Sums Of Variables Research Articles

Estimation of variance of partial sums of an associated sequence of random variables

Let X n , n ⩾ 1 be a stationary sequence of associated random variables satisfying E( X 1) = μ, E( X 2 1) < ∞ and ( Var S n) n → σ 2 as n → ∞. In this paper, an estimator of σ 2 based on the subseries values using overlapping blocks is studied. A central limit theorem related to this estimator is obtained.

Variance of Interaction Effects of Sire and Herd for Yield Traits of Holsteins in California, New York, and Pennsylvania with an Animal Model

An animal model with a REML algorithm was used to estimate variances of additive genetic effects and interaction effects of sire and herd. Milk and fat yields were analyzed for first, second, and third lactations of Holsteins from California, New York, and Pennsylvania. Twenty samples of data were used in the study: 10 from California, 4 from New York, and 6 from Pennsylvania. Mean number of lactations per sample was 36,820 from 18,189 cows in 156 herds. Mean fractions of phenotypic variance of interaction effects of sire and herd for milk and fat yields were .015 and .019 for first lactation and .019 and .021 for all (up to three) lactations rather than the .14 used for national genetic evaluations in the US. Mean heritability estimates for milk and fat yields were .26 and .24 for first lactation and .21 and .21 for all lactations in California and .34 and .35 for first lactations and .28 and .29 for all lactations in New York. Sums of variances of permanent environmental and interaction effects of sire and herd were similar to those used for national genetic evaluations in the US.Analysis of another 10 samples from California and 10 samples from New York showed only slightly different fractions of phenotypic variance for milk yield for interaction effects of sire by herd, sire by herd by year, and sire by herd by year by season: .023, .027, and .037 for California and .023, .017, and .023 for New York, respectively.

Estimation of the Variance of Partial Sums for ρ-Mixing Random Variables

Let { X n , n ≥ 1} be a stationary sequence of ρ-mixing random variables satisfying EX n = μ, EX 2 n < ∞, Var Sn/ n → σ 2 > 0. This paper presents a class of estimators of σ and investigates their weak consistency as well as their asymptotic normality. Applications to the self-normalizing central limit theorem and confidence for the sample mean are also discussed.

Two-stage suboptimal dual controller for systems with stochastic parameters using optimal predictors

A suboptimal dual controller is presented for a class of linear systems with unknown and randomly varying parameters. The loss function which is minimised consists of the sum of output variances up to two steps ahead in time, conditioned on past values of the control and output signals. Optimal predictors are used to replace the future outputs which are needed to compute the control signal at each step. The behaviour of the two-stage controller is illustrated by two examples.

Why expect a horse to fly?

