A boundary crossing probability for the Bessel process

1998 ◽  
Vol 30 (3) ◽  
pp. 807-830 ◽  
Author(s):  
Rebecca A. Betensky
Keyword(s):  
Nonparametric Test ◽  
Bessel Process ◽  
Boundary Crossing ◽  
Test Statistic ◽  
Boundary Crossing Probability ◽  
Significance Level ◽  
Straight Line ◽  
Crossing Probability ◽  
Nonparametric Test Statistic ◽  
First Crossing Time

Analytic approximations are derived for the distribution of the first crossing time of a straight-line boundary by a d-dimensional Bessel process and its discrete time analogue. The main ingredient for the approximations is the conditional probability that the process crossed the boundary before time m, given its location beneath the boundary at time m. The boundary crossing probability is of interest as the significance level and power of a sequential test comparing d+1 treatments using an O'Brien-Fleming (1979) stopping boundary (see Betensky 1996). Also, it is shown by DeLong (1980) to be the limiting distribution of a nonparametric test statistic for multiple regression. The approximations are compared with exact values from the literature and with values from a Monte Carlo simulation.

A boundary crossing probability for the Bessel process

1998 ◽  
Vol 30 (03) ◽  
pp. 807-830
Author(s):  
Rebecca A. Betensky
Keyword(s):  
Nonparametric Test ◽  
Bessel Process ◽  
Boundary Crossing ◽  
Test Statistic ◽  
Boundary Crossing Probability ◽  
Significance Level ◽  
Straight Line ◽  
Crossing Probability ◽  
Nonparametric Test Statistic ◽  
First Crossing Time

Analytic approximations are derived for the distribution of the first crossing time of a straight-line boundary by a d-dimensional Bessel process and its discrete time analogue. The main ingredient for the approximations is the conditional probability that the process crossed the boundary before time m, given its location beneath the boundary at time m. The boundary crossing probability is of interest as the significance level and power of a sequential test comparing d+1 treatments using an O'Brien-Fleming (1979) stopping boundary (see Betensky 1996). Also, it is shown by DeLong (1980) to be the limiting distribution of a nonparametric test statistic for multiple regression. The approximations are compared with exact values from the literature and with values from a Monte Carlo simulation.

scholarly journals Two-dimensional stochastic modeling for predicting bankruptcy for manufacturing companies

2021 ◽  
Vol 54 (2) ◽  
pp. 123-129
Author(s):  
James C. Fu ◽  
Winnie H. W. Fu
Keyword(s):  
Brownian Motion ◽  
Real Data ◽  
Bankruptcy Prediction ◽  
Boundary Crossing ◽  
Time Interval ◽  
Two Dimensional ◽  
Manufacturing Companies ◽  
Boundary Crossing Probability ◽  
Data Set ◽  
Crossing Probability

Increasing accuracy of the model prediction on business bankruptcy helps reduce substantial losses for owners, creditors, investors and workers, and, further, minimize an economic and social problem frequently. In this study, we propose a stochastic model of financial working capital and cashflow as a two-dimensional Brownian motion X(t) = (X1(t),X2(t)) on the business bankruptcy prediction. The probability of bankruptcy occurring in a time interval [0,T] is defined by the boundary crossing probability of the two-dimensional Brownian motion entering a predetermined threshold domain. Mathematically, we extend the result in Fu and Wu (2016) on the boundary crossing probability of a high dimensional Brownian motion to an unbounded convex hull. The proposed model is applied to a real data set of companies in US and the numerical results show the proposed method performs well.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (1) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

Stochastic Boundary Crossing Probabilities for the Brownian Motion

2013 ◽  
Vol 50 (02) ◽  
pp. 419-429 ◽  
Author(s):  
Xiaonan Che ◽  
Angelos Dassios
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Simulation Experiments ◽  
Crossing Probability ◽  
Crossing Probabilities ◽  
Banks Set ◽  
Stochastic Boundary

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

A Nonparametric Test Statistic for the General Linear Model

1989 ◽  
Vol 14 (4) ◽  
pp. 351-371 ◽  
Author(s):  
Michael R. Harwell ◽  
Ronald C. Serlin
Keyword(s):  
Linear Model ◽  
Type I Error ◽  
Nonparametric Test ◽  
General Linear Model ◽  
Error Rates ◽  
General Linear ◽  
Type I ◽  
Test Statistic ◽  
Rank Statistics ◽  
Nonparametric Test Statistic

Puri and Sen (1969Puri and Sen (1985) presented a nonparametric test statistic based on a general linear model approach that is appropriate for testing a wide class of hypotheses. The two forms of this statistic, pure- and mixed-rank, differ according to whether the original predictor values or their ranks are used. Both forms permit the use of standard statistical packages to perform the analyses. The applicability of these statistics in testing a number of hypotheses is highlighted, and an example of their use is given. A simulation study for the multivariate-multiple-regression case is used to examine the distributional behavior of the pure- and mixed-rank statistics and an important competitor, the rank transformation of Conover and Iman (1981). The results suggest that the pure- and mixed-rank statistics are superior with respect to minimizing liberal Type I error rates, whereas the Conover and Iman statistic produces larger power values.

scholarly journals Stochastic Boundary Crossing Probabilities for the Brownian Motion

2013 ◽  
Vol 50 (2) ◽  
pp. 419-429 ◽  
Author(s):  
Xiaonan Che ◽  
Angelos Dassios
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Simulation Experiments ◽  
Crossing Probability ◽  
Crossing Probabilities ◽  
Banks Set ◽  
Stochastic Boundary

Using martingale methods, we derive a set of theorems of boundary crossing probabilities for a Brownian motion with different kinds of stochastic boundaries, in particular compound Poisson process boundaries. We present both the numerical results and simulation experiments. The paper is motivated by limits on exposure of UK banks set by CHAPS. The central and participating banks are interested in the probability that the limits are exceeded. The problem can be reduced to the calculation of the boundary crossing probability from a Brownian motion with stochastic boundaries. Boundary crossing problems are also very popular in many fields of statistics.

A boundary-crossing result for Brownian motion

1992 ◽  
Vol 29 (2) ◽  
pp. 448-453 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Brownian Motion ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (01) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

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