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  • 1
    Electronic Resource
    Electronic Resource
    An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes (1992)
    Springer
    Probability theory and related fields 94 (1992), S. 119-133 
    Details
    ISSN: 1432-2064
    Keywords: 60 G 15 ; 60 F 15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let {X(t),t ≧ 0} be a stationary Gaussian process withEX(t)=0,EX 2(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t |α + o (|t|α)ast→0 for someC〉0, 0〈α≤2; (ii)r(t)=0(t −2γ) as t→∞ for some γ〉0 and (iii) supt≥s|r(t)|〈1 for eachs〉0. Put ξ(t)= sup {s:0 ≦s ≦t,X(s)≧ (2logs)1/2}. The law of the iterated logarithm implies $$lim sup_{t \to \infty } (\xi (t) - t) = 0$$ a.s. This paper gives the lower bound of ζ(t) and obtains an Erdős-Rèvèsz type LIL, i.e., $$\lim \inf _{t \to \infty } (\xi (t) - t)/(t(\log t)^{{{(\alpha - 2)} \mathord{\left/ {\vphantom {{(\alpha - 2)} {(2}}} \right. \kern-\nulldelimiterspace} {(2}}\alpha )} \log \log t) = - (2 + \alpha )\sqrt \pi /(\alpha {\rm H}_\alpha (2C)^{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} )$$ a.s. if 0〈α〈2 and $$\lim \inf _{t \to \infty } \log (\xi (t)/t)/log \log t = - 2\pi /\sqrt {2C} $$ . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01222513
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  • 2
    Electronic Resource
    Electronic Resource
    On almost sure limit inferior for B-valued stochastic processes and applications (1994)
    Springer
    Probability theory and related fields 99 (1994), S. 29-54 
    Details
    ISSN: 1432-2064
    Keywords: 60G17 ; 60G15 ; 60G10 ; 60J65
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01199589
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  • 3
    Electronic Resource
    Electronic Resource
    A Cramér Type Large Deviation Result for Student's t-Statistic (1999)
    Springer
    Journal of theoretical probability 12 (1999), S. 385-398 
    Details
    ISSN: 1572-9230
    Keywords: t-statistic ; asymptotic distribution ; self-normalized large deviation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let X, X 1, X 2,... be independent and identically distributed random variables with a finite third moment, and let T n be the Student's t-statistic. This paper shows that lim n→∞ P(T n〉x)/P(t n〉x)=1 holds uniformly in 0≤x≤o(n 1/6), where t n has a t-distribution with n−1 degrees of freedom. An example is also given to show that a finite third moment is necessary for this result.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021626127372
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  • 4
    Electronic Resource
    Electronic Resource
    A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables (2000)
    Springer
    Journal of theoretical probability 13 (2000), S. 343-356 
    Details
    ISSN: 1572-9230
    Keywords: negative dependence ; independent random variables ; comparison theorem ; moment inequality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {X i, 1≤i≤n} be a negatively associated sequence, and let {X* i , 1≤i≤n} be a sequence of independent random variables such that X* i and X i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(∑ n i=1 X i)≤Ef(∑ n i=1 X* i ) for any convex function f on R 1 and that Ef(max1≤k≤n ∑ n i=k X i)≤Ef(max1≤k≤n ∑ k i=1 X* i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007849609234
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  • 5
    Electronic Resource
    Electronic Resource
    Self-normalized central limit theorem for sums of weakly dependent random variables (1994)
    Springer
    Journal of theoretical probability 7 (1994), S. 309-338 
    Details
    ISSN: 1572-9230
    Keywords: Central limit theorem ; weakly dependent random variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {X n,n≥1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX n=μ,EX n 2 〈∞,Var S n /n→σ2 and the central limit theorem. This paper presents two estimators of σ2. Their weak and strong consistence as well as their rate of convergence are obtained for α-mixing, ρ-mixing and associated sequences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02214272
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  • 6
    Electronic Resource
    Electronic Resource
    Small Ball Estimates for Gaussian Processes under Sobolev Type Norms (1999)
    Springer
    Journal of theoretical probability 12 (1999), S. 699-720 
    Details
    ISSN: 1572-9230
    Keywords: Gaussian process ; Wiener process ; fractional Brownian motion ; Sobolev norm ; Chung's LIL
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation theory is developed for small ball estimates. As an application the Chung's LIL for fractional Brownian motions is given in this setting.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021675731663
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  • 7
    Electronic Resource
    Electronic Resource
    A note on small ball probability of a Gaussian process with stationary increments (1993)
    Springer
    Journal of theoretical probability 6 (1993), S. 595-602 
    Details
    ISSN: 1572-9230
    Keywords: Small ball probability ; Gaussian process ; fractional Wiener process
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {X(t), 0≤t≤1} be a Gaussian process with mean zero and stationary increments. Let σ2(h) =EX 2(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01066719
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  • 8
    Electronic Resource
    Electronic Resource
    Small ball probabilities for Gaussian processes with stationary increments under Hölder norms (1995)
    Springer
    Journal of theoretical probability 8 (1995), S. 361-386 
    Details
    ISSN: 1572-9230
    Keywords: Gaussian processes ; small ball probabilities ; the law of the iterated logarithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Small ball probabilities are estimated for Gaussian processes with stationary increments when the small balls are given by various Hölder norms. As an application we establish results related to Chung's functional law of the iterated logarithm for fractional Brownian motion under Hölder norms. In particular, we identify the points approached slowest in the functional law of the iterated logarithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02212884
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  • 9
    Electronic Resource
    Electronic Resource
    Existence of Bayesian Estimates for the Polychotomous Quantal Response Models (1999)
    Springer
    Annals of the Institute of Statistical Mathematics 51 (1999), S. 637-656 
    Details
    ISSN: 1572-9052
    Keywords: Improper prior ; logit model ; log-log model ; posterior distribution ; probit model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper investigates the existence of Bayesian estimates for polychotomous quantal response models using a uniform improper prior distribution on the regression parameters. Necessary and sufficient conditions for the propriety of the posterior distribution with a general link function are established. In addition, the sufficient conditions for the existence of the posterior moments and the posterior moment generating function are obtained. It is also found that the propriety guarantees the existence of the maximum likelihood estimate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004027028943
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  • 10
    Electronic Resource
    Electronic Resource
    Bahadur efficiency and robustness of studentized score tests (1996)
    Springer
    Annals of the Institute of Statistical Mathematics 48 (1996), S. 295-314 
    Details
    ISSN: 1572-9052
    Keywords: Bahadur slope ; efficiency ; influence function ; score test
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We derive the exact Bahadur slopes of studentized score tests for a simple null hypothesis in a one-parameter family of distributions. The Student's t-test is included as a special case for which a recent result of Rukhin (1993, Sankhyā Ser. A, 55, 159–163) was improved upon. It is shown that locally optimal Bahadur efficiency for one-sample location models with a known or estimated scale parameter is attained within the class of studentized score tests. The studentized test has an asymptotic null distribution free of the scale parameter, and the optimality of likelihood scores does not depend on the existence of a moment generating function. We also consider the influence function and breakdown point of such tests as part of our robustness investigation. The influence of any studentized score test is bounded from above, indicating certain degree of robustness of validity, but a bounded score function is needed to cap the influence from below and to ensure a high power breakdown point. We find that the standard Huber-type score tests are not only locally minimax in Bahadur efficiency, but also very competitive in global efficiency at a variety of location models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00054792
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