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Bernstein inequality for the parameter of the pth order autoregressive process AR(p)

Samir Benaissa (2006)

Applicationes Mathematicae

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: X ̃ t = θ ̃ X ̃ t - 1 + ε ̃ t . In this paper we study the convergence in distribution of the linear operator I ( θ ̃ T , θ ̃ ) = ( θ ̃ T - θ ̃ ) θ ̃ T - 2 for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.

Binary segmentation and Bonferroni-type bounds

Michal Černý (2011)

Kybernetika

We introduce the function Z ( x ; ξ , ν ) : = ∫ - ∞ x ϕ ( t - ξ ) · Φ ( ν t ) d t , where ϕ and Φ are the pdf and cdf of N ( 0 , 1 ) , respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables...

Bounded double square functions

Jill Pipher (1986)

Annales de l'institut Fourier

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For f ∈ L l o c 1 ( R 2 ) , we determine the sharp order of local integrability obtained when the square function of f is in L ∞ . The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of C p in ∥ f ∥ p ≤ C p ∥ S f ∥ p .

Bounds on tail probabilities for negative binomial distributions

Peter Harremoës (2016)

Kybernetika

In this paper we derive various bounds on tail probabilities of distributions for which the generated exponential family has a linear or quadratic variance function. The main result is an inequality relating the signed log-likelihood of a negative binomial distribution with the signed log-likelihood of a Gamma distribution. This bound leads to a new bound on the signed log-likelihood of a binomial distribution compared with a Poisson distribution that can be used to prove an intersection property...

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