Cumulative distribution of chi square

2011 ◽  
Author(s):  
ASBC Technical Committe
Keyword(s):  
Cumulative Distribution ◽  
Chi Square

Percentiles of von Mises Stress from Combined Random Vibration and Static Loading by Approximate Noncentral Chi Square Distribution

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Patrick. A Tibbits
Keyword(s):  
Lower Bounds ◽  
Cumulative Distribution ◽  
Von Mises Stress ◽  
Upper And Lower Bounds ◽  
Linear Structures ◽  
Chi Square ◽  
Mean Values ◽  
Random Loads ◽  
Nonzero Mean ◽  
Von Mises

Firstly, a calculation for percentiles of von Mises stress in linear structures subjected to Gaussian random loads is extended to the case of Gaussian random loads having nonzero mean values, i.e.,the inclusion of static loads. The development is restricted to the case of plane stress. The method includes calculation of a given percentile of von Mises stress to any desired accuracy, a rapid estimate of the percentile, and upper and lower bounds on the von Mises stress. The calculation expands the cumulative distribution function of the von Mises stress as a series of noncentral chi-square distributions. Summation of a sufficient number of terms of the series calculates the percentile to the desired accuracy. The rapid estimate of the percentile interpolates the distribution of the von Mises stress in a small number of inverse noncentral chi-square2 distribution functions. The upper and lower bounds on the percentiles take advantage of the noncentral chi-square distribution of summations of normally distributed stress components. Second and third calculation methods arise from approximations of the distribution of quadratic forms of noncentral normal variables, or equivalently, linear combinations of noncentral chi-square variables. These methods provide rapid estimates of percentiles of von Mises stress in linear structures under random loads having nonzero mean values. The accuracy and computational efficiency of the methods are reviewed and compared. The methods are expected to have wide application in design of and prognostics for components subjected to constant structural loads coupled with random loading arising from vibrations caused by wind, waves, seismic events, engines, turbulence, acoustic noise, etc.

scholarly journals Goodness of Fit Tests in Modeling the Distribution of the Daily Rate of Return of the WIG20 Companies

2012 ◽  
Vol 10 (2) ◽  
pp. 103-113
Author(s):  
Kamila Bednarz
Keyword(s):  
Goodness Of Fit ◽  
Laplace Distribution ◽  
Cumulative Distribution ◽  
Rate Of Return ◽  
Likelihood Method ◽  
Return Distribution ◽  
Chi Square ◽  
Goodness Of Fit Tests ◽  
Daily Rate ◽  
Chi Square Test

Goodness of Fit Tests in Modeling the Distribution of the Daily Rate of Return of the WIG20 Companies In this paper a classic rate of return was examined. Due to a limited quantitative range, the study included only the modeling of the rate of return distribution of the WIG20 index and its companies by means of the Laplace distribution and the Gaussian distribution. Additionally, the goodness of fit tests and methods of estimating the aforementioned distributions parameters were thoroughly covered. When applying the Laplace distribution to modeling the rate of return distribution the parameters were determined by means of two methods: the method of moments and the maximum likelihood method. The maximum period was determined, for which usefulness of the distribution in modeling the rates of return distribution was observed, as well as the results of the chi-square test for class intervals with varying length ensuring equal probability, and for intervals with identical length considering two methods of determining the theoretical size: in accordance with the cumulative distribution function as well as on the basis of the probability density function.

scholarly journals CHI-SQUARE SIMULATION OF THE CIR PROCESS AND THE HESTON MODEL

2013 ◽  
Vol 16 (03) ◽  
pp. 1350014 ◽  
Author(s):  
SIMON J. A. MALHAM ◽  
ANKE WIESE
Keyword(s):  
Degrees Of Freedom ◽  
Transition Probability ◽  
Decimal Place ◽  
Heston Model ◽  
Cumulative Distribution ◽  
Inversion Method ◽  
Chi Square ◽  
Sums Of Powers ◽  
Cir Process ◽  
Gaussian Sampling

The transition probability of a Cox–Ingersoll–Ross process can be represented by a non-central chi-square density. First, we establish a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second, we show that Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance–rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third, we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley–Springer–Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.

scholarly journals A Study on Rain Rate Prediction of Southwestern Nigeria

2020 ◽  
pp. 1-12
Author(s):  
Folasade Abiola Semire ◽  
Adeyanju Adekunle ◽  
Robert Olayimika Abolade
Keyword(s):  
Rain Rate ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Percentage Error ◽  
Accurate Estimation ◽  
Volume Data ◽  
Chi Square ◽  
Southwestern Nigeria ◽  
Scientific Methods ◽  
Rainfall Volume

Rainfall parameters can be utilized to investigate the effect of climate change through scientific methods. However, data on rainfall rate exceeded for a fraction of an average year is grossly unavailable over Nigeria’s climate, thereby diminishing the capability of existing models to adequately estimate the effect of degradation due to rain. Hence, more accurate estimation is required for better predictions. Rainfall volume data for six different locations in the south-western region of Nigeria were obtained for rain rate computation using Semire and Rosmiwati model. The curve-fitted Cumulative Distribution Functions were compared with the ITU-R rain rate model (Recommendation P.837-6) and compensation function was obtained using error analysis while the performance was evaluated with respect to existing models using Chi-square, and Percentage Error and Root Mean Square Error (RMSE) metrics. The outcome of this study can be adopted for better understanding of spatial rainfall intensity in this region and other climatic zones of similar rainfall characteristics.

