Number of observations near the maximum in an F α-scheme

Let K _n (a) be the number of observations in the interval (M _n ,?a, M _n ), where M _n is the maximum value in a sequence of size n. We study the asymptotic properties of K _n (a) under the F ^α-scheme and discuss the influence of the associated sequence α _n on the limit behaviour of this random variable.     

Weak and strong laws of large numbers for sub-linear expectation

Abstract

In this paper, we derive a new form of weak laws of large numbers for sub-linear expectation and establish the equivalence relation among this new form and the other two forms of weak laws of large numbers for sub-linear expectation. Moreover, we obtain the strong laws of large numbers for sub-linear expectation under a general moment condition by applying our new weak laws of large numbers.     

On the formulation of uniform laws of large numbers: a truncation approach

The paper develops a general framework for the formulation of generic uniform laws of large numbers. In particular, we introduce a basic generic uniform law of large numbers that contains recent uniform laws of large numbers by Andrews [2] and Hoadley [9] as special cases. We also develop a truncation approach that makes it possible to obtain uniform laws of large numbers for the functions under consideration from uniform laws of large numbers for truncated versions of those functions. The point of the truncation approach is that uniform laws of large numbers for the truncated versions are typically easier to obtain. By combining the basic uniform law of large numbers and the truncation approach we also derive generalizations of recent uniform laws of large numbers introduced in Pötscher and Prucha [15, 16].     

Diagnostic plots for identifying max domains of attraction under power normalization

Diagnostic plots for determining the max domains of attraction of power normalized partial maxima are proposed. A test to ascertain the veracity of the claim that data distribution belongs to a max domain of attraction under power normalization is given. The performance of this test is demonstrated using data simulated from many well-known distributions. Furthermore, two real-world datasets are analysed using the proposed procedure.     

On the max-semistable limit of maxima of stationary sequences with missing values

Let {_Xn}$\{X_n\}$ be a stationary sequence with marginal distribution in the domain of attraction of a max-semistable distribution. This includes all distributions in the domain of attraction of any max-stable distribution and also other distributions like some integer-valued distributions with exponential type tails such as the Negative Binomial case. We consider the effect of missing values on the distribution of the maximum term. The pattern of occurrence of the missing values must be either iid or strongly mixing. We obtain the expression of the extremal index for the resulting sequence.     

On the number of near-records after the maximum observation in a continuous sample

{X_n, n≥1} are independent and identically distributed random variables with continuous distribution function F(x). For j=1,…,n, X_j is called a near-record up to time n if X_j ∈ (M_na, M_n], where M_n = max_1≤j≤n {X_j} and a is a positive constant. Let Z_n(a) denote the number of near-records after, and including the maximum observation of the sequence. In this paper, the distributional results of Z_n(a) are considered and its asymptotic behaviours are studied.     

A strong law of large numbers for independent random variables under non-additive probabilities

Abstract

Under non‐additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.     

DISTRIBUTIONAL CHARACTERIZATIONS THROUGH SCALING RELATIONS

Investigated here are aspects of the relation between the laws of X and Y where X is represented as a randomly scaled version of Y. In the case that the scaling has a beta law, the law of Y is expressed in terms of the law of X. Common continuous distributions are characterized using this beta scaling law, and choosing the distribution function of Y as a weighted version of the distribution function of X, where the weight is a power function. It is shown, without any restriction on the law of the scaling, but using a one‐parameter family of weights which includes the power weights, that characterizations can be expressed in terms of known results for the power weights. Characterizations in the case where the distribution function of Y is a positive power of the distribution function of X are examined in two special cases. Finally, conditions are given for existence of inverses of the length‐bias and stationary‐excess operators.     

ON THE NUMBER OF RECORDS NEAR THE MAXIMUM

Recent work has considered properties of the number of observations X_j, independently drawn from a discrete law, which equal the sample maximum X_(n) The natural analogue for continuous laws is the number K_n(a) of observations in the interval (X_(n)–a, X_(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of K_n(a), mentioning examples where evaluations can be given. It seeks limit laws for n→ and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X_(n) - X_(n-j) with j fixed.     

Exact Distributional Results for Random Resistance Trees

With a view to the study of, for instance, arterial trees, this paper presents some exact distributional results on finite trees with (reciprocal) inverse Gaussian and gamma resistances. In particular, it is shown that under the specified model the conditional distribution of the minimal sufficient statistic given the total resistance of the tree is a convolution of gamma distributions and two-dimensional reciprocal inverse Gaussian distributions.     

