On the maximal inequalities for the average of pairwise i.i.d. random variables

We obtain two sided inequalities for the tail of the maximal function of the averages of a multiple sequence of pairwise i.i.d. random variables taking values in a separable Banach space. We then use the results to establish a necessary and sufficient con¬dition, in terms of the common distribution of the norm of the random variables, for the maximal function to be in L , 1< p << &z.rdang;     

Redundant observations at testing i.i.d. random variables

n = 2 and 3. Here we characterize all testing problems with i.i.d. random variables where an additional observation fails to improve the power. Received: August 31, 2000; revised version: January 10, 2001     

Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions

In this article, we introduce new asymptotic expansions for probability functions of sums of independent and identically distributed random variables. Results are obtained by efficiently employing information provided by lower-order convolutions. In comparison with Edgeworth-type theorems, advantages include improved asymptotic results in the case of symmetric random variables and ease of computation of main error terms and asymptotic crossing points. The first-order estimate can perform quite well against the corresponding renormalized saddlepoint approximation and, pointwise, requires evaluation of only a single convolution integral. While the new expansions are fairly straightforward, the implications are fortuitous and may spur further related work.     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

Extrapolation of subsampling distribution estimators: The i.i.d. and strong mixing cases

Politis & Romano (1994) proposed a general subsampling methodology for the construction of large‐sample confidence regions for an arbitrary parameter under minimal conditions. Nevertheless, the subsampling distribution estimators may sometimes be inefficient (in the case of the sample mean of i.i.d. data, for instance) as compared to alternative estimators such as the bootstrap and/or the asymptotic normal distribution (with estimated variance). The authors investigate here the extent to which the performance of subsampling distribution estimators can be improved by interpolation and extrapolation techniques, while at the same time retaining the robustness property of consistent distribution estimation even in nonregular cases; both i.i.d. and weakly dependent (mixing) observations are considered.     

Correction to: Comparison of joint control schemes for multivariate normal i.i.d. output

AStA Advances in Statistical Analysis - In the original paper, we incorrectly stated that...     

Complete and complete moment convergence for i.i.d. random variables under exponential moment conditions

In this paper, we establish a complete convergence result and a complete moment convergence result for i.i.d. random variables under moment condition which is slightly weaker than the existence of the moment generating function. The main results extend and improve the related known results of Lanzinger (1998 Lanzinger, H. (1998). A Baum-Katz theorem for random variables under exponential moment conditions. Stat. Probab. Lett. 39(2):89–95.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut and Stadtmüller (2011 Gut, A., Stadtmüller, U. (2011). An intermediate Baum-Katz theorem. Stat. Probab. Lett. 81(10):1486–1492.[Crossref], [Web of Science ®] , [Google Scholar]).     

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.     

The variance and the range of i.I.D. random variables

Chew and Teugels (1978) discussed the joint asynptotic behaviour, as n→∞ , of the sum and the maximum of n i.i.d. randan variables. Here it is noted that it is more natural to include also the minimum in the discussion, and that one can extend their argument to allcw this; of course this means that the range can also be dealt with. Moreover the variance, like many U-statistics, can be approximated by a sum of transforms of the basic randan variables, and similar methods can be used to study the joint asynptotic behaviour of, for example, the variance and the maximum and minimum.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

Interval estimation for the Sharpe Ratio when returns are not i.i.d. with special emphasis on the GARCH(1,1) process with symmetric innovations

In this paper, assuming that returns follows a stationary and ergodic stochastic process, the asymptotic distribution of the natural estimator of the Sharpe Ratio is explicitly given. This distribution is used in order to define an approximated confidence interval for the Sharpe ratio. Particular attention is devoted to the case of the GARCH(1,1) process. In this latter case, a simulation study is performed in order to evaluate the minimum sample size for reaching a good coverage accuracy of the asymptotic confidence intervals.     

Misspecified T2 tests. i. location and scale

Properties of Hotelling's (1931) T ² are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T ² and further properties of the T ² test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.     

Estimation of tail parameters under type i censoring

This paper is a continuation of previous work concerning the estimation of tail-parameters under Type II censoring (Weissman 1978). The same estimation problem is considered here, this truip under Type I censoring. A sample of size n is censored below aE a given level x₀it is assumed that che underlying distriibution .function (df)belogs to the domain of attraction of a known extreme-value distribution and that K - K(x_o) , the number of observed values, remains finite as on - ∞ . We offer here estimators, which are asymptotically maximum likelihood estimators (MLE's), for quantiles associated with the tail of F such as location and scale parameters, quantiles and F(x) itself (for x in the tail). The results are applied to two illustrative examples.     

Experimental design and optimal decision i general remarks

The problems of an optimal selection of decision function, the design, and the sample size are worked out mainly in separate theories with different objective functions. In applications a unique objective function is given and with respect to this the three components of the statistical approach must be choosen simultaneously. The paper contents initial proposals for such an approach which is demonstrated by the example of parameter estimating in normal distribution. Besides a general theorem of separability and a theorem of duality of two optimization problems of the complete statistical problem are given.     

Information theoretic criterion approach to dimensionality reduction in multinomial logistic regression models. part i: theory

We discuss the issue of dimensionality reduction in multinomial logistic models as problems arising in variable selection, collapsibility of responses and linear restrictions in the parameter matrix. A method using the information theoretic criterion suggested by Bai, Krishnaiah and Zhao, a variation of Akaike information criterion, is used to estimate the rank of the parameter matrix. The same procedure is used for the selection of variables and collapsibility of response categories. The strong consistency of this procedure is established in all the problems.     

On the duality between long-run relations and common trends in the i(1) versus(2)model.an application to aggregate money holdings

Long-run relations and common trends are discussed in terms of the multivariate cointegration model given in the autoregressive and the moving average form. The basic results needed for the analysis of I(1) and 1(2)processes are reviewed and the results applied to Danish monetary data. The test procedures reveal that nominal money stock is essentially I(2). Long-run price homogeneity is supported by the data and imposed on the system. It is found that the bond rate is weakly exogenous for the long-run parameters and therefore act as a driving trend. Using the nonstationarity property of the data, “excess money” is estimated and its effect on the other determinants of the system is investigated. In particular, it is found that “excess money” has no effect on price inflation.     

Multiplicative-interaction logit models for i×j×2 three-way tables

Multiplicative-interaction (M-I) logit models are proposed for three-way IxJx2 contingency tables where the third variable constitutes a binary response. Models are derived by assigning unknown scores to the categories and forming product interactions from them. Asymptotic results under special sampling constraints are derived for maximum likelihood estimates and the goodness-of-fit statistics. The class of models proposed in this paper are found to be useful when no obvious scores are available. An example is included.