BIAS REDUCTION AND LIKELIHOOD-BASED ALMOST EXACTLY SIZED HYPOTHESIS TESTING IN PREDICTIVE REGRESSIONS USING THE RESTRICTED LIKELIHOOD

2009 ◽  
Vol 25 (5) ◽  
pp. 1143-1179 ◽  
Author(s):  
Willa W. Chen ◽  
Rohit S. Deo
Keyword(s):  
Random Variable ◽  
Ordinary Least Squares ◽  
Bias Reduction ◽  
Small Error ◽  
Cumulative Distribution ◽  
Ratio Test ◽  
Chi Square ◽  
Predictive Regressions ◽  
Restricted Likelihood ◽  
Size Distortions

Difficulties with inference in predictive regressions are generally attributed to strong persistence in the predictor series. We show that the major source of the problem is actually the nuisance intercept parameter, and we propose basing inference on the restricted likelihood, which is free of such nuisance location parameters and also possesses small curvature, making it suitable for inference. The bias of the restricted maximum likelihood (REML) estimates is shown to be approximately 50% less than that of the ordinary least squares (OLS) estimates near the unit root, without loss of efficiency. The error in the chi-square approximation to the distribution of the REML-based likelihood ratio test (RLRT) for no predictability is shown to be $({\textstyle{3 \over 4}} - \rho ^2)n^{ - 1} (G_3 (\cdot) - G_1 (\cdot)) + O(n^{ - 2}),$ where |ρ| < 1 is the correlation of the innovation series and Gs(·) is the cumulative distribution function (c.d.f.) of a $\chi _s^2 $ random variable. This very small error, free of the autoregressive (AR) parameter, suggests that the RLRT for predictability has very good size properties even when the regressor has strong persistence. The Bartlett-corrected RLRT achieves an O(n−2) error. Power under local alternatives is obtained, and extensions to more general univariate regressors and vector AR(1) regressors, where OLS may no longer be asymptotically efficient, are provided. In simulations the RLRT maintains size well, is robust to nonnormal errors, and has uniformly higher power than the Jansson and Moreira (2006, Econometrica 74, 681–714) test with gains that can be substantial. The Campbell and Yogo (2006, Journal of Financial Econometrics 81, 27–60) Bonferroni Q test is found to have size distortions and can be significantly oversized.

scholarly journals Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 129
Author(s):  
Árpád Baricz ◽  
Dragana Jankov Maširević ◽  
Tibor K. Pogány
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Degrees Of Freedom ◽  
Cumulative Distribution Function ◽  
Mean Value ◽  
Cumulative Distribution ◽  
Chi Square ◽  
Mean Value Theorems ◽  
Definite Integrals ◽  
Du Bois

The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.

Percentiles of von Mises Stress From Combined Random Vibration and Static Loading by Approximate Noncentral Chi Square Distribution

2011 ◽  
Author(s):  
Patrick A. Tibbits
Keyword(s):  
Lower Bounds ◽  
Cumulative Distribution ◽  
Von Mises Stress ◽  
Upper And Lower Bounds ◽  
Linear Structures ◽  
Chi Square ◽  
Mean Values ◽  
Random Loads ◽  
Nonzero Mean ◽  
Von Mises

Firstly, a calculation for percentiles of von Mises stress in linear structures subjected to Gaussian random loads is extended to the case of Gaussian random loads having nonzero mean values, i.e., the inclusion of static loads. The development is restricted to the case of plane stress. The method includes calculation of a given percentile of von Mises stress to any desired accuracy, a rapid estimate of the percentile, and upper and lower bounds on the von Mises stress. The calculation expands the cumulative distribution function of the von Mises stress as a series of noncentral chi square distributions. Summation of a sufficient number of terms of the series calculates the percentile to the desired accuracy. The rapid estimate of the percentile interpolates the distribution of the von Mises stress in a small number of inverse noncentral chi-square distribution functions. The upper and lower bounds on the percentiles take advantage of the noncentral chi-square distribution of summations of normally distributed stress components. Second and third calculation methods arise from approximations of the distribution of quadratic forms of noncentral normal variables, or equivalently, linear combinations of noncentral chi square variables. These methods provide rapid estimates of percentiles of von Mises stress in linear structures under random loads having nonzero mean values. The accuracy and computational efficiency of the methods are reviewed and compared. The methods are expected to have wide application in design of and prognostics for components subjected to constant structural loads coupled with random loading arising from vibrations caused by wind, waves, seismic events, engines, turbulence, acoustic noise, etc.

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