Contributors to a multivariate statistical process control chart signal

A significant practical disadvantage of multivariate statistical process control is that it is difficult to determine which of the monitored variables is responsible for the out-of-control signal. While there have been several proposed techniques and diagnostics, most of these have disadvantages. Rather than an ad hoc selection, we use three approaches to develop a diagnostic for a chi-square chart and generate the same statistic from each approach. Consequently, intuitive interpretations of the diagnostic are provided and previous proposals are simplified by a focus on a single metric.     

The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

Asymptotic results for empirical and partial-sum processes: A review

Two processes of importance in statistics and probability are the empirical and partial-sum processes. Based on d-dimensional data X₁, … X_a the empirical measure is defined for any A ⊂ R^d by the sample proportion of observations in A. When normalized, F_n yields the empirical process W_n: = n^1/2 (F_n - F), where F denotes the “true” probability measure. To define partial-sum processes, one needs data that are assigned to specified locations (in contrast to the above, where specified unit masses are assigned to random locations). A suitable context for many applications is that of data attached to points of a lattice, say {X_j:j ϵ J^d} where J = {1, 2,…}, for which the partial sums are defined for any A ⊂ R^d by Thus S(A) is the sum of the data contained in A. When normalized, S yields the partial-sum process. This paper provides an overview of asymptotic results for empirical and partial-sum processes, including strong laws and central limit theorems, together with some indications of their inferential implications.     

On the behaviour of some transforms of the sample correlation coefficient in samples from the bivariate t and the bivariate X2distribution

The present paper studies the normality of five transformations suggested in the literature to normalize the sample correlation coefficient. The parent populations are the bivariate t and the bivariate X ²The results in the previous work of Subrahmaniam and Gajjar are exploited to assess their performance. The density estimation procedure of Tarter and Kronmal is used to provide empiric support to the asymptotic results     

Spherical symmetry: An elementary justification

Summary The present paper includes characterizations of the conditions of spherical symmetry and of centered spherical symmetry. These characterizations provide an empirical justification for the above mentioned conditions of symmetry.     

On confidence regions for the mean of a multivariate time series

We consider the problem of setting up a confidence region for the mean of amultivariate timeseries ont he basis of a part-realisation of that series.A procedure for setting up a confidence interval for the mean of a univariate time series Is implicitin Jones(1976).We present an analogous procedure for setting up a confidence region for the mean of a multivariatet ime series.This procedure is base donastatistic which is an analogue of Hotelling'sT'.Our results are applied to a comparison of climate means obtained from experiments with a General Circulation Model of the earth's atmosphere.     

ON CHARACTERIZATIONS THROUGH MIXED SUMS

Let X, X‘ and X“ be independent and have the same law, and let U be uniform on [0,1] and independent of X’ and XI”. A problem which has received considerable attention is to identify the laws of X for which X =c U(X‘+X“), the equality being in law. If X is positive it has an exponential law, and mixture solutions exist without the sign restriction. The problem has been extended to allow more summands on either side and to let U have a special beta law. In the positive case X then has a gamma law. This problem is tackled in full generality by considering an integral equation for the characteristic function of X. Uniqueness results and necessary conditions are given. The general case is reduced to that where X is positive, and the solution of the corresponding integral equation is constructed under the necessary conditions. This is shown to characterize all possible laws of X when there is a single summand on the left and U has an arbitrary law on [0,1]. Examples are given and known particular results are sharpened.     

The strong laws of large numbers for weighted sums of extended negatively dependent random variables

In this paper, the strong laws of large numbers for maximum value of weighted sums of extended negatively dependent random variables are obtained, which improve and extend the corresponding ones for independent random variables and some dependent random variables.     

On Max-Multiscaling Distributions as Extended Max-Semistable Ones

Abstract

We introduce max-multiscaling distributions as solutions to a functional equation which, in a natural way, extends the one fulfilled by max-semistable distributions. We establish that strictly max-multiscaling distributions are products of at most two max-semistable distributions. Next, we show how to obtain these solutions as limit laws of normalized maximum of suitable independent sequences of random variables when sample size has geometric growth.     

A General Class of Distributions: Properties and Applications

ABSTRACT

In this article, we derive a general class of distributions and establish its relationship to χ² distribution. The proposed class includes normal, inverse Gaussian, lognormal, gamma, Rayleigh, and Maxwell distributions. Various statistical properties of the class are discussed. Some applications of the class are given.