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This paper is devoted to the numerical analysis of abstract parabolic differential equations in -spaces. The presentation uses general approximation scheme and is based on semigroup theory and a functional analysis approach. For the solutions of the first and second order of accuracy difference schemes, the almost coercive inequality in spaces with the factor are obtained. In the case of UMD space En we establish a coercive inequality in under the condition of R-boundedness. [ABSTRACT FROM AUTHOR]

This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail-most often needed in practical work. [ABSTRACT FROM AUTHOR]

This paper investigates the distribution-free outer confidence interval for the quantile interval introduced by Wilks. Besides exact expressions and a recurrence formula some bounds are derived for the probability of correct coverage of the quantile interval which improve bounds known up to now, and which are tested in several examples. Asymptotic approximations are discussed and applications of the developed methods to other statistics based on order statistics are indicated. [ABSTRACT FROM AUTHOR]

This paper proposes the use of approximate posterior distributions resulting from operational prior distributions chosen with regard to the realized likelihood function. L.J. Savage's "precise measurement" is generalized for approximation in terms of an arbitrary operational prior density, including mixed-type prior distributions with positive probabilities on singular subsets. A new approximation is also given relating such distributions to absolutely continuous distributions with high local concentrations of density, Mixed-type distributions constructed from the natural conjugate prior distributions are proposed and illustrated in the normal-sampling case for unified Bayesian inference in testing and estimation contexts. [ABSTRACT FROM AUTHOR]

The article presents a discussion on the nature of approximation in the context of open-channel flow and the application of critical flow principle in studying the flow features of the circular-crested weir. It notes that the paper's potential approach is based on the velocity distribution around a circular cylinder drowned in a fluid at infinite depth. This shows that the streamlines are concentric at the weir crest. The Boussinesq type approach is considered as the fundamental of a more general method in treating weir flow problem. It points out that the flow features of the circular-crested weir can be predicted through the use of critical flow theory.

In applications of SEM, investigators obtain and interpret parameter estimates that are computed so as to produce optimal model fit in the sense that the obtained model fit would deteriorate to some degree if any of those estimates were changed. This property raises a question: to what extent would model fit deteriorate if parameter estimates were changed? And which parameters have the greatest influence on model fit? This is the idea of parameter influence. The present paper will cover two approaches to quantifying parameter influence. Both are based on the principle of likelihood displacement (LD), which quantifies influence as the discrepancy between the likelihood under the original model and the likelihood under the model in which a minor perturbation is imposed (Cook, 1986). One existing approach for quantifying parameter influence is a vector approach (Lee & Wang, 1996) that determines a vector in the parameter space such that altering parameter values simultaneously in this direction will cause maximum change in LD. We propose a new approach, called influence mapping for single parameters, that determines the change in model fit under perturbation of a single parameter holding other parameter estimates constant. An influential parameter is defined as one that produces large change in model fit under minor perturbation. Figure 1 illustrates results from this procedure for three different parameters in an empirical application. Flatter curves represent less influential parameters. Practical implications of the results are discussed. The relationship with statistical power in structural equation models is also discussed. [image omitted]FIGURE 1 Influence mapping for single parameters. [ABSTRACT FROM AUTHOR]

In their paper, Good and Gaskins determine empirically the number of terms required in Hermite and Fourier expansions to obtain equivalent degrees of approximation. The argument that a sign pattern in the coefficients in the Hermite expansion of a function slows the convergence is ingenious, but there is another reason for the better approximation by Fourier series than for Hermite series that is probably more important.