Sums Of Variables Research Articles

Certain non-uniform rates of convergence to normality for a restricted class of martingales

Non-uniform rates of convergence to normality are derived for standardized sums of random variables which form a martingale difference array, with the sum of conditional variances being a constant in each row. As a consequence, we obtain probabilities of moderate deviations, convergence of absolute moments and (nonuniform) L p versions of the Berry-Esseen theorem.

Confidence intervals on the intraclass correlation in the unbalanced one-way classification

Several methods are compared for constructing confidence intervals on the intraclass correlation coefficient in the unbalanced one-way classification. The results suggest that a conservative approximation of the exact procedure developed by Wald (1940) can be used for hand calculations, When the exact solution is desired, a solution procedure is recommended that is computationally convenient and allows the investigator to determine the precision of the estimate. In cases where a prior estimate of the correlation is available, researchers may select intervals based on either the analysis of variance or unweighted sums of squares estimator.

A functional central limit theorem for strongly mixing sequences of random variables

We prove a functional central limit theorem for a class of strongly mixing sequences of random variables. Stationarity is not assumed, but the variances of the partial sums must grow linearly. Our theorem extends previous results by supplying sufficient conditions for the weak convergence of the partial sum process to the Wiener process under less restrictive moment assumptions. For instance, if \(\mathop {\sup }\limits_n {\text{ }}EX_n^2 |\log |X_n \parallel ^{1 + \varepsilon } 0, and the mixing rate is exponential, then this functional c.l.t. holds. Under the weaker assumption with e=0, the c.l.t. may fail to hold, and it is possible that the c.l.t. is satisfied, but the sequence of partial sum processes is not tight.

TheFStatistic in the Two-Way Layout With Rank–Score Transformed Data

Abstract The limiting null distribution of the usual F statistic for main effects in the two-way layout is shown to have the same limiting distribution when applied to ranks and scores based on ranks as when applied to normal data. The limit is taken as the cell size N increases without bound. The denominator of the F statistic times (J − 1)/J is shown to provide an unbiased and consistent estimator of the limit (as N → ∞) of N -1 times the variance of the treatment sum of scores when the null hypothesis is true. The test based on the scores statistic is demonstrated to be consistent for a wide class of fixed alternative hypotheses.

Anova and minque type of estimators for the one-way random effects model

The one-way random effects model with unequal variances and unequal sample sizes is considered. Estimation of the variances, variance of a single observation (total variance), and the standard error of the unweighted mean are considered. Precision of the Analysis of Variance and Unweighted Sums of Squares type of estimators and the Minimum Norm Quadratic Unbiased Estimators with a priori weights are examined.

Concordance among self-regulated responses: is self-regulatory ability a psychophysiological "trait"?

The purposes of this study were to examine whether or not self-regulation of physiological responses demonstrates day-to-day reliability, to determine the degree of individual subject consistency (or concordance) in the ability to self-regulate across several different physiological responses, and, finally, to explore the impact of biofeedback training on interresponse concordance. Twenty normal subjects participated in six bidirectional self-regulation sessions-the first and last sessions involving instructions only, and the remainder, biofeedback. Self-regulation scores consisted of the absolute difference between increase and decrease trial means. The average test-retest reliability coefficients (rs) for the self-regulation scores, across the four biofeedback sessions, were a highly significant .50, .68, .30, and .47 for EEG, EMG, HR, and SCL, respectively. By contrast, the average concordance among the self-regulation scores for the four feedback sessions, estimated by Kendall's coefficient of concordance, was a marginally significant 39% of the possible variance of the rank sums. This corresponds to an average between-response rs value only of .19. The concordance level from the initial no-feedback (i.e., instructions only) session was not significant. Multivariate concordance levels did increase during the first three feedback sessions, but declined at the fourth, and again was nonsignificant during the final no-feedback session. Among the individual self-regulation response pairings, only the EEG/EMG combination was consistently associated during the no-feedback sessions. The present results suggest that self-regulatory ability is neither a highly unitary "trait"-like phenomenon nor an entirely response-specific event, but may vary considerably as a function of subject factors, or the situational circumstances, under which it is measured.

