Switching Order Statistics through Random Power Contractions

This paper investigates a new random contraction scheme which complements the length‐biasing and convolution contraction schemes considered in the literature. A random power contraction is used with order statistics, leading to new and elegant characterizations of the power distribution. In view of Rossberg's counter‐example of a non‐exponential law with exponentially distributed spacings of order statistics, possibly the most appealing consequence of the result is a characterization of the exponential distribution via an independent exponential shift of order statistics.     

The move-to-partner rule for self-organizing task allocation on a linear array

The well-known Move-to-Front rule for self-organizing lists is modified to a so-called Move-to-Partner rule for dynamic task allocation on a chain of processors. We enable an analysis of the sequence of expected communication costs by reducing their computation to that of three-dimensional arrays of certain probabilities, for which a recursion formula and an asymptotic expansion can be given. The costs depend on the underlying communication profile; two types of such profiles are investigated in more detail, namely a two-dimensional version of Zipf's law, and profiles based on a class structure of tasks. In the considered cases, the Move-to-Partner rule effects fast convergence to a stationary state, but comparably high expected stationary costs.     

Asymptotic normality for the estimator of non parametric regression model under ϕ-mixing errors

In this article, by using the Rosenthal-type inequality and the Bernstein's big-block and small-block procedure, we establish the asymptotic normality for the estimators of non parametric regression model based on ?-mixing errors. The result obtained in the article generalizes some corresponding ones for some dependent random variables.     

A note on stratification and gain in precision in estimating diversity from large samples

Several indices of entropy have been suggested in the literature as weighted diversity measures of a population with respect to a classification process. Among them, Shannon's entropy and Havrda -Charvát's non-additive entropies of order a, have been exhaustively used.

When the population is finite but too large to be censused, the diversity with respect to a given classification process must be estimated from a sample.

In this note, on the basis of an asymptotic study of the sample indices in the stratified random sampling, we are going to confirm that when we deal with large samples one can guarantee a gain in precision from stratified random over simple random sampling. This gain becomes considerable when the ‘inaccuracy" (as intended by Kerridge and Rathie and Kannapan) between the frequency vector in each stratum and that in the whole population, varies greatly from stratum to stratum.     

Lexigenesis in ancestral south-east Australian Aboriginal language

The 1/x frequency distribution is known to researchers ranging from economists and biologists to electronic engineers. It is known to linguists as Zipf's Law (Zipf, 1949) and has recently been shown not to be a consequence of the Central Limit Theorem (Troll & Graben, 1998)--leaving an "unsolved problem' in information theory (Jones, 1999). This 1/x distribution, associated with scale-invariant physical systems (Machlup & Hoshiko, 1980), is a special case of the general power law x^λ arising from the Lagrangian L(x,[Fdot](x)) = ½x^1-λ[Fdot]² and, as λ need not be an integer, some related research understandably involves fractals (Allison et al. , 2001). The present paper generalizes this Lagrangian to include a van der Waals effect. It is argued that ancestral Aboriginal language consisted of root-morphemes that were built up into, and often condensed within, subsequent words or lexemes. Using discrete-optimization techniques pioneered elsewhere (Illert, 1987; Reverberi, 1985), and the new morpho-statistics, this paper models lexeme-condensation in ancestral south-east Australian Aboriginal language.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Serial correlation or random subject effects

In longitudinal data analysis with random subject effects, there is often within subject serial correlation and possibly unequally spaced observations. This serial correlation can be partially confounded with the random between subject effects. In real data, it is often not clear whether there is serial correlation, random subject effects or both. Using inference based on the likelihood function, it is not always possible to identify the correct model, especially in small samples. However, it is important that some effort be made to attempt to find a good model rather than just making assumptions. This often means trying models with random coefficients, with serial correlation, and with both. Model selection criteria such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used. The problem of modelling serial correlation with unequally spaced observations is addressed. A real data example is presented where there is an apparent heterogeneity of variances, possible serial correlation and between subject random effects. In this example, it turns out that the random subject effects explains both the serial correlation and the variance heterogeneity.     

Generation of random matrices with orthonormal columns and multivariate normal wariates with given sample mean and covariance

In this paper, an algorithm for generating random matrices with orthonormal columns is introduced. As pointed out by a referee, the algorithm is almost identical to Wedderburn's (1975) unpublished method. The method can also be considered as an extension of Stewart's (1980) method, which was designed to generate random orthogonal matrices. It is found outperforming a simple extension of the QR factorization method and that of Heiberger's (1978) method. This paper also demonstrates how the algorithm can be used in generating multivariate normal variates with given sample mean and sample covariance matrix.     

