scholarly journals Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Wayan Somayasa
Keyword(s):  
Multivariate Regression ◽  
Continuous Functions ◽  
Upper And Lower Bounds ◽  
Boundary Crossing ◽  
Partial Sums ◽  
Power Functions ◽  
Regression Residuals ◽  
Kolmogorov Smirnov ◽  
Von Mises ◽  
Crossing Probabilities

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value α is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.

scholarly journals Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

2016 ◽  
Vol 27 (6) ◽  
pp. 1513-1523 ◽  
Author(s):  
Nino Kordzakhia ◽  
Alexander Novikov ◽  
Bernard Ycart
Keyword(s):  
Boundary Crossing ◽  
Kolmogorov Smirnov ◽  
Crossing Probabilities

A note on boundary - crossing probabilities for the Brownian motion

1972 ◽  
Vol 9 (04) ◽  
pp. 857-861 ◽  
Author(s):  
C. S. Smith
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Simultaneous Equations ◽  
Boundary Crossing ◽  
Direct Solution ◽  
Kolmogorov Smirnov ◽  
Alternative Procedures ◽  
Crossing Probabilities ◽  
Special Case ◽  
Smirnov Test

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob. 8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

scholarly journals Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Wayan Somayasa ◽  
Gusti N. Adhi Wibawa ◽  
La Hamimu ◽  
La Ode Ngkoimani
Keyword(s):  
Asymptotic Theory ◽  
Spatial Regression ◽  
Test Procedure ◽  
Real Data ◽  
Partial Sums ◽  
Finite Sample ◽  
Partial Sums Processes ◽  
Finite Sample Size ◽  
Kolmogorov Smirnov ◽  
Von Mises

We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.

A note on boundary - crossing probabilities for the Brownian motion

1972 ◽  
Vol 9 (4) ◽  
pp. 857-861 ◽  
Author(s):  
C. S. Smith
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Simultaneous Equations ◽  
Boundary Crossing ◽  
Direct Solution ◽  
Kolmogorov Smirnov ◽  
Alternative Procedures ◽  
Crossing Probabilities ◽  
Special Case ◽  
Smirnov Test

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob.8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals A new metric between distributions of point processes

2008 ◽  
Vol 40 (03) ◽  
pp. 651-672 ◽  
Author(s):  
Dominic Schuhmacher ◽  
Aihua Xia
Keyword(s):  
Point Processes ◽  
Relative Difference ◽  
Continuous Functions ◽  
Upper And Lower Bounds ◽  
Point Pattern ◽  
Multiple Point ◽  
Finite Point ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Comprehensive Collection

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

Detecting Nonlinear Associations, Plus Comments on Testing Hypotheses About the Correlation Coefficient

2001 ◽  
Vol 26 (1) ◽  
pp. 73-83 ◽  
Author(s):  
Rand R. Wilcox
Keyword(s):  
T Test ◽  
Test Statistic ◽  
The Social ◽  
Zero Correlation ◽  
Common Strategy ◽  
Kolmogorov Smirnov ◽  
Nonlinear Associations ◽  
Student’S T ◽  
Von Mises ◽  
Student’S T Test

Let (Yi,Xi ), i = 1, . . . , n, be a random sample from some p + 1 variate distribution where Xi is a vector of length p. In the social sciences, the most common strategy for detecting an association between Y and the marginal distributions is to test the hypothesis that the corresponding correlations are zero using a standard Student’s t test. There are two practical problems with this strategy. First, for reasons described in the article, there are situations where the correlation between two random variables is zero, but Student’s t test is not even asymptotically correct. In fact, the probability of rejecting can approach one as the sample size gets large, even though the hypothesis of a zero correlation is true. Of course, one can also apply standard methods based on a linear regression model and the least squares estimator, but the same practical problems arise. Second, Student’s t test can miss nonlinear associations. This latter problem is the main motivation for this article. Results of a former study suggest an approach that avoids both of the difficulties just described. Based on simulations, it is found that the Cramér-von Mises form of the test statistic is generally better than the Kolmogorov-Smirnov form. Situations arise where this method has less power than Student’s t test, but this is due in part to t test’s use of an incorrect estimate of the standard error.

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