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.

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 (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.

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.

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.

scholarly journals Radial Velocity Profiles for Anisotropic Spherically Symmetric Clusters: An Example

1988 ◽  
Vol 126 ◽  
pp. 691-692
Author(s):  
Herwig Dejonghe
Keyword(s):  
Distribution Function ◽  
Radial Velocity ◽  
Velocity Dispersion ◽  
Mass Density ◽  
Distribution Functions ◽  
Spherically Symmetric ◽  
Line Profiles ◽  
Anisotropic Models ◽  
The Family

A 1-parameter family of anisotropic models is presented. They all satisfy the Plummer law in the mass density, but have different velocity dispersions. Moreover, the stars are not confined to a particular subset of the total accessible phase space. This family is mathematically simple enough to be explored analytically in detail. The family is rich enough though to allow for a 3-parameter generalization which illustrates that even when both the mass density and the velocity dispersion profiles are required to be the same, a degeneracy in the possible distribution functions persists. The observational consequences of the degeneracy can be studied by calculating the observable radial velocity line profiles obtained with different distribution functions. It turns out that line profiles are relatively sensitive to changes in the distribution function. They therefore can be considered to be more natural observables when a determination of the distribution function is desired.

Approximating gamma distributions by normalized negative binomial distributions

1994 ◽  
Vol 31 (02) ◽  
pp. 391-400 ◽  
Author(s):  
José A. Adell ◽  
Jesús De La Cal
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Upper And Lower Bounds ◽  
Exact Order ◽  
Bernstein Type ◽  
Birth Process ◽  
Charged Multiplicity ◽  
Binomial Distributions ◽  
Gamma Distributions ◽  
Multiplicity Distributions

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

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.

Approximating gamma distributions by normalized negative binomial distributions

1994 ◽  
Vol 31 (2) ◽  
pp. 391-400 ◽  
Author(s):  
José A. Adell ◽  
Jesús De La Cal
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Upper And Lower Bounds ◽  
Exact Order ◽  
Bernstein Type ◽  
Birth Process ◽  
Charged Multiplicity ◽  
Binomial Distributions ◽  
Gamma Distributions ◽  
Multiplicity Distributions

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

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