scholarly journals Mean and variance of balanced Pólya urns

2020 ◽  
Vol 52 (4) ◽  
pp. 1224-1248
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Largest Eigenvalue ◽  
Asymptotically Normal ◽  
Pólya Urn ◽  
Polya Urn ◽  
Mean And Variance ◽  
The Mean ◽  
The Largest Eigenvalue

AbstractIt is well-known that in a small Pólya urn, i.e., an urn where the second largest real part of an eigenvalue is at most half the largest eigenvalue, the distribution of the numbers of balls of different colours in the urn is asymptotically normal under weak additional conditions. We consider the balanced case, and then give asymptotics of the mean and the covariance matrix, showing that after appropriate normalization, the mean and covariance matrix converge to the mean and covariance matrix of the limiting normal distribution.

scholarly journals Limit Theorems for Random Triangular URN Schemes

2009 ◽  
Vol 46 (03) ◽  
pp. 827-843 ◽  
Author(s):  
Rafik Aguech
Keyword(s):  
Limit Theorems ◽  
Embedding Method ◽  
Largest Eigenvalue ◽  
Dominant Class ◽  
Pólya Urn ◽  
Polya Urn ◽  
The Mean ◽  
Generalized Pólya Urn ◽  
The Largest Eigenvalue

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

scholarly journals Limit Theorems for Random Triangular URN Schemes

2009 ◽  
Vol 46 (3) ◽  
pp. 827-843 ◽  
Author(s):  
Rafik Aguech
Keyword(s):  
Limit Theorems ◽  
Embedding Method ◽  
Largest Eigenvalue ◽  
Dominant Class ◽  
Pólya Urn ◽  
Polya Urn ◽  
The Mean ◽  
Generalized Pólya Urn ◽  
The Largest Eigenvalue

In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

Applications of Sequentially Estimating the Mean in a Normal Distribution Having Equal Mean and Variance

2004 ◽  
Vol 23 (4) ◽  
pp. 625-665 ◽  
Author(s):  
Nitis Mukhopadhyay ◽  
Greg Cicconetti
Keyword(s):  
Normal Distribution ◽  
Mean And Variance ◽  
The Mean

scholarly journals Estimating the mean and variance from the five-number summary of a log-normal distribution

2020 ◽  
Vol 13 (4) ◽  
pp. 519-531
Author(s):  
Jiandong Shi ◽  
Tiejun Tong ◽  
Yuedong Wang ◽  
Marc G. Genton
Keyword(s):  
Normal Distribution ◽  
Mean And Variance ◽  
The Mean ◽  
Log Normal ◽  
Log Normal Distribution

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (4) ◽  
pp. 938-951 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

Comparison among Some Methods for Estimating the Parameters of Truncated Normal Distribution

2021 ◽  
Vol 20 ◽  
pp. 79-95
Author(s):  
Hilmi Kittani ◽  
Mohammad Alaesa ◽  
Gharib Gharib
Keyword(s):  
Normal Distribution ◽  
Moment Method ◽  
Truncated Normal Distribution ◽  
Moment Methods ◽  
Distribution Parameters ◽  
Mean And Variance ◽  
Truncated Normal ◽  
The Mean ◽  
Censored Sampling ◽  
Mean Square Errors

The aim of this study is to investigate the effect of different truncation combinations on the estimation of the normal distribution parameters. In addition, is to study methods used to estimate these parameters, including MLE, moments, and L-moment methods. On the other hand, the study discusses methods to estimate the mean and variance of the truncated normal distribution, which includes sampling from normal distribution, sampling from truncated normal distribution and censored sampling from normal distribution. We compare these methods based on the mean square errors, and the amount of bias. It turns out that the MLE method is the best method to estimate the mean and variance in most cases and the L-moment method has a performance in some cases.

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (04) ◽  
pp. 938-951
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

Sign in / Sign up

Export Citation Format

Share Document