Test of two hypotheses explaining the size of populations in a system of cities

Two classical hypotheses are examined about the population growth in a system of cities: Hypothesis 1 pertains to Gibrat's and Zipf's theory which states that the city growth–decay process is size independent; Hypothesis 2 pertains to the so-called Yule process which states that the growth of populations in cities happens when (i) the distribution of the city population initial size obeys a log-normal function, (ii) the growth of the settlements follows a stochastic process. The basis for the test is some official data on Bulgarian cities at various times. This system was chosen because (i) Bulgaria is a country for which one does not expect biased theoretical conditions; (ii) the city populations were determined rather precisely. The present results show that: (i) the population size growth of the Bulgarian cities is size dependent, whence Hypothesis 1 is not confirmed for Bulgaria; (ii) the population size growth of Bulgarian cities can be described by a double Pareto log-normal distribution, whence Hypothesis 2 is valid for the Bulgarian city system. It is expected that this fine study brings some information and light on other usually considered to be more pertinent countries of city systems.     

The random robin problem for laplace's equation

A random generalized solution to the Robin problem for Laplace's equation is defined in terms of the sections of the random boundary data. Existence, uniqueness, and properties are established for such a solution. Particularizations to the Dirichlet and Newmann problems as well as generalization to the Robin-Poisson problem are mentioned. Applications of the results are provided.     

The Standard Error of the Mean and the Difference between Means for Finite Populations

This article discusses a representation of Pearson's chi-square for independence in two-way contingency tables in terms of conditional probabilities of two categorical random variables and proposes a functional interpretation of Pearson's chi-square. This representation is suggested for use in the teaching of statistical independence between categorical variables.     

A consistent estimator for logistic mixed effect models

We propose a consistent and locally efficient method of estimating the model parameters of a logistic mixed effect model with random slopes. Our approach relaxes two typical assumptions: the random effects being normally distributed, and the covariates and random effects being independent of each other. Adhering to these assumptions is particularly difficult in health studies where, in many cases, we have limited resources to design experiments and gather data in long‐term studies, while new findings from other fields might emerge, suggesting the violation of such assumptions. So it is crucial to have an estimator that is robust to such violations; then we could make better use of current data harvested using various valuable resources. Our method generalizes the framework presented in Garcia & Ma (2016) which also deals with a logistic mixed effect model but only considers a random intercept. A simulation study reveals that our proposed estimator remains consistent even when the independence and normality assumptions are violated. This contrasts favourably with the traditional maximum likelihood estimator which is likely to be inconsistent when there is dependence between the covariates and random effects. Application of this work to a study of Huntington's disease reveals that disease diagnosis can be enhanced using assessments of cognitive performance. The Canadian Journal of Statistics 47: 140–156; 2019 © 2019 Statistical Society of Canada     

Some exact results on the sample autocovariances of a seasonal ARIMA model

A seasonal random walk is an ARIMA process such that the first difference of order s (s ≥ 1) is a white noise. Given a series of observations from a particular linear transformation of a seasonal random walk, we study the autocovariances c'(k) based on uncentered data and the autocovariances c(k) based on centered data. In both cases, we provide exact, explicit formulae for the mean, variance, and covariance of the sample autocovariances. It is seen that the moments of the c(k)'s are different from those of the c'(k)'s, even asymptotically. Several analytical results presented in the paper were derived by using a symbolic manipulation program.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.     

Comparison of experiments for a class of positively dependent random variables

Using Blackwell's definition for comparison of experiments, it is shown that some sets of positively dependent random variables are less informative than similar sets of independent random variables. It is also shown that the information content of symmetric multivariate normal random vectors with a common known variance increases as the common correlation coefficient decreases. Some results which compare members of two-parameter exponential families are also included.     

Maximal inequalities and strong law of large numbers for sequences of m-asymptotically almost negatively associated random variables

In this paper, we establish some inequalities for maximum of partial sums of m-asymptotically almost negatively associated random variables. With the help of these inequalities we prove some strong law of large numbers.     

On Confidence Intervals for Process Capability Indices in a One-Way Random Model

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C _pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.     

On Large Deviations of Sums of Independent Random Variables

Extensions of some limit theorems are proved for tail probabilities of sums of independent identically distributed random variables satisfying the one-sided or two-sided Cramér's condition. The large deviation x-region under consideration is broader than in the classical Cramér's theorem, and the estimate of the remainder is uniform with respect to x. The corresponding asymptotic expansion with arbitrarily many summands is also obtained.     

Complete moment convergence for weighted sums of extended negatively dependent random variables

Complete moment convergence for weighted sums of sequence of extended negatively dependent (END) random variables is discussed. Some new sufficient and necessary conditions of complete moment convergence for weighted sums of END random variables are obtained, which improve and extend some well-known results in the literature.     

Limit theorems for a general weighted process under random censoring

Necessary and sufficient conditions for weak and strong convergence are derived for the weighted version of a general process under random censoring. To be more explicit, this means that for this process complete analogues are obtained of the Chibisov-O'Reilly theorem, the Lai-Wellner Glivenko-Cantelli theorem, and the James law of the iterated logarithm for the empirical process. The process contains as special cases the so-called basic martingale, the empirical cumulative hazard process, and the product-limit process. As a tool we derive a Kiefer-process-type approximation of our process, which may be of independent interest.