Caricature, conflation, and confusion -- these are primary means by which Wahlsten in Intelligence of Heritability seeks to upgrade our education concerning concept of heritability. heritability coefficient (h'Symbol not transcribed'2) is an estimate of proportion of phenotypic variance in a trait attributable to genotypic variance and has been of practical value in artificial selection of desirable phenotypic characteristics in domestic strains. Because heritability has broader implications concerning amenability of traits to both natural selection and environmental influence heritability continues to be a concept much misunderstood and abused by those still ensnared in politics of nature/nurture debate.In this context, Wahlsten is perhaps both thought - provoking and contentious: thought - provoking because during process of debunking certain myths about heritability analysis some even greater misconceptions are revealed; contentious because his criticism of concept of heritability involves construction of a caricature or strawman, conflation of a statistical test with a conceptual model, and confusion of levels of analysis. Nevertheless, if object of scientific discourse is scientific progress, then ensuing discussion of Wahlsten's critique should ultimately result in raising our collective intelligence about heritability.What's All Fuss About Heritability?Broad sense heritability (h'Symbol not transcribed'2), accurately described by Wahlsten, estimates proportion of total phenotypic variance in a trait (V'Symbol not transcribed'Y) in a breeding population that is attributed to genotypic variance (V'Symbol not transcribed'G). conceptual model of h'Symbol not transcribed'2 is represented thus:h'Symbol not transcribed'2 = V'Symbol not transcribed'G/V'Symbol not transcribed'YA primary assumption of this model according to Wahlsten is that the measured score (or phenotype) of an individual on a psychological test (Y'Symbol not transcribed'i) is sum of only two components, G'Symbol not transcribed'i determined by genes and E'Symbol not transcribed'i specified by person's environment: that is G and E must not interact. Therefore according to Wahlsten:V'Symbol not transcribed'Y = V'Symbol not transcribed'G + V'Symbol not transcribed'EPresenting this as basic conceptual model of heritability, where effects of genes and environment are strictly additive, Wahlsten subsequently and somewhat predictably proceeds to highlight its inadequacies. For example, in citing a number of important examples of gene - environment interactions, including McGill maze bright and dull rats (Hughes & Zubek, 1956), Wahlsten reveals inherent inability of this model to accommode gene - environment interactions. resultant statement that The additive model is not biologically realistic. is a foregone conclusion, but whole exercise a bit puzzling.My initial exposure to behaviour genetics, which I am certain was not unique, included presentation of a conceptual model of heritability where total phenotypic variance (V'Symbol not transcribed'Y) equals sum of genotypic variance (V'Symbol not transcribed'G) and environmental variance (V'Symbol not transcribed'E) plus interaction of two terms (V'Symbol not transcribed'GxE). original conceptual model as presented in various texts (e.g., Carlson, 1988; Drickamer & Vessey, 1982; Farnsworth, 1978) is:h'Symbol not transcribed'2 = V'Symbol not transcribed'G/V'Symbol not transcribed'Y, where V'Symbol not transcribed'Y = V'Symbol not transcribed'G + V'Symbol not transcribed'E + V'Symbol not transcribed'GxEWhile a number of other authors have presented simplified model as depicted by Wahlsten they usually specify that this particular model only holds if genotype and environment do not interact (e.g., Plomin, DeFries & Fulkner, 1988; Wilson, 1975). …

Relationships Between Kendall's Coefficient of Concordance and a Nonparametric Measure of Phenotypic Stability with Implications for the Consistency in Rankings as Affected by Variance Components

AbstractFor a two‐way classification with rows (= genotypes) and columns (= environments) relationships between Kendall's coefficient of concordance W and a nonparametric measure of phenotypic stability S (= variance among the ranks over the environments) are presented.This provides an analogy between Wricke's ecovalence and the sum of squares of genotype‐environment interactions on the parametric side and S and Kendall's W on the nonparametric side: S is a genotype's contribution to the discordance (1 — W) in the data set, while Wricke's ecovalence is a genotype's contribution to the genotype‐environment interaction sum of squares.Genotype x environment interactions may lead, but must not necessarily lead to non‐identical rank orders of the genotypes in different environments (crossover versus noncrossover interactions). When does interaction become rank interaction?The similarity of the rank orders (measured by W) can be approximately expressed by the ratio of the genotypic variance and sum of genotypic variance plus residual variance. For the consistency of rankings this relation leads to an approximate test of significance which is based on variance components.Finally, a numerical example is given using grain yield data of 20 genotypes (varieties) of winter wheat in 10 environments (locations) from German registration trials.

Lower bounds for the variance of sums of weakly dependent variables

Lower bounds for the variance of sums of weakly dependent variables

Energy-based smoothing of data

Given a set of large noisy data, a technique which generates line segments and provides an optimal smooth connection of the line segments to form a smoothing function is proposed. In this technique data are recursively subdivided into two subsets, means of the data in each subset are computed and a line segment is defined between the means of two adjacent subsets. An energy function is defined as the sum of variances of all the subsets added with a weighted (by a weighting parameter) sum of squaring the difference of means between two adjacent subsets. Optimal connections of line segments are obtained by the simulated annealing technique. The weighting parameter is introduced in the energy function to express the relative importance of minimizing local variances of data versus the smoothness of connections of line segments. By adjusting the weighting parameter, a second optimization is obtained in the sense that the smoothing function formed by the connected line segments provides the best compromise between the closeness of data to the smoothing function and the optimal smoothness of the function.