A Comparison of Peanut Multilines and Component Lines across Environments 1

In peanuts (Arachis hypogaea L.) a high degree of phenotypic similarity among components of multiline cultivars is necessary in order to meet the demands of industry. To achieve this, the compositional scheme for peanut multilines has utilized sibling lines composited between the F4 and F8 generations. As a result, the alleged enhancement of stability may be reduced due to a lack of genetic heterogeneity among component lines, which abates an important advantage of multiline cultivars. This study was conducted to examine the relationship among sibling components of two peanut multilines and to determine if the multilines perform in a more stable manner across environments than any single component. Partitioning of genotype ✕ environment (GE) interaction sum of squares, regression analyses, and stability variances were used to assess stability over environments (2 years, 4 locations) for two multilines and their four components. For one multiline, three of four component lines were not significantly different from the multiline for yield nor did deviations from regression and stability variances differ among components. Conversely, components of the second multiline displayed significant variability for yield, regression coefficients and deviations from regression. However, the multiline performed in a stable manner with relatively high yields and low deviations from regression. Mean yields, regression coefficients, and deviations from regression were shown to be useful parameters in the formation of a multiline. Joint regression analysis and GE partitioning techniques were useful in determining the nature and origin of the GE among groups of late generation segregates. Although additional studies are needed, these data indicate that the compositional scheme for peanut multilines is a feasible method to circumvent GE effects. This study, however, suggests that pure lines of peanuts can be identified that are as stable across environments as multilines.

On optimal parameter estimation in credibility

The present paper is concerned with optimal estimation of the 1st and 2nd order structural moments appearing in credibility formulas. In a recent paper De Vylder has treated the problem in the case of multinormal conditional distributions under quite restrictive assumptions. He minimizes, within a certain restricted class of unbiased estimators, the variance (or the sum of variances if the estimand is a matrix) and next replaces all structural moments (up to fourth order) in the solution by estimates based on the data. This paper is an attempt to simplify the method and extend it so as to make it applicable in more general situations. By suitable choice of a (sufficient) set of statistics and a suitable parametrization, the powerful theory of estimation in linear models can be employed, which makes cumbersome minimization procedures superfluous. The theory is applied to the cases with binomial. Poisson, compound Poisson, and multinormal conditional distributions. Some simulation studies have been performed to assess the performance of the estimators.

Properties of minimum mean squared error block quantizers (Corresp.)

Two results in minimum mean square error quantization theory are presented. The first section gives a simplified derivation of a well-known upper bound to the distortion introduced by a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -dimensional optimum quantizer. It is then shown that an optimum multidimensional quantizer preserves the mean vector of the input and that the mean square quantization error is given by the sum of the component variances of the input minus the sum of the variances of the output.

A standard determination of paired air- and bone-conduction thresholds under different masking noise conditions.

Air-(ac) and bone-conduction (bc) thresholds were determined for 96 subjects in the age range 16-30 years. The bc tests were carried out using a B-71 vibrator applied to left and right mastoid under three conditions at the contralateral ear: unoccluded, masked at 25 dB and at 40 dB sensation level; ac audiometry and masking noise used a TDH-39 earphone. By a systematic process of otological classification of the subjects, the bc threshold for 'normal hearing' could be estimated with an uncertainty of less than 1 dB and interpolated for masking noise levels from zero to 40 dB sensation level. Results are expressed as alternating force levels in an artificial mastoid of current commercial design. Variance of the air-bone difference for individual ears is found to be less than the sum of ac and bc threshold variances calculated separately, implying strong air-bone correlation. Comparison is made with other recent studies using B-71 and KH-70-type vibrators, and average results are presented in the context of a proposed international standard for the normal threshold of hearing by bc.

On a class of extremal problems in statistics

Let m denote the infimum of the Integral of a function q w r t all probability measures with given marginals. The determination of m is of interest for a series of stochastic problems. In the present paper we prove a duality theorem for the determination of m and give some examples for its application. We consider especially the problem of extremal variance of sums of random variables and prove a theorem for the existence of random variables with given marginal distributions, such that their sum has variance zero.

Analysis of a multifactor nonrandom mating model in population genetics

The complete time dependent behavior of a population under the system of mixed random mating and selfing is discussed, assuming multiple unlinked loci with multiple alleles. Expressions are obtained for the distribution of genotypes. The genotypic mean and variance under the assumption that the effects of different loci are additive are considered. The covariance between a pair of factors depends upon the dominance deviations; the parameter of nonrandom mating, β; and the gene frequencies. It vanishes in the absence of dominance or when β=0 or 1. The usual assumption that the variance due to several independently segregating loci is equal to the sum of variances due to each locus separately is not valid in the present case.

