An approximation of F distribution by binomial probabilities (original) (raw)
Related papers
Exact distribution for the generalized F tests
Discussiones Mathematicae-Probability and …, 2002
Generalized F statistics are the quotients of convex combinations of central chi-squares divided by their degrees of freedom. Exact expressions are obtained for the distribution of these statistics when the degrees of freedom either in the numerator or in the denominator are even. An example is given to show how these expressions may be used to check the accuracy of Monte-Carlo methods in tabling these distributions. Moreover, when carrying out adaptative tests, these expressions enable us to estimate the p-values whenever they are available.
An approximation to the F distribution using the chi-square distribution
Computational Statistics & Data Analysis, 2002
For the cumulative distribution function (c.d.f.) of the F distribution, F(x; k; n), with associated degrees of freedom, k and n, a shrinking factor approximation (SFA), G( kx; k), is proposed for large n and any ÿxed k, where G(x; k) is the chi-square c.d.f. with degrees of freedom, k, and = (kx; n) is the shrinking factor. Numerical analysis indicates that for n=k ¿ 3, approximation accuracy of the SFA is to the fourth decimal place for most small values of k. This is a substantial improvement on the accuracy that is achievable using the normal, ordinary chi-square, and Sche à e-Tukey approximations. In addition, it is shown that the theoretical approximation error of the SFA, |F(x; k; n) − G( kx; k)|, is O(1=n 2 ) uniformly over x.
On the computation of the noncentral F and noncentral beta distribution
2008
Unfortunately many of the numerous algorithms for computing the cdf and noncentrality parameter of the noncentral F and beta distributions can return completely incorrect results as demonstrated in the paper by examples. Existing algorithms are scrutinized and those parts that involve numerical difficulties are identified. As a result, a pseudo code is presented in which all the known numerical problems are resolved. This pseudo code can be easily implemented in programming language C or FORTRAN without understanding the complicated mathematical background. Symbolic evaluation of a finite and closed formula is proposed to compute exact cdf values. This approach makes it possible to check quickly and reliably the values returned by professional statistical packages over an extraordinarily wide parameter range without any programming knowledge. This research was motivated by the fact that a most useful table for calculating the size of detectable effects for ANOVA tables contains suspicious values in the region of large noncentrality parameter values compared to the values obtained by Patnaik's 2-moment central-F approximation. The reason is identified and the corrected form of the table for ANOVA purposes is given. The accuracy of the approximations to the noncentral-F distribution is also discussed.
—Evaluation of probability distribution functions and their inverse functions plays a primarily important role in a statistical analysis and inference. In addition to statistically critical values, it produces probability values required to complete such inferential tasks. Therefore Chi-squared (2) distribution function was evaluated via its related incomplete gamma function and by integrating the power series and continued fraction expansion with the left-to-right and right-to-left approaches. The standard normal z, Student's t, F, and 2 values were computed using the step number-skipping and binary bisection search. With the optimized algorithm and a computer-based convergence technique as well as their simplicity, they resulted in a tremendous improvement of computational precision and efficiency. Other novel notion and important information on integrating probability distribution functions with computing tricks has also been introduced.
2020
The beta-binomial model that is generated by a simple mixture model has been commonly applied in the health, physical, and social sciences. In clinical and public health, overdispersion occurs due to biological variation between the subjects of interest. Both the binomial and beta-binomial models are applied to different problems occurring in rational test theory. In this study, we focused on modeling overdispersion for binomial distribution. The main aim was to show a complete and extensive understanding of the beta-binomial model and updated form by broaden its practical applications in the field of breast cancer with hormone medication. It is observed in different independent Bernoulli trials yes/no (xi=1, 0) experiments with success probabilities 0<p i< 1 and compare the model in a sequence of ni. The performance of the maximum likelihood estimates technique that is used in moderate and small samples ni by a Newton-Raphson iterative method using Matlab package. We have fou...
THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRI BUTION – AN EXPLORATORY STUDY
K Y PUBLICATION, 2021
The present paper explore empirically the relationship between the Binomial and the Normal distribution. The paper also attempts to find out the values of "n" and "p" above which the Binomial distribution can safely be approximated to the Normal distribution? For different combinations of n (20 and 50), p (0.1, 0.2, 0.5) and the number of successes (X = 0,1,2,3…n), the probabilities are calculated using both the Binomial and the Normal distribution approach and comparisons are made. For all the combinations of n (20, 50) and P(0.1, 0.2, 0.5), the expected frequency distributions of the Binomial and the Normal are found to be comparable. The Chi-square test revealed no significant differences between the expected frequencies of the Binomial and the Normal distribution. According to the literature available, for a good approximation of Binomial distributrion by the Normal distribution, n should be atleast 50. However, the results of the present study suggest that for the Normal approximation of the Binomial distribution n can be as low as 20 and P ≥ 0.1, thus giving rise to a new condition that is np ≥ 2. There is no need to put any condition on nq. For P=0.5, the Normal and the Binomial distributions are found be comparable even when n=7.
Computation of the Generalized F Distribution
Australian <html_ent glyph="@amp;" ascii="&"/> New Zealand Journal of Statistics, 2001
Exact expressions for the distribution function of a random variable of the form ((α1χ 2 m 1 + α2χ 2 m2)/|m|)/(χ 2 ν /ν) are given where the chi-square distributions are independent with degrees of freedom m1, m2, and ν respectively. Applications to detecting joint outliers and Hotelling's misspecified T 2 distribution are given.
2016 International Conference on Computational Science and Computational Intelligence (CSCI), 2016
Evaluation of probability distribution functions and their inverse functions play a central role in finding a critical value or threshold for a statistical inference and analysis. A polynomial fitting of the required number of recurrences for evaluating standard normal function with a continued fraction expansion was created for a desirable precision of computational results. Student's t and the F distribution functions were evaluated by their related incomplete beta function and by combining the left-to-right approach with the right-to-left one to a continued fraction expansion. They result in a tremendous improvement of evaluation precision and efficiency with the optimized algorithm and a computing convergence technique as well as their simplicity. Other novel notion and important information on integrating probability distribution functions with computing tricks have also been provided.
Journal of Statistical Planning and Inference, 2007
Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.