Variation of anatomical and centroid points in the human fetal skull.

To overcome the problem of variation of reference points in cephalometric analysis two methods have been used: 1) fixed relations between consistently recognizable anatomical points and 2) centers of gravity. The linear distance between each point and every other point was calculated from their coordinates measured on radiographs of 60 fetuses (49-212 mm crown-rump length). For each distance between every permutation of points the variance was calculated for the series of 60 fetuses. The sum of the variances for each individual point against all other points was added together and the mean variance for each point derived by dividing by the total number of points. On the basis of mean variance the centroid of the total skull outline showed least variation and was therefore the most stable point measured. This was closely followed by the origin of the coordinate reference grid based on fixed relations theory. All the centroid points showed less variation than the anatomical points upon which the origin of the grid was based.

Frequency-domain localization of intracerebral dipolar sources

A frequency-domain localization of intracerebral dipolar sources can be carried out basically in the same way as a time-domain localization, since the minimization problems to be solved in the two cases have exactly the same structure. The quantity to be minimized can be interpreted as a sum of residual variances of independent linear minimization problems. In the time domain there is one linear problem per sampling time, whereas in the frequency domain there are two linear problems per frequency. It is demonstrated that algorithms readily available for the solution of the time-domain inverse problem can also be used for the solution of the frequency-domain inverse problem. In this way it is not only possible to localize multiple dipoles, but also to combine information from different epochs and from different frequencies.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Performance of efficient balanced codes

The problem of appraising the spectral performance of codes based on a new algorithm for generating zero-disparity codewords presented by D.E. Knuth (1986) is addressed. In order to get some insight into the efficiency of Knuth's construction technique, the authors evaluate the spectral properties of its code streams. The structure of Knuth codes allows the derivation a simple expression for (an approximation to) the sum of variance of these codes. This quantity plays a key role in the spectral performance characterization of DC-balanced codes. The authors evaluate this expression and compare the sum variance of Knuth codes with the sum variance of the polarity bit codes for fixed redundancy. Under the premise that the sum variance can serve as a quantity to judge the width of the spectral notch, the authors conclude that codes based on Knuth's algorithm offer less spectral suppression than polarity bit codes with the same redundancy. >

Quantum theory of optical phase correlations.

We extend the theory of the Hermitian optical phase operator to analyze the quantum phase properties of pairs of electromagnetic field modes. The operators representing the sum and difference of the two single-mode phases are simply the sum and difference of the two single-mode phase operators. The eigenvalue spectra of the sum and difference operators have widths of 4\ensuremath{\pi}, but phases differing by 2\ensuremath{\pi} are physically indistinguishable. This means that the phase sum and difference probability distributions must be cast into a 2\ensuremath{\pi} range. We obtain mod(2\ensuremath{\pi}) probability distributions for the phase sum and difference that unambiguously reveal the signatures of randomness, phase correlations, and phase locking. We use our approach to investigate the phase sum and difference properties for uncorrelated modes in random and partial phase states and the phase-locked properties of the two-mode squeezed vacuum states. We reveal the fundamental property of two-mode squeezed states that the phase sum is locked to the argument of the squeezing parameter. The variance of the phase sum depends dilogarithmically on 1+tanhr, where r is the magnitude of the squeezing parameter, vanishing in the large squeezing limit.

