Estimation of subcritical Galton Watson processes with correlated immigration

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications Pub Date : 2025-02-27 DOI:10.1016/j.spa.2025.104614

Yacouba Boubacar Maïnassara , Landy Rabehasaina

{"title":"Estimation of subcritical Galton Watson processes with correlated immigration","authors":"Yacouba Boubacar Maïnassara , Landy Rabehasaina","doi":"10.1016/j.spa.2025.104614","DOIUrl":null,"url":null,"abstract":"<div><div>We consider an observed subcritical Galton Watson process <span><math><mrow><mo>{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span> with correlated stationary immigration process <span><math><mrow><mo>{</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span>. Two situations are presented. The first one is when <span><math><mrow><mtext>Cov</mtext><mrow><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> for <span><math><mi>k</mi></math></span> larger than some <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when <span><math><mrow><mo>{</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span> has general correlation structure: under mixing assumptions, we exhibit an estimator for the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104614"},"PeriodicalIF":1.1000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414925000559","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}

引用次数: 0

Abstract

We consider an observed subcritical