Sums Of Iid Random Variables Research Articles

Extreme eigenvalues of nonlinear correlation matrices with applications to additive models

The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.

Estimating the distribution of a stochastic sum of IID random variables

Abstract Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

Remarks on the Rényi Entropy of a Sum of IID Random Variables

In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the Renyi entropy of the sum of independent random variables. Through a variational analysis, we show that the generalized Gaussian fails to be a minimizer for the problem.

Linear dimension reduction approximately preserving a function of the 1-norm

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are not too close, the distance between their images is a strictly concave increasing function of their original distance, up to multiplicative error. The target dimension $k$ need only be quadratic in the logarithm of the size of the point set to ensure the result holds with high probability. The linear embeddings are random matrices composed of standard Cauchy random variables, and the proofs rely on Chernoff bounds for sums of iid random variables. The new metric is translation invariant, but is not induced by a norm.

Review of median stable distributions and Schröder’s equation

Median stable distributions are an extension of traditional (mean) stable distributions. The traditional definition of stability (in terms of sums of iid random variables) is recast as a condition on the sampling distribution of an estimator. For the traditional (mean) stable distribution, the sample mean’s (rescaled) sampling distribution is identical to the distribution of the iid data. Median stable distributions are defined similarly by replacing the sample mean with the sample median. Since the sampling distribution of the median is a functional its stable distribution is the solution to a functional equation. It turns out that this defining functional equation is an instance of a famous equation due to Schröder from 1870. The fame of the equation is due to the way it incorporates iteration of functions, a key feature of what many years later would become dynamic systems analysis. The current paper reviews median stable distributions in light of its connection to Schröder’s functional equation.

Karamata Method and the Maximum of Sums of Independent Random Variables

We revisit the classical problem of convergence of the maximum of cumulative sums of IID random variables by introducing ideas from the Karamata's celebrated proof of the Hardy-Littlewood Tauberian Theorem [J. Karamata, Uber die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319-320]. As a main result we present an extension of the Darling-Heyde convergence device, and use it to provide a direct analytic proof of convergence in distribution of normalized cumulative sums of IID random variables. The introduction of Karamata's ideas makes the proof self-contained and relying only on basic regular variation results.

REFINEMENT OF CONVERGENCE RATES FOR WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

In this paper, we establish one general result on precise asymptotics of weighted sums for i.i.d. random variables. As a corollary, we have the results of Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368], Gut and Spătaru [Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000) 1870–1883; Precise asymptotics in the Baum–Katz and Davis laws of large numbers, J. Math. Anal. Appl. 248 (2000) 233–246], Gut and Steinebach [Convergence rates and precise asymptotics for renewal counting processes and some first passage times, Fields Inst. Comm. 44 (2004) 205–227] and Heyde [A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975) 173–175]. Meanwhile, we provide an answer for the possible conclusion pointed out by Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368].

Deriving Skewness and Excess Kurtosis of the Sum of iid Random Variables

Deriving Skewness and Excess Kurtosis of the Sum of iid Random Variables

The convergence of the Rényi entropy of the normalized sums of IID random variables

We prove that the Rényi entropy of order α ( α > 1 ) of the normalized sums of IID random variables with continuous differentiable density is convergent to the Rényi entropy of order α of the standard Gaussian distribution, and obtain the corresponding rates of convergence.

Reliable inference for the Gini index

Although attention has been given to obtaining reliable standard errors for the plugin estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but eective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen’s poverty index and its modification known as the Sen-Shorrocks-Thon index.

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.

Convolution equivalence and distributions of random sums

A serious gap in the Proof of Pakes’s paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev’s theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Levy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin’s theorem is proved.

Some statistical applications of Faa di Bruno

The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503–520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.

Law of the iterated logarithm for self-normalized sums and their increments

LetX1,X2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper usesMn=max1?i?n|Xi| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

Refined Baum–Katz laws for weighted sums of iid random variables

We consider weighted sums ∑ k p nk X k with iid random variables ( X n ) and compare the tail probabilities of these sums with the moment conditions on X 1 . If X 1 is in the domain of attraction of a stable law then refined Baum–Katz laws generalizing results of Heyde, Gut and other authors are presented. Some special examples of weights p nk originating from summability are discussed.

On the LIL for Self-Normalized Sums of IID Random Variables

Let \(X,X_i ,i \in \mathbb{N},\) be i.i.d. random variables and let, for each \(n \in \mathbb{N},S_n = \sum\nolimits_{i = 1}^n {X_i }\) and \(V_n^2 = \sum\nolimits_{i = 1}^n {X_i^2 }\). It is shown that \(\lim \sup _{n \to \infty } {{|S_n |} \mathord{\left/ {\vphantom {{|S_n |} {(V_n \sqrt {\log \log n} ) < \infty }}} \right. \kern-\nulldelimiterspace} {(V_n \sqrt {\log \log n} ) < \infty }}\) a.s. whenever the sequence of self-normalized sums Sn/Vn is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if \(\lim \sup _{t \to \infty } {{[t|\mathbb{E}XI(|X| \leqslant t)|} \mathord{\left/ {\vphantom {{[t|\mathbb{E}XI(|X| \leqslant t)|} {\mathbb{E}X^2 I(|X| \leqslant t)] < \infty }}} \right. \kern-\nulldelimiterspace} {\mathbb{E}X^2 I(|X| \leqslant t)] < \infty }}\)

Rates of clustering in Strassen's LIL for partial sum processes

We provide a detailed description of the rate of clustering in Strassen's functional law of the iterated logarithm for partial sum processes. Necessary and sufficient conditions are given for certain clustering rates in terms of moment-type conditions. In the process, we also derive a new strong approximation of sums of iid random variables by a Wiener process.

Inference for Events with Dependent Risks in Multiple Endpoint Studies

Abstract Kaplan–Meier and cumulative incidence functions are not sufficient descriptive devices for studies that have multiple time-to-event endpoints. For example, in cancer treatment research the probability of tumor recurrence conditional on not having died from treatment-related toxicities and the prevalence of graft-versus-host disease among leukemia-free patients surviving a bone marrow transplant are of interest. These quantities can be estimated nonparametrically using simple functions of several Kaplan–Meier and cumulative incidence estimates for events with possibly dependent risks. We derive asymptotic distribution theory for such functions by representing Kaplan–Meier, cumulative incidence, and cumulative hazard estimators as sums of iid random variables. Variance estimation also follows directly from this representation. Two-sample test statistics with asymptotic null distribution theory are presented. Several examples illustrate the utility of these results.

On the Almost Sure Behavior of Sums of IID Random Variables in Hilbert Space

We study the almost sure behavior of sums of $\operatorname{iid}$ random variables satisfying the bounded LIL in Hilbert space. We show that the almost sure behavior is different from the Gaussian case, whenever the second strong moments are infinite. A law of the $k$ times iterated logarithm is established which refines the bounded LIL. The interesting feature here is that contrary to the known conditions for the bounded LIL, one needs not only moment type conditions but also a nice structure of the covariance operator.

Characterizations of the exponential distribution by weighted sums of iid random variables

The exponential distribution is characterized by means of the distribution of a weighted sum of independent, identically distributed random variables. The conditions considered turn out to be sufficient in the case of two random variables only.