Spectral null codes

The servo position information of magnetic tape or disk recorders is often recorded as low-frequency components usually called pilot tracking tones. Binary codes giving rise to a spectral null at an arbitrary frequency are used to provide space for the allocation of auxiliary pilot tones. Here, encoding methods are treated in which binary data are mapped into constrained binary sequences for shaping the spectrum. The rate and power spectral density function of memoryless codes that exhibit spectral nulls are computed. The relationship between the code redundancy and spectral notch width is quantified with a parameter called the sum variance. It is found that twice the product of the spectral notch width times the sum variance is approximately unity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Chick feeding in the diving petrels Pelecanoides georgicus and P. urinatrix exsul

Chick feeding in common diving petrels (Pelecanoides urinatrix exsul) and South Georgia diving petrels (P. georgicus) was studied on Bird Island, South Georgia. Complete chick meals removed from the proventriculus of adults averaged 25.5 g (n = 32) for common diving petrels (17.6% of adult mass) and 23.3 g (n = 24) for South Georgia diving petrels (20.2% of adult mass); neither contained stomach oils. The sum of the positive mass increments during overnight weighings (SUM) averaged 48.6 g for common diving petrel chicks (n = 78 chick nights) and 41.6 g for South Georgia diving petrel chicks (n = 78 chick nights). Average adult feeding frequencies were 0.95 meals day−1 and 0.92 meals day−1, respectively. Relative meal size in diving petrels was similar to that of other procellariiforms, but SUM averaged about twice that of other petrels. The lower conversion efficiency of meals to body mass in diving petrel chicks reflects the absence of stomach oils in the diet. Higher chick feeding frequency and lower variance in SUM are consistent with the hypothesis that diving petrels forage nearshore on reliable food supply compared with other procellariiforms.

Combined cumulative sum and shewhart variance charts

The process variance is usually assumed to be a constant for a control chart for the process mean. However, both changes in the process mean and process variance ought to be regarded as indications that the process is out of control. A combined cumulative sum (CUSUM) chart and Shewhart variance chart provides simultaneous control of the process mean and process variance. Exact expressions for the moments of the combined CUSUM chart and Shewhart variance chart are given as integral equations. These integral equations can be solved by using numerical methods for the tabulation of average run lengths (ARLs) and higher moments of the combined CUSUM chart and Shewhart variance chart. A control design strategy is given and is illustrated with an example

Contrast of an integrated fringe pattern generated by partially developed speckle

The contrast of a totally integrated fringe pattern generated by a partially developed gaussian speckle is studied. It is shown that fluctuations of the integrated fringe pattern obtained in Young's interference experiment are determined by the sum of variance and covariance of the speckle pattern intensity. In this way we have studied indirectly the second-order statistics of the partially developed speckle in the far-field region. The theoretical results are compared with experimental data.

The present status of confidence interval estimation on variance components in balanced and unbalanced random models

Much research has been conducted to develop confidence Intervals on linear combinations and ratios of variance components in balanced and unbalanced random models.This paper first presents confidence intervals on functions of variance components in balanced designs.These results assume that classical analysis of variance sums of squares are independent and have exact scaled chi-squared distributions.In unbalanced designs, either one or both of these assumptions are violated, and modifications to the balanced model intervals are required.We report results of some recent work that examines various modifications for some particular unbalanced designs.

Variance of Set-Indexed Sums of Mixing Random Variables and Weak Convergence of Set-Indexed Processes

A uniform bound is found for the variance of a partial-sum set-indexed process under a mixing condition. Sufficient conditions are given for a sequence of partial-sum set-indexed processes to converge to Brownian motion. The requisite tightness follows from hypotheses on the metric entropy of the class of sets and moment and mixing conditions on the summands. The proof uses a construction of Bass [2]. Convergence of finite-dimensional laws in this context is studied in [16].

A Variance Bounds Test of the Linear Quadratic Inventory Model

This paper develops and applies a novel test of the Holt et al. linear quadratic inventory model. It is shown that a central property of the model is that a certain weighted sum of variances and covariances of production, sales, and inventories must be nonnegative. The weights are the basic structural parametersof the model. The model may be tested by seeing whether this sum is in fact nonnegative. When the test is applied to some nondurables data aggregated to the two-digit SIC code level, it almost always rejects the model, even though the model does well by traditional criteria.