽

Conditional Distribution ◽

Unobserved Heterogeneity ◽

Random Effect ◽

Data Models ◽

Panel Data Model ◽

Panel Data Models ◽

Likelihood Functions ◽

Correlated Random Effects

Under a Mundlak-type correlated random effect (CRE) specification, we first show that the average likelihood of a parametric nonlinear panel data model is the convolution of the conditional distribution of the model and the distribution of the unobserved heterogeneity. Hence, the distribution of the unobserved heterogeneity can be recovered by means of a Fourier transformation without imposing a distributional assumption on the CRE specification. We subsequently construct a semiparametric family of average likelihood functions of observables by combining the conditional distribution of the model and the recovered distribution of the unobserved heterogeneity, and show that the parameters in the nonlinear panel data model and in the CRE specification are identifiable. Based on the identification result, we propose a sieve maximum likelihood estimator. Compared with the conventional parametric CRE approaches, the advantage of our method is that it is not subject to misspecification on the distribution of the CRE. Furthermore, we show that the average partial effects are identifiable and extend our results to dynamic nonlinear panel data models.

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HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788709000378 ◽

2010 ◽

Vol 88 (1) ◽

pp. 93-102 ◽

Cited By ~ 12

Author(s):

MARGARYTA MYRONYUK

Keyword(s):

Abelian Group ◽

Automorphism Group ◽

Real Line ◽

Linear Form ◽

Conditional Distribution ◽

Random Variables ◽

Characterization Theorem ◽

Gaussian Distributions ◽

Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

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Asymptotic normality of conditional distribution estimation in the single index model

Acta Universitatis Sapientiae Mathematica ◽

10.1515/ausm-2017-0010 ◽

2017 ◽

Vol 9 (1) ◽

pp. 162-175

Author(s):

Diaa Eddine Hamdaoui ◽

Amina Angelika Bouchentouf ◽

Abbes Rabhi ◽

Toufik Guendouzi

Keyword(s):

Distribution Function ◽

Confidence Interval ◽

Asymptotic Normality ◽

Conditional Distribution ◽

Conditional Distribution Function ◽

Single Index ◽

Index Model ◽

Single Index Model ◽

Distribution Estimation

AbstractThis paper deals with the estimation of conditional distribution function based on the single-index model. The asymptotic normality of the conditional distribution estimator is established. Moreover, as an application, the asymptotic (1 − γ) confidence interval of the conditional distribution function is given for 0 < γ < 1.

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A nonlinear filtering algorithm based on an approximation of the conditional distribution

IEEE Transactions on Automatic Control ◽

10.1109/9.847749 ◽

2000 ◽

Vol 45 (3) ◽

pp. 580-585 ◽

Cited By ~ 38

Author(s):

H.J. Kushner ◽

A.S. Budhiraja

Keyword(s):

Conditional Distribution ◽

Nonlinear Filtering ◽

Filtering Algorithm

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On the Conditional Distribution of Goodness-of-Fit Tests

Communication in Statistics- Theory and Methods ◽

10.1080/03610920500476622 ◽

2006 ◽

Vol 35 (3) ◽

pp. 541-549 ◽

Cited By ~ 7

Author(s):

Federico O'Reilly ◽

Leticia Gracia-Medrano

Keyword(s):

Conditional Distribution ◽

Goodness Of Fit ◽

Goodness Of Fit Tests

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Some calculations relevant to thermal annealing of fission tracks in apatite

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0109 ◽

1988 ◽

Vol 419 (1857) ◽

pp. 305-321 ◽

Cited By ~ 17

Keyword(s):

Conditional Distribution ◽

Fission Track ◽

Length Distribution ◽

Track Length ◽

Apatite Crystal ◽

Routine Practice ◽

Fission Tracks ◽

Line Segments ◽

Segment Model ◽

Arbitrary Plane

Calculations in stochastic geometry are applied to the geological problem of analysing the statistical distribution of fission tracks in an apatite crystal, when information is available only by plane sampling. The feature of particular interest is the effect of anisotropy, in the sense of dependence of track length on orientation. Using a realistic Poisson line-segment model, we obtain formulae for the density of line segments intersecting an arbitrary plane and for the length distributions of confined tracks, semi-tracks and projected semi-tracks in terms of the conditional distribution of length given orientation. These formulae are used to explain and quantify the effect of anisotropy seen in experimental data from fission track annealing studies. We argue that track orientations, in addition to lengths, carry potentially useful information. For confined tracks, we recommend that both length and angle to the c -axis be measured as routine practice. For projected semi-tracks, where it is much harder to extract useful information from the observed length distribution, the measurement of angle, in addition to length, may prove fruitful, particularly when confined tracks are scarce.

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