Monte Carlo comparisons of selected clustering procedures

Monte Carlo methods were used to estimate the percent misclassification of 13 clustering methods for six types of parameterizations of two bivariate normal populations. The clustering methods were compared by using the probabilities of misclassification and incidence matrices. It was determined that correlations and differences in population sizes adversely influenced all clustering methods, where differences in the variance structure did not appreciably affect the results. The k-means partitioning method was the overall best method. Considering only agglomerative methods, the sum of squares, variance, furthest neighbor and rank score methods were generally superior to the other non-partitioning methods considered. The overall poorest methods were judged to be nearest neighbor and maximum likelihood. However, as the complexity of the distributions increased, the differences between all of the methods decreased.

Maximum likelihood estimation of variance components in a completely random bib design

ABSTRACT The purposes of this paper are to abstract from a number of articles variance component estimation procedures which can be used for completely random balanced incomplete block designs, to develop an iterated least squares (ITLS) computing algorithm for calculating maximum likelihood estimates, and to compare these procedures by use of simulated experiments. Based on the simulated experiments, the estimated mean square errors of the ITLS estimates are generally less than*those for previously proposed analysis of variance and symmetric sums estimators.

A Generalized χ2 for the Significance of Differences in Repeated, Related Measures Applied To Different Samples

When repeated measures are not independent, sample data can be used to estimate the average covariance among the measures, and this can be used to calculate an appropriate estimate of the variance in a sum of measures. When such an estimate of the variance is employed, a χ2 statistic can be used to test the significance of differences in repeated, related measures applied to different samples.

The optimal linear alternative to price and quantity controls in the multifirm case

Linear control schedules in output have been shown superior in the one-firm case to either of the extreme controls—price or quantity; they optimally trade off the desirable characteristics of both extremes. When many firms are regulated, however, that superiority fades. Then total output affects expected benefits and can display a larger (or smaller) variance than the sum of individual firms' output variances (upon which expected costs depend) if costs are positively (negatively) correlated. Output variation must be discouraged (encouraged), therefore, and the linear schedule rotates toward the quantity (price) extreme. The better extreme might thereby become the best choice among all three alternatives.

On a unified approach to the law of the iterated logarithm for martingales

There are two distinct approaches in the literature to framing a version of the law of the iterated logarithm for martingales. One involves norming by constants, using the martingale variance and the other involves norming by random variables, using the sums of conditional variances of the increments, given their past. In this paper a portmanteau approach is provided, still based on the Skorokhod representation of the martingale, but involving normalization by more general random variables. This extends the functional forms of all the previously existing results.

Power Sum Distributions

Abstract Let X 1, …, X n be jointly distributed random variates. The quantity log Σexp{X i } is called the “power sum” of the variates X 1, …, X n . Previous articles [1, 4] find the moment generating function, expectation, and variance of the power sum of two normally distributed X i 's. We generalize these results to the case of n jointly distributed variates.

Suboptimal input signals for linear system identification

It is well known that inputs for identification experiments can be synthesized on the basis of optimizing a certain performance criterion, such as variances of parameters, estimates of weighting functions and probabilities of transmission error. Such signals give the best possible performance for given constraints of input energy, observation time and nature of noise. The question arises whether one can choose signals on the basis of other possibly more intuitive or physical criteria and see what kind of performance they give w.r.t. optimal inputs and common signals, as unit step inputs. These signals, which fit between the two extremes are here referred to as suboptimal. The measure of performance is here taken to be the sum of parameter variances. It is shown that suboptimal inputs on the basis of maximizing the output signal/noise ratio, matching the system bandwidth and of pseudo-random binary noise nature can be selected for a given system. It is, however, also indicated that this is not a general me...

Minimum-Variance Method for Potential Scattering

In variational solutions of differential or integral equations the variance integral and variance sum provide measures of the inadequacy of the trial function. For certain potential-scattering problems the variance integral can be used to derive rigorous upper and lower bounds to the phase shift. In more general scattering problems, the study of the variance sum or integral gives a check on the convergence of the standard variational calculations, and an alternative criterion for choosing the best trial function. Results are shown for an attractive exponential potential.