Rth Moment Research Articles

A kappa distribution with a hydrological application

A description of the mathematical properties of three-parameter kappa distribution (K3D) is provided. We consider the hazard rate function, the rth moment, the rth L-moment, and the asymptotic distribution of the extreme order statistics and their graphical illustrations. It turns out that this distribution is close to a generalized Pareto distribution, so that the K3D is suitable for modeling threshold excesses or modeling data that has a lower bound. In addition we studied three estimation methods (method of moments, method of L-moments, and maximum likelihood). The Monte Carlo simulation for performance evaluation shows that the maximum likelihood estimator and the L-moment estimator worked equally well. The method of moments estimator worked poorly for all cases. We illustrate its applicability for daily rainfall data.

Measuring Process Performance Based on Expected Loss with Asymmetric Tolerances

By approaching capability from the point of view of process loss similar to C pm , Johnson (1992) provided the expected relative loss L e to consider the proximity of the target value. Putting the loss in relative terms, a user needs only to specify the target and the distance from the target at which the product would have zero worth to quantify the process loss. Tsui (1997) expressed the index L e as L e = L ot + L pe , which provides an uncontaminated separation between information concerning the process relative off-target loss (L ot ) and the process relative inconsistency loss (L pe ). Unfortunately, the index L e inconsistently measures process capability in many cases, particularly for processes with asymmetric tolerances, and thus reflects process potential and performance inaccurately. In this paper, we consider a generalization, which we refer to as , to deal with processes with asymmetric tolerances. The generalization is shown to be superior to the original index L e . In the cases of symmetric tolerances, the new generalization of process loss indices , and reduces to the original index L e , L ot , and L pe , respectively. We investigate the statistical properties of a natural estimator of and when the underlying process is normally distributed. We obtained the rth moment, expected value, and the variance of the natural estimator , , and . We also analyzed the bias and the mean squared error in each case. The new generalization measures process loss more accurately than the original index L e .

Optimal lower bounds on the dilatational strain inside random two-phase composites subjected to hydrostatic loading

Composites made from two linear isotropic elastic materials are considered. It is assumed that only the volume fraction of each elastic material is known. The composite is subjected to a uniform hydrostatic strain. For this case lower bounds on all rth moments of the dilatational strain field inside each phase are obtained for r ⩾ 2. A lower bound on the maximum value of the dilatational strain field is also obtained. These bounds are given in terms of the volume fractions of the component materials. All of these bounds are shown to be the best possible as they are attained by the dilatational strain field inside the Hashin–Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of the local dilatational strain in terms of a statistical description of the microstructure.

The overflow process from a state-dependent queue

The interoverflow time distribution is explicitly obtained as a power series for a finite-state queueing system with general state-dependent arrival and service rates. This can be expressed as a hyper-exponential distribution in closed form for specific constant, linear and quadratic arrival and service rates. Further, we obtain an explicit formula for the rth moment of the overflow process. We also analyse the inverse problem of finding the arrival and service rates from the given spectral data. Numerical illustrations are presented.

A random interval estimation of the estimated process accuracy index

Process accuracy index Ca has been proposed in the manufacturing industry to provide a numerical measure to assess process performance with respect to accuracy. Information regarding process centering (the ability to cluster around the specification mid-point) in existing engineering statistics and quality assurance literature is relatively rare. In this paper, a natural estimator of the accuracy index is considered under the normality assumption. The exact probability density function and the rth moment raised from a non-central chi-squared distribution of the estimated process accuracy index are derived. A decision-making procedure based on a random interval estimation of the considered index is also constructed. A practical example is demonstrated to illustrate how the proposed reliable procedure may be applied for in-plant applications.

