An Inverse of the Incomplete Beta Function (F-(Variance Ratio) Distribution Function)

2005 ◽  
Author(s):  
Armido Didonato
Keyword(s):  
Distribution Function ◽  
Beta Function ◽  
Variance Ratio ◽  
Incomplete Beta Function ◽  
Ratio Distribution

Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter

2021 ◽  
Vol 127 (1) ◽  
pp. 111-130
Author(s):  
Dimitris Askitis
Keyword(s):  
Distribution Function ◽  
Beta Distribution ◽  
Probability Distributions ◽  
Beta Function ◽  
Incomplete Beta Function ◽  
Univariate Function ◽  
Convexity Properties ◽  
Logarithmic Concavity ◽  
Two Parameter ◽  
Monotonicity Results

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.

On the asymptotic normality of the number of replications of a paired comparison

1983 ◽  
Vol 20 (03) ◽  
pp. 554-562 ◽  
Author(s):  
V. V. Menon ◽  
N. K. Indira
Keyword(s):  
Distribution Function ◽  
Probability Function ◽  
Negative Binomial ◽  
Paired Comparison ◽  
Beta Function ◽  
Incomplete Beta Function ◽  
Normal Approximations ◽  
The Mean ◽  
Mean Variance ◽  
Made In

Consider the number Xm of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.

The Cummulants and an Exact Expression of Generalized Logistic Distribution

1989 ◽  
Vol 38 (3-4) ◽  
pp. 213-218 ◽  
Author(s):  
Karan P. Singh
Keyword(s):  
Distribution Function ◽  
Data Analysis ◽  
Beta Function ◽  
Exact Expression ◽  
Logistic Distribution ◽  
Incomplete Beta Function ◽  
Shape Parameters ◽  
Generalized Logistic Distribution

A generalization of the logistic distribution proposed by Prentice (1975) is very useful for approximating the continuoui distributions and for data analysis. In this paper, an expression for the rth cumulant of the generalization is derived. The generalized logistic (GL) distribution of Prentice (1975) can be expressed as an incomplete beta function. In particular when both shape parameters are integers, the cumulative GL distribution function can be simply expressed as a sum of binomial probabilities. The exact expression of the cumulative GL distribution function when one of the shape parameters is not an integer is derived.

A generalised incomplete beta function and its application to multi-line stock control

1973 ◽  
Vol 10 (04) ◽  
pp. 748-760 ◽  
Author(s):  
J. C. Gittins ◽  
M. J. Maher
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Beta Function ◽  
Distribution Functions ◽  
Incomplete Beta Function ◽  
Stock Control ◽  
Binomial Distributions ◽  
The Family ◽  
Optimal Values ◽  
Poisson Demands

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

On the asymptotic normality of the number of replications of a paired comparison

1983 ◽  
Vol 20 (3) ◽  
pp. 554-562 ◽  
Author(s):  
V. V. Menon ◽  
N. K. Indira
Keyword(s):  
Distribution Function ◽  
Probability Function ◽  
Negative Binomial ◽  
Paired Comparison ◽  
Beta Function ◽  
Incomplete Beta Function ◽  
Normal Approximations ◽  
The Mean ◽  
Mean Variance ◽  
Made In

Consider the number Xm of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.

A generalised incomplete beta function and its application to multi-line stock control

1973 ◽  
Vol 10 (4) ◽  
pp. 748-760
Author(s):  
J. C. Gittins ◽  
M. J. Maher
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Beta Function ◽  
Distribution Functions ◽  
Incomplete Beta Function ◽  
Stock Control ◽  
Binomial Distributions ◽  
The Family ◽  
Optimal Values ◽  
Poisson Demands

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

scholarly journals The incomplete beta function and its ratio to the complete beta function

1968 ◽  
Vol 22 (101) ◽  
pp. 159-159 ◽  
Author(s):  
David Osborn ◽  
Richard Madey
Keyword(s):  
Beta Function ◽  
Incomplete Beta Function

scholarly journals Extended precision computation of the incomplete beta function

1983 ◽  
Author(s):  
Leigh Allen Ihnen
Keyword(s):  
Beta Function ◽  
Incomplete Beta Function
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