Cramer Condition Research Articles

Cramer Large Deviations when the Extreme Conjugate Distribution is Heavy-tailed

The classical large deviations problem is considered under the assumption that the unilateral Cramer condition holds on a bounded interval. We assume that the extreme conjugate distribution is from the domain of attraction of a stable law. A limit is established up to which the asymptotic Cramer--Petrov representation is valid.

Cramer's condition and Sanov's theorem

We discuss whether Sanov's theorem can be extended to a topology that renders the mapping v → ↦ f d v continuous, for a given measurable function f. We show that this is possible if and only if f possesses all exponential moments with respect to the underlying law μ. AMS classification: 60F10

Large deviations for sums of lattice random variables under the Cramer conditions

Large deviations for sums of lattice random variables under the Cramer conditions

Malliavin calculus and asymptotic expansion for martingales

We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Cramer condition did in the case of independent observations. Applications to statistics are presented.

Local and Global Upper Functions for Random Fields

We introduce and calculate the local and global upper functions for arbitrary, i.e., not necessarily Gaussian, random fields. Only the Cramer condition is assumed to be fulfilled for the fields under consideration. Despite the generality we show by examples that the results obtained are precise for the Gaussian fields studied earlier. Possible applications are described.

Probabilities of Large Deviations on the Whole Axis

Probabilities of Large Deviations on the Whole Axis

Bernstein-type large deviations inequalities for partial sums of strong mixing processes

By using an approximation of partial sums of aprocess by independent r.v.'s we obtain large deviation inequalities. In particular we show that if is a centered real valued process, if there exist constants m and M such that and if 1≥p n ≥n/2 then, for every ϵ>0, we have where (α(p), p ≥ 1) denote ihe strong mixing coefficients of (Xt). This inequality is simpler and more explicit than those which are actually available in the literature. Similarly we obtain an inequality for processes satisfying Cramer's conditions. Applications to the rate of convergence in the law of large numbers and to nonparametric regression with dependent errors are given.

Lower Bounds for Large Deviations in Rd, d > 1

AbstractWe consider the sums Z(n) of i.i.d. random vectors taking values in a d‐dimensional Euclidean space. It is assumed that the so‐called CRAMER condition holds in a neighbourhood of the origin. We establish lower bounds for the large deviation probabilities P(Z(n) ∉ A) with A belonging to a large class of sets.

Some Estimates for Probabilities of One-Sided Large Deviations

Previous article Next article Some Estimates for Probabilities of One-Sided Large DeviationsL. V. RozovskiiL. V. Rozovskiihttps://doi.org/10.1137/1130108PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. N. Amosova, The rate of convergence in the one-sided law of large numbers, Litovsk. Mat. Sb., 16 (1976), 5–12, 231, (In Russian.) 54:8813 0345.60018 Google Scholar[2] M. U. Gafurov, On estimates of the convergence rate in the law of the iterated logarithm, Abstracts of Reports of the 4th Japanese-Soviet Symposium on Probability Theory and Mathematical Statistics, Vol. I, Metsniereba, Tbilisi, 1982, 213–214, (In Russian.) Google Scholar[3] M. U. Gafurov and , A. D. Slastnikov, On the distribution of the time of last exit, the excess and the number of crossings of a random walk over a curvilinear boundary, Dokl. Akad. Nauk SSSR, 257 (1981), 526–529, (In Russian.) 82d:60125 Google Scholar[4] T. Avery Davis, Convergence rates for the law of the iterated logarithm, Ann. Math. Statist., 39 (1968), 1479–1485 40:6626 0174.49902 CrossrefGoogle Scholar[5] B. F. Skovoroda, On the convergence rate in the law of the iterated logarithm, Manuscript deposited in the VINITI, No. 4706-4783. (In Russian.) Google Scholar[6] L. V. Rozovskii˘, Estimates of the rate of convergence in the weak law of large numbers, Litovsk. Mat. Sb., 20 (1980), 147–163, 211, (In Russian.) 83i:60030 0463.60029 Google Scholar[7] M. U. Gafurov and , V. I. Rotar', On the exit of a random walk from curvilinear boundary, Theory Prob. Appl., 28 (1983), 179–184 0531.60065 LinkGoogle Scholar[8] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346 52:9335 0322.60042 CrossrefGoogle Scholar[9] I. F. Pinelis, Limit theorems on large deviations with Cramer's condition violated for sums of infinite-dimensional random variables, Manuscript deposited in the VINITI, No. 1674-1681. (In Russian.) Google Scholar[10] A. I. Martikainen, One-sided versions of the law of large numbers: strong law and convergence rate, Sixteenth All-union School-Colloquium on Probability Theory and Mathematical Statistics, Bakuriani, Feb. 26-Mar. 5, 1982, Summaries of Reports, izd-vo TGU, Tbilisi, 1982, 48–54, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Small Deviation Probabilities for Maximum of Partial SumsL. V. Rozovsky11 November 2010 | Theory of Probability & Its Applications, Vol. 54, No. 4AbstractPDF (156 KB) Volume 30, Issue 4| 1986Theory of Probability & Its Applications History Submitted:29 March 1984Published online:28 July 2006 InformationCopyright © 1986 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130108Article page range:pp. 850-855ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

An Inconsistent Maximum Likelihood Estimate

Abstract An example is given of a family of distributions on [— 1, 1] with a continuous one-dimensional parameterization that joins the triangular distribution (when Θ = 0) to the uniform (when Θ = 1), for which the maximum likelihood estimates exist and converge strongly to Θ = 1 as the sample size tends to infinity, whatever be the true value of the parameter. A modification that satisfies Cramer's conditions is also given.