A Bayesian Approach to Obtain a Lower Bound for the Cpm Capability Index

AbstractThe Taguchi capability index Cpm, which incorporates the departure of the process mean from the target value, has been proposed to the manufacturing industry for measuring manufacturing capability. A Bayesian procedure has been considered for testing process performance assuming $\mu =T$, which was generalized without assuming $\mu = T$. Statistical properties of the natural estimator of the index Cpm for normal processes have been investigated extensively. However, the investigation was restricted to processes with symmetric tolerances. Recently, a generalized Cpm, referred to as $C''_{{\rm p}m}$, was proposed to cover processes with asymmetric tolerances. Under the normality assumption, the statistical properties of the estimated $C''_{{\rm p}m}$ including the exact sampling distribution, the rth moment, expected value, variance, and the mean‐squared error were obtained. In this paper, we use a Bayesian approach to obtain the interval estimation for the generalized Taguchi capability index $C''_{{\rm p}m}$. Consequently, the manufacturing capability testing can be performed for quality assurance. Copyright © 2005 John Wiley & Sons, Ltd.

Estimating Process Capability Indices Based on Subsamples for Asymmetric Tolerances

Abstract Chen and Pearn (2001) proposed a new generalization of process capability indices (PCIs) for processes with asymmetric tolerances. (u, v), is superior to the original index Cp (u, v) and other existing generalizations by being closely related to actual the process yield, more sensitive to the process centering for given values of μ and σ2, and the on-target process characteristic with the maximal value. In this article, Cp″ (u, v) is presented as the function of the accuracy index δ″ and the precision index γ″. We investigate the relationships of δ″ and γ″ with the process yield. We obtain the exact cumulative distribution functions and explicit forms of probability density functions of the natural estimators of δ″ and Cp″ (u, v) based on small subsamples data collecting from past “in-control” and S control charts. In addition, we derive the rth moments of and (u, v) and the expected values and the variances for , , and (u, v). We also analyze the statistical properties of the estimated indices ,...

Estimating Process Capability Indices Based on Subsamples for Asymmetric Tolerances

Chen and Pearn (2001) proposed a new generalization of process capability indices (PCIs) for processes with asymmetric tolerances. (u, v), is superior to the original index Cp (u, v) and other existing generalizations by being closely related to actual the process yield, more sensitive to the process centering for given values of μ and σ2, and the on-target process characteristic with the maximal value. In this article, Cp″ (u, v) is presented as the function of the accuracy index δ″ and the precision index γ″. We investigate the relationships of δ″ and γ″ with the process yield. We obtain the exact cumulative distribution functions and explicit forms of probability density functions of the natural estimators of δ″ and Cp″ (u, v) based on small subsamples data collecting from past “in-control” and S control charts. In addition, we derive the rth moments of and (u, v) and the expected values and the variances for , , and (u, v). We also analyze the statistical properties of the estimated indices , , and (u, v) assuming the process is normally distributed.

Optimal lower bounds on the electric-field concentration in composite media

Composites made from two linear-isotropic-dielectric materials are considered. It is assumed that only the volume fraction and the two-point correlation function of each dielectric material are known. Lower bounds on all rth moments of the electric-field intensity inside each phase are obtained for r⩾2. A lower bound on the maximum field intensity inside the composite is also obtained. The bounds are given in terms of the one- and two-point statistics of the microgeometry. All of these bounds are shown to be the best possible as they are attained by the electric-field associated with a suitably constructed space-filling confocal-ellipsoid assemblage. The bounds provide a new opportunity for the assessment of local field behavior in terms of a statistical description of the microstructure.

Cumulants are universal homomorphisms into Hausdorff groups

This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let ProbOpen image in new window denote the convolution semigroup of all probability measures on Open image in new window with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence Open image in new window regarded as a function from ProbOpen image in new window into the additive topological group Open image in new window ofall real sequences, is universal among continuous homomorphisms from ProbOpen image in new window into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through κ. An analogous result, referring to just the first Open image in new window cumulants,holds for the semigroup Open image in new window of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from Open image in new window the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.

Long‐Range Dependence of Markov Renewal Processes

SummaryThis paper examines long‐range dependence (LRD) and asymptotic properties of Markov renewal processes generalizing results of Daley for renewal processes. The Hurst index and discrepancy function, which is the difference between the expected number of arrivals in (0, t] given a point at 0 and the number of arrivals in (0, t] in the time stationary version, are examined in terms of the moment index. The moment index is the supremum of the set of r > 0 such that the rth moment of the first return time to a state is finite, employing the solidarity results of Sgibnev. The results are derived for irreducible, regular Markov renewal processes on countable state spaces. The paper also derives conditions to determine the moment index of the first return times in terms of the Markov renewal kernel distribution functions of the process.

