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Flesch-Kincaid Reading Grade Level Re-Examined: Creating A Uniform Method For Calculating Readability On A Certification Exam, Emily Neuhoff, Kristiana M. Feeser, Kayla Sutherland, Thomas Hovatter

Online Journal for Workforce Education and Development

Abstract

Objective: This study attempted to establish a consistent measurement technique of the readability of a state-wide Certified Nursing Assistant’s (CNA) certification exam. Background: Monitoring the readability level of an exam helps ensure all test versions do not exceed the maximum reading level of the exam, and that knowledge of the subject matter, rather than reading ability, is being assessed. Method: A two part approach was used to specify and evaluate readability. First, two methods (Microsoft Word^® (MSW) software and published readability formulae) were used to calculate Flesch Reading Ease (FRE) and Flesch-Kincaid Reading Grade Level (FKRGL) for multiple …

The Weak Euler Scheme For Stochastic Delay Equations, Evelyn Buckwar, Rachel Kuske, Salah-Eldin A. Mohammed, Tony Shardlow

Articles and Preprints

We study weak convergence of an Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …

The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao

Articles and Preprints

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.

In Part 1, we prove general existence and compactness theorems for C^k-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the …

Variable Selection For 1d Regression Models, David J. Olive, Douglas M. Hawkins

Articles and Preprints

Variable selection, the search for j relevant predictor variables from a group of p candidates, is a standard problem in regression analysis. The class of 1D regression models is a broad class that includes generalized linear models. We show that existing variable selection algorithms, originally meant for multiple linear regression and based on ordinary least squares and Mallows’ C_p, can also be used for 1D models. Graphical aids for variable selection are also provided.

Two Simple Resistant Regression Estimators, David J. Olive

Articles and Preprints

Two simple resistant regression estimators with O_P(n^−1/2) convergence rate are presented. Ellipsoidal trimming can be used to trim the cases corresponding to predictor variables x with large Mahalanobis distances, and the forward response plot of the residuals versus the fitted values can be used to detect outliers. The first estimator uses ten forward response plots corresponding to ten different trimming proportions, and the final estimator corresponds to the “best” forward response plot. The second estimator is similar to the elemental resampling algorithm, but sets of O(n) cases are used instead of randomly …

A Resistant Estimator Of Multivariate Location And Dispersion, David J. Olive

Articles and Preprints

This paper presents a simple resistant estimator of multivariate location and dispersion. The DD plot is a plot of Mahalanobis distances from the classical estimator versus the distances from a resistant estimator and can be used to detect outliers and as a diagnostic for multivariate normality. The new estimator can be used in the DD plot, is easy to compute and provides insights about several useful robust algorithm techniques.

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Robust Regression With High Coverage, David J. Olive, Douglas M. Hawkins

Articles and Preprints

An important parameter for several high breakdown regression algorithm estimators is the number of cases given weight one, called the coverage of the estimator. Increasing the coverage is believed to result in a more stable estimator, but the price paid for this stability is greatly decreased resistance to outliers. A simple modification of the algorithm can greatly increase the coverage and hence its statistical performance while maintaining high outlier resistance.

Inconsistency Of Resampling Algorithms For High Breakdown Regression Estimators And A New Algorithm, Douglas M. Hawkins, David J. Olive

Articles and Preprints

Since high breakdown estimators are impractical to compute exactly in large samples, approximate algorithms are used. The algorithm generally produces an estimator with a lower consistency rate and breakdown value than the exact theoretical estimator. This discrepancy grows with the sample size, with the implication that huge computations are needed for good approximations in large high-dimensioned samples

The workhorse for HBE has been the ‘elemental set’, or ‘basic resampling’ algorithm. This turns out to be completely ineffective in high dimensions with high levels of contamination. However, enriching it with a “concentration” step turns it into a method that is able …

Applications Of Robust Distances For Regression, David J. Olive

Articles and Preprints

The DD plot, introduced by Rousseeuw and Van Driessen (1999), is a plot of classical vs robust Mahalanobis distances: MD_i vs RD_i. The DD plot can be used as a diagnostic for multivariate normality and elliptical symmetry, and to assess the success of numerical transformations towards elliptical symmetry. In the regression context, many procedures can be adversely affected if strong nonlinearities are present in the predictors. Even if strong nonlinearities are present, the robust distances can be used to help visualize important regression models such as generalized linear models.

Orthogonal Arrays Of Strength Three From Regular 3-Wise Balanced Designs, Charles J. Colbourn, D. L. Kreher, John P. Mcsorley, D. R. Stinson

Articles and Preprints

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.

A Note On Visualizing Response Transformations In Regression, R. Dennis Cook, David J. Olive

Articles and Preprints

A new graphical method for assessing parametric transformations of the response in linear regression is given. Simply regress the response variable Y on the predictors and find the fitted values. Then dynamically plot the transformed response Y^(λ) against those fitted values by varying the transformation parameter λ until the plot is linear. The method can also be used to assess the success of numerical response transformation methods and to discover influential observations. Modifications using robust estimators can be used as well.

Stochastic Functional Differential Equations On Manifolds, Rémi Léandre, Salah-Eldin A. Mohammed

Articles and Preprints

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters.

High Breakdown Analogs Of The Trimmed Mean, David J. Olive

Articles and Preprints

Two high breakdown estimators that are asymptotically equivalent to a sequence of trimmed means are introduced. They are easy to compute and their asymptotic variance is easier to estimate than the asymptotic variance of standard high breakdown estimators.

Applications And Algorithms For Least Trimmed Sum Of Absolute Deviations Regression, Douglas M. Hawkins, David Olive

Articles and Preprints

High breakdown estimation (HBE) addresses the problem of getting reliable parameter estimates in the face of outliers that may be numerous and badly placed. In multiple regression, the standard HBE's have been those defined by the least median of squares (LMS) and the least trimmed squares (LTS) criteria. Both criteria lead to a partitioning of the data set's n cases into two “halves” – the covered “half” of cases are accommodated by the fit, while the uncovered “half”, which is intended to include any outliers, are ignored. In LMS, the criterion is the Chebyshev norm of the residuals of the …

Improved Feasible Solution Algorithms For High Breakdown Estimation, Douglas M. Hawkins, David J. Olive

Articles and Preprints

High breakdown estimation allows one to get reasonable estimates of the parameters from a sample of data even if that sample is contaminated by large numbers of awkwardly placed outliers. Two particular application areas in which this is of interest are multiple linear regression, and estimation of the location vector and scatter matrix of multivariate data. Standard high breakdown criteria for the regression problem are the least median of squares (LMS) and least trimmed squares (LTS); those for the multivariate location/scatter problem are the minimum volume ellipsoid (MVE) and minimum covariance determinant (MCD). All of these present daunting computational problems. …

The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based …

Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We study the behavior for large |x| of Kunita-type stochastic flows φ(t, ω, x) on R^d, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large |x|, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.

Lyapunov Exponents Of Linear Stochastic Functional-Differential Equations. Ii. Examples And Case Studies, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overlineλ₁$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ₁$(σ) behaves like -σ² /2, while for large σ, it grows like logσ. In the regular case, it is shown that a discrete Oseledec …

Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed

Articles and Preprints

We establish the existence of smooth densities for solutions of R^d-valued stochastic hereditary differential systems of the form

dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).

In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in R^d and g is an n x d matrix-valued function defined on [0, ∞) x R^d, such that gg^* has …