Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation

2020 ◽  
Vol 30 (6) ◽  
pp. 417-437
Author(s):  
Valeriy A. Voloshko ◽  
Yuriy S. Kharin
Keyword(s):  
Long Memory ◽  
Asymptotic Variance ◽  
Sufficient Conditions ◽  
Statistical Parameter ◽  
Asymptotically Normal ◽  
Discrete Random Variables ◽  
Recursive Computation ◽  
Computation Algorithm ◽  
Nonlinear Autoregressive ◽  
Nonlinear Autoregressive Models

AbstractWe introduce a new model ${\mathscr{P}}$-CNAR(s) of sequences of discrete random variables with long memory determined by semibinomial conditionally nonlinear autoregression of order s ∈ ℕ with small number of parameters. Probabilistic properties of this model are studied. For parameters of the model ${\mathscr{P}}$-CNAR a family of consistent asymptotically normal statistical FB-estimates is suggested and the existence of an efficient FB-estimate is proved. Computational advantages of FB-estimate w.r.t. maximum likelihood estimate are shown: less restrictive sufficient conditions for uniqueness, explicit form of FB-estimate, fast recursive computation algorithm under extension of the model ${\mathscr{P}}$-CNAR. Subfamily of “sparse” FB-estimates that use some subset of frequencies of s-tuples is constructed, the asymptotic variance minimization problem within this subfamily is solved.

TESTING FOR LONG MEMORY IN VOLATILITY

2002 ◽  
Vol 18 (6) ◽  
pp. 1291-1308 ◽  
Author(s):  
Clifford M. Hurvich ◽  
Philippe Soulier
Keyword(s):  
Long Memory ◽  
Null Hypothesis ◽  
Asymptotic Variance ◽  
Wald Test ◽  
Stochastic Volatility Models ◽  
Asymptotically Normal ◽  
Volatility Models ◽  
Regression Estimators ◽  
Mean And Variance ◽  
Short Memory

We consider the asymptotic behavior of log-periodogram regression estimators of the memory parameter in long-memory stochastic volatility models, under the null hypothesis of short memory in volatility. We show that in this situation, if the periodogram is computed from the log squared returns, then the estimator is asymptotically normal, with the same asymptotic mean and variance that would hold if the series were Gaussian. In particular, for the widely used GPH estimator [d with circumflex above]GPH under the null hypothesis, the asymptotic mean of m1/2[d with circumflex above]GPH is zero and the asymptotic variance is π2/24 where m is the number of Fourier frequencies used in the regression. This justifies an ordinary Wald test for long memory in volatility based on the log periodogram of the log squared returns.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

scholarly journals PARAMETER ESTIMATION IN NONLINEAR AR–GARCH MODELS

2011 ◽  
Vol 27 (6) ◽  
pp. 1236-1278 ◽  
Author(s):  
Mika Meitz ◽  
Pentti Saikkonen
Keyword(s):  
Asymptotic Normality ◽  
Autoregressive Models ◽  
Garch Models ◽  
Martingale Difference ◽  
First Order ◽  
First Results ◽  
Consistency And Asymptotic Normality ◽  
Quasi Maximum Likelihood ◽  
Nonlinear Autoregressive ◽  
Nonlinear Autoregressive Models

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

On the weak invariance principle for non-adapted stationary random fields under projective criteria

2021 ◽  
Author(s):  
Han-Mai Lin
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Long Memory ◽  
Sufficient Conditions ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Projective Criteria ◽  
Stationary Random Fields ◽  
Do So

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

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