Law of large numbers for sample mean of random weighting estimate

For the random weighting mean, a strong law of large numbers is obtained under the assumption of finiteness of the rth moment, for some r>1, and weak law of large numbers is obtained under the finiteness of the first moment.

A unified approach to the analysis of Horton‐Strahler parameters of binary tree structures

AbstractThe Horton‐Strahler number naturally arose from problems in various fields, e.g., geology, molecular biology and computer science. Consequently, detailed investigations of related parameters for different classes of binary tree structures are of interest. This paper shows one possibility of how to perform a mathematical analysis for parameters related to the Horton‐Strahler number in a unified way such that only a single analysis is needed to obtain results for many different classes of trees. The method is explained by the examples of the expected Horton‐Strahler number and the related rth moments, the average number of critical nodes, and the expected distance between critical nodes. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 252–277, 2002

How are moments and moments of spacings related to distribution functions?

It is shown how rth moments of random variables and rth product moments of spacings between random variables can be written as r-dimensional integrals involving only the distribution function, and not the density or any other function of the variables of integration. These fundamental formulae are the main focus of the paper and appear to be new for r⩾3, as well as for the second cross-moment of spacings. The formulae for ordinary moments are derived separately for r=3 and 4. Those for moments of spacings allow an expression for quite general r. Links with L-moments and with expressions for the moments of the sample range are explored. Extensions to the independent but not identically distributed case are given.

Spatial scaling properties of annual streamflow in the United States

The spatial scaling properties of annual average streamflow is examined using records from 1 433 river basins across the continental United States. The log-linear relationship ln(E[Qr i]) = a + br ln(Ai) is representative throughout the United States, where E[Qr i] represents the expectation of the rth moment of annual streamflow at site i, and Ai represents drainage area. The scaling model parameters ar and br follow nearly perfect linear relationships ar = rα and br = rβ throughout the continental United States. We conclude that the probability distribution of annual streamflow follows simple scaling relationships in all regions of the United States. In temperate regions where climate is relatively homogeneous, scale alone describes most of the variability in the moments of annual streamflow. In the more climatically heterogeneous regions, such as in the Upper Colorado and Missouri river basins, scale alone is a poor predictor of the moments of annual flow.

Discrete Approximations to Continuous Univariate Distributions—an Alternative to Simulation

Summary A method to replace a continuous univariate distribution with a discrete distribution that takes MN different values is analysed. Both distributions share the same rth moments for r=0, . . ., 2N−1 and their corresonding distribution functions coincide at least at M+1 points. Several statistical and engineering examples are considered in which the discrete approximation may be used to avoid a simulation study that would be much more demanding computationally.

A paradox in least-squares estimation of linear regression models

This note considers a paradox arising in the least-squares estimation of linear regression models in which the error terms are assumed to be i.i.d. and possess finite rth moment, for r∈[1,2). We give a concrete example to show that the least-squares estimator of the slope parameter is inconsistent when the intercept parameter of the model is given. However, surprisingly this estimator is consistent when the intercept parameter is intendedly assumed to be unknown and re-estimated simultaneously with the slope parameter.

Asymptotic optimality of the wald sequential test

We give a generalization to the case of m hypotheses of a theorem of Lai and derive an asymptotic optimality property of the Wald sequential test for general, possibly dependent, random variables with respect to the rth moment of observation time, under some r-quick convergence condition. We also extend the definition of this convergence and give an application to sequential analysis.

Moments of random walk with fixed end point

It is shown that, for d-dimensional random walks with fixed end points, the rth moments at the ith step are rth order polynomials of i. The results are applicable to (1 + 1)-dimensional SOS models of interfaces.

Estimation of moments under koziol-green model for random censorship

The Koziol-Green model for random censorship is considered (the model of two competing risks with proportional hazards) and an estimate of the rth moment under this model is proposed. Some asymptotic properties of the suggested estimate are studied. The estimate is shown to be consistent and asymptotically normally distributed. Moreover, the joint asymptotic normality of estimated moments is proved. The results are applied to the estimated variance.