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.

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.

Asymptotic properties of the number of replications of a paired comparison

1974 ◽  
Vol 11 (1) ◽  
pp. 43-52 ◽  
Author(s):  
V. R. R. Uppuluri ◽  
W. J. Blot
Keyword(s):  
Paired Comparison ◽  
Beta Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Incomplete Beta Function ◽  
Discrete Random Variable ◽  
World Series ◽  
Mean And Variance ◽  
The Mean ◽  
Made In

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

Asymptotic properties of the number of replications of a paired comparison

1974 ◽  
Vol 11 (01) ◽  
pp. 43-52 ◽  
Author(s):  
V. R. R. Uppuluri ◽  
W. J. Blot
Keyword(s):  
Paired Comparison ◽  
Beta Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Incomplete Beta Function ◽  
Discrete Random Variable ◽  
World Series ◽  
Mean And Variance ◽  
The Mean ◽  
Made In

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

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.

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.

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 Bayesian geostatistical modelling for mapping schistosomiasis transmission

Parasitology ◽  
2009 ◽  
Vol 136 (13) ◽  
pp. 1695-1705 ◽  
Author(s):  
P. VOUNATSOU ◽  
G. RASO ◽  
M. TANNER ◽  
E. K. N'GORAN ◽  
J. UTZINGER
Keyword(s):  
Negative Binomial ◽  
Infection Intensity ◽  
Spatial Process ◽  
Economic Data ◽  
Excess Zeros ◽  
Risk Profiling ◽  
Morbidity Control ◽  
The Mean ◽  
Binomial Models ◽  
Made In

SUMMARYProgress has been made in mapping and predicting the risk of schistosomiasis using Bayesian geostatistical inference. Applications primarily focused on risk profiling of prevalence rather than infection intensity, although the latter is particularly important for morbidity control. In this review, the underlying assumptions used in a study mapping Schistosoma mansoni infection intensity in East Africa are examined. We argue that the assumption of stationarity needs to be relaxed, and that the negative binomial assumption might result in misleading inference because of a high number of excess zeros (individuals without an infection). We developed a Bayesian geostatistical zero-inflated (ZI) regression model that assumes a non-stationary spatial process. Our model is validated with a high-quality georeferenced database from western Côte d'Ivoire, consisting of demographic, environmental, parasitological and socio-economic data. Nearly 40% of the 3818 participating schoolchildren were infected with S. mansoni, and the mean egg count among infected children was 162 eggs per gram of stool (EPG), ranging between 24 and 6768 EPG. Compared to a negative binomial and ZI Poisson and negative binomial models, the Bayesian non-stationary ZI negative binomial model showed a better fit to the data. We conclude that geostatistical ZI models produce more accurate maps of helminth infection intensity than the spatial negative binomial ones.

scholarly journals VSS: Variance-stabilized signals for sequencing-based genomic signals

2020 ◽  
Author(s):  
Faezeh Bayat ◽  
Maxwell Libbrecht
Keyword(s):  
Negative Binomial ◽  
Empirical Relationship ◽  
Data Sets ◽  
Data Set ◽  
Variance Stabilization ◽  
Mean And Variance ◽  
The Mean ◽  
Genomic Signals ◽  
Mean Variance ◽  
Inverse Hyperbolic Sine

AbstractMotivationA sequencing-based genomic assay such as ChIP-seq outputs a real-valued signal for each position in the genome that measures the strength of activity at that position. Most genomic signals lack the property of variance stabilization. That is, a difference between 100 and 200 reads usually has a very different statistical importance from a difference between 1,100 and 1,200 reads. A statistical model such as a negative binomial distribution can account for this pattern, but learning these models is computationally challenging. Therefore, many applications—including imputation and segmentation and genome annotation (SAGA)—instead use Gaussian models and use a transformation such as log or inverse hyperbolic sine (asinh) to stabilize variance.ResultsWe show here that existing transformations do not fully stabilize variance in genomic data sets. To solve this issue, we propose VSS, a method that produces variance-stabilized signals for sequencingbased genomic signals. VSS learns the empirical relationship between the mean and variance of a given signal data set and produces transformed signals that normalize for this dependence. We show that VSS successfully stabilizes variance and that doing so improves downstream applications such as SAGA. VSS will eliminate the need for downstream methods to implement complex mean-variance relationship models, and will enable genomic signals to be easily understood by [email protected]://github.com/faezeh-bayat/Variance-stabilized-units-for-sequencing-based-genomic-signals.

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