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.

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.

scholarly journals Asymptotic Properties of a Leader Election Algorithm

2011 ◽  
Vol 48 (02) ◽  
pp. 569-575 ◽  
Author(s):  
Ravi Kalpathy ◽  
Hosam M. Mahmoud ◽  
Mark Daniel Ward
Keyword(s):  
Geometric Distribution ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Leader Election ◽  
Wasserstein Metric ◽  
Coin Tossing ◽  
Mean And Variance ◽  
The Mean ◽  
Election Algorithm ◽  
Leader Election Algorithm

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

Asymptotic Properties of a Leader Election Algorithm

2011 ◽  
Vol 48 (2) ◽  
pp. 569-575 ◽  
Author(s):  
Ravi Kalpathy ◽  
Hosam M. Mahmoud ◽  
Mark Daniel Ward
Keyword(s):  
Geometric Distribution ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Leader Election ◽  
Wasserstein Metric ◽  
Coin Tossing ◽  
Mean And Variance ◽  
The Mean ◽  
Election Algorithm ◽  
Leader Election Algorithm

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

On certain coloring parameters of Mycielski graphs of some graphs

2018 ◽  
Vol 10 (03) ◽  
pp. 1850030
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
K. A. Germina ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Graph Coloring ◽  
Random Variable ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Variable Formula ◽  
Statistical Measures ◽  
Random Experiment ◽  
Mean And Variance

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text]. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph coloring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the coloring parameters of the Mycielskian of certain standard graphs.

The Largest Missing Value in a Sample of Geometric Random Variables

2014 ◽  
Vol 23 (5) ◽  
pp. 670-685 ◽  
Author(s):  
MARGARET ARCHIBALD ◽  
ARNOLD KNOPFMACHER
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Missing Value ◽  
Asymptotic Expressions ◽  
Mean And Variance ◽  
The Mean ◽  
Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

STATISTICAL INFERENCE FOR K INDEPENDENT, HOMOGENEOUS ARITHMETIC PROCESSES

2000 ◽  
Vol 07 (03) ◽  
pp. 223-236 ◽  
Author(s):  
FRANCIS KIT-NAM LEUNG
Keyword(s):  
Stochastic Process ◽  
Linear Regression ◽  
Renewal Process ◽  
Random Variable ◽  
Simple Linear Regression ◽  
Variable Formula ◽  
Mean And Variance ◽  
The Mean ◽  
Time Continuum ◽  
Area Of Application

For k=1,…, K, a stochastic process {An,k, n =1, 2,…} is an arithmetic process (AP) if there exists some real number, d, so that {An,k +(n-1)d, n =1, 2,…} is a renewal process (RP). AP is a stochastically monotonic process and can be used to model a point process, i.e., point events occurring in a haphazard way in time (or space), especially with a trend. For example, the events may be failures arising from a deteriorating machine; and such a series of failures is distributed haphazardly along a time continuum. In this paper, we discuss estimation procedures for K independent, homogeneous APs. Two statistics are suggested for testing whether K given processes come from a common AP. If this is so, we can estimate the parameters d, [Formula: see text] and [Formula: see text] of the AP based on the techniques of simple linear regression, where [Formula: see text] and [Formula: see text] are the mean and variance of the first average random variable [Formula: see text], respectively. In this paper, the procedures are, for the most part, discussed in reliability terminology. Of course, the methods are valid in any area of application, in which case they should be interpreted accordingly.

scholarly journals Estimation of Hazard Rate and Mean Residual Life Ordering for Fuzzy Random Variable

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
S. Ramasubramanian ◽  
P. Mahendran
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Mean Residual Life ◽  
Mean And Variance ◽  
The Mean ◽  
Hazard Rate Ordering ◽  
Fuzzy Random

L2-metric is used to find the distance between triangular fuzzy numbers. The mean and variance of a fuzzy random variable are also determined by this concept. The hazard rate is estimated and its relationship with mean residual life ordering of fuzzy random variable is investigated. Additionally, we have focused on deriving bivariate characterization of hazard rate ordering which explicitly involves pairwise interchange of two fuzzy random variablesXandY.

scholarly journals Modelling Assumed Metric Paired Comparison Data - Application to Learning Related Emotions

2014 ◽  
Vol 44 (1) ◽  
pp. 3-15 ◽  
Author(s):  
Alexandra Grand ◽  
Regina Dittrich
Keyword(s):  
Regression Model ◽  
Beta Distribution ◽  
Paired Comparison ◽  
Random Variable ◽  
Open Unit ◽  
Student Survey ◽  
Data Set ◽  
Linear Predictor ◽  
Data Application ◽  
The Mean

In this article we suggest a beta regression model that accounts for the degree of preference in paired comparisons measured on a bounded metric paired comparison scale. The beta distribution for bounded continuous random variables assumes values in the open unit interval (0,1). However, in practice we will observe paired comparison responses that lie within a fixed or arbitrary fixed interval [-a,a] with known value of a. We therefore transform the observed responses into the interval (0,1) and assume that these transformed responses are each a realization of a random variable which follows a beta distribution. We propose a simple paired comparison regression model for beta distributed variables which allows us to model the mean of the transformed response using a linear predictor and a logit link function -- where the linear predictor is defined by the parameters of the logit-linear Bradley-Terry model. For illustration we applied the presented model to a data set obtained from a student survey of learning related emotions in mathematics.

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