scholarly journals Coalescence in Critical and Subcritical Galton-Watson Branching Processes

2012 ◽  
Vol 49 (03) ◽  
pp. 627-638 ◽  
Author(s):  
K. B. Athreya
Keyword(s):  
Random Sampling ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Sampling Without Replacement ◽  
Coalescence Time ◽  
Lines Of Descent

In a Galton-Watson branching process that is not extinct by the nth generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation X n a pairwise coalescence time. Similarly, let Y n denote the coalescence time for the whole population of the nth generation conditioned on the event that it is not extinct. In this paper the distributions of X n and Y n , and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.

scholarly journals Coalescence in Critical and Subcritical Galton-Watson Branching Processes

2012 ◽  
Vol 49 (3) ◽  
pp. 627-638 ◽  
Author(s):  
K. B. Athreya
Keyword(s):  
Random Sampling ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Sampling Without Replacement ◽  
Coalescence Time ◽  
Lines Of Descent

In a Galton-Watson branching process that is not extinct by the nth generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation Xn a pairwise coalescence time. Similarly, let Yn denote the coalescence time for the whole population of the nth generation conditioned on the event that it is not extinct. In this paper the distributions of Xn and Yn, and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.

scholarly journals Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

2013 ◽  
Vol 50 (2) ◽  
pp. 576-591
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Single Type ◽  
Last Common Ancestor ◽  
Death Time ◽  
Age Dependent ◽  
Lines Of Descent ◽  
First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

2013 ◽  
Vol 50 (02) ◽  
pp. 576-591 ◽  
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Single Type ◽  
Last Common Ancestor ◽  
Death Time ◽  
Age Dependent ◽  
Lines Of Descent ◽  
First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={a t,i : 1≤ i≤ Z(t)}, where a t,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let D k(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of D k(t) and its limit distribution as t→∞.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Coalescence on critical and subcritical multitype branching processes

2016 ◽  
Vol 53 (3) ◽  
pp. 802-817
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Joint Distribution ◽  
Branching Process ◽  
Common Ancestor ◽  
Branching Processes ◽  
Recent Common Ancestor ◽  
Most Recent Common Ancestor ◽  
Generation Number ◽  
Multitype Branching Processes ◽  
Lines Of Descent

AbstractConsider a d-type (d<∞) Galton–Watson branching process, conditioned on the event that there are at least k≥2 individuals in the nth generation, pick k individuals at random from the nth generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.

scholarly journals Estimation of general parameters using ancillary information in simple random sampling without replacement

2021 ◽  
pp. 101754
Author(s):  
Nitesh Kumar Adichwal ◽  
Abdullah Ali H. Ahmadini ◽  
Yashpal Singh Raghav ◽  
Rajesh Singh ◽  
Irfan Ali
Keyword(s):  
Random Sampling ◽  
Simple Random Sampling ◽  
Sampling Without Replacement ◽  
Ancillary Information

Simultaneous Estimation of Proportions in a Finite Population

1993 ◽  
Vol 43 (1-2) ◽  
pp. 65-74
Author(s):  
N. Mukhopadhyay ◽  
S. Chattopadhyay
Keyword(s):  
Random Sampling ◽  
Finite Population ◽  
Simple Random Sampling ◽  
Subject Classification ◽  
Simultaneous Estimation ◽  
Sampling Strategies ◽  
Sampling Without Replacement ◽  
First Order ◽  
Multistage Sampling

Sequential and multistage sampling strategies via simple random sampling without replacement, are proposed for simultaneously estimating several proportions in a finite population. Various asymptotic first-order properties are addressed, while some limited moderate sample performance have also been included. AMS (1980) Subject Classification: Primary 62L99; Secondary 62L12

Item Count Technique in Ranked Set Sampling

2022 ◽  
pp. 26-41
Author(s):  
Beatriz Cobo ◽  
Elvira Pelle
Keyword(s):  
Simulation Study ◽  
Random Sampling ◽  
Simple Random Sampling ◽  
Ranked Set Sampling ◽  
Alternative Methods ◽  
Randomized Response Technique ◽  
Randomized Response ◽  
Sampling Without Replacement ◽  
Questioning Techniques ◽  
Item Count Technique

In situations where the estimation of the proportion of sensitive variables relies on the observations of real measurements that are difficult to obtain, there is a need to combine indirect questioning techniques. In the present work, the authors will focus on the item count technique, with alternative methods of sampling, such as the ranked set sampling. They are based on the idea proposed by Santiago et al., which combines the randomized response technique proposed by Warner together with ranked set sampling. The authors will carry out a simulation study to compare the item count technique under ranked set sampling and under simple random sampling without replacement.

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