ON ESTIMATES AND THE ASYMPTOTIC BEHAVIOR OF THE PROBABILITY OF NONINTERSECTION OF MOVING BOUNDARIES BY SUMS OF INDEPENDENT RANDOM VARIABLES

This paper studies estimates and the asymptotic behavior as for the probabilities , and , , where , the being independent identically distributed random variables with mean zero, and and are nonrandom functions. Under certain restrictions on the boundaries and logarithmic asymptotes of these probabilities are found in the case when the satisfy (respectively) a two-sided or a one-sided Cramer condition. The method is based on an absolutely continuous substitution for the original probability measure. Bibliography: 18 titles.

Jackknifing Maximum Likelihood Estimates

This paper proves the apparently outstanding conjecture that the maximum likelihood estimate (m.l.e.) "behaves properly" when jackknifed. In particular, under the usual Cramer conditions (1) the jackknifed version of the consistent root of the m.l. equation has the same asymptotic distribution as the consistent root itself, and (2) the jackknife estimate of the variance of the asymptotic distribution of the consistent root is itself consistent. Further, if the hypotheses of Wald's consistency theorem for the m.l.e. are satisfied, then the above claims hold for the m.l.e. (as well as for the consistent root).

Asymptotic expansions in the central limit theorem under moment conditions

Let X 1, X 2,... be a sequence of independent k-dimensional random vectors with mean 0, covariance matrix I, and finite absolute moments of order s≧3. Let, furthermore, s 0 be the integral part of s and ψ be the formal Edgeworth expansion for $$S_n = n^{ - 1/2} \sum\limits_{j = 1}^n {X_j } $$ of length s 0 −2 and Q n be the distribution of S n. Under a Lindeberg type condition we prove that (i) $$\int {fd(Q_n - \Psi ) = o(n^{ - (s_0 - 2)/2} } $$ for smooth functions f; (ii) $$\int {\left\| x \right\|^r d(Q_n - \Psi ) = o(n^{ - (s - 2)/2} ) + O(n^{ - (r + k)/2} )} $$ for 0≦r≧s; and (iii) $$\int {\left\| x \right\|^s 1_{\left\{ {\left\| x \right\| > ((s - 2)\log n)^{1/2} } \right\}} (x)dQ_n = o(n^{ - (s - 2)/2} )} $$ . We do not assume that the distributions of X 1, X 2,... are lattice or satisfy a Cramer condition. Hence our results hold also for random vectors X 1, X 2,... having discrete nonlattice distributions.

Estimates of the higher-order spectral densities of stationary processes which satisfy the Cramer condition with ?Rosenblatt? mixing

Estimates of the higher-order spectral densities of stationary processes which satisfy the Cramer condition with ?Rosenblatt? mixing

A Survey of Maximum Likelihood Estimation: Part 2

In Part 2 of this survey the discussion is mainly confined to the problem of comparing maximum likelihood estimation with other estimation procedures and the related problems that arise by reason of the existence of inconsistent maximum likelihood estimates, and of other estimators which are superefficient, with regard to which special reference is made to the work of C. R. Rao (1961a and 1962a). Initially, however, there is a further discussion of Cramer's conditions as they relate to consistency and efficiency, and there is also a section on the general multiparameter situation. All references are contained within the bibliography at the end of Part 1 (Int. Statist. Rev., Vol. 40, No. 3, 1972, pp. 329-354), but a number of important definitions are given separately in the Appendix at the end of this paper.

Asymptotic expansions for large deviations when Cramer's condition fails

Asymptotic expansions for large deviations when Cramer's condition fails

Rates of Weak Convergence and Asymptotic Expansions for Classical Central Limit Theorems

Rates of Weak Convergence and Asymptotic Expansions for Classical Central Limit Theorems

On the conditions for consistency and asymptotic efficiency of maximum likelihood estimates

Abstract Extract Cramer [1] has given sufficient conditions for consistency and asymptotic efficiency of the root of the likelihood equation (Theorem 1 below). In this paper we give two new theorems concerning the same asymptotic properties. Theorem 2 is a direct generalization of Cramer's theorem, in that one of his conditions is replaced by a weaker one. Theorem 3 contains an alternative set of conditions: in contrast to Theorems 1 and 2 it does not assume that the third partial derivative of the density function with respect to the parameter exists, and the consistency property does not even require the existence of the second partial derivative. Also considered is an example which does not satisfy one of Cramer's conditions but which satisfies both the weaker set of conditions in the generalized Theorem 2 and the new set of conditions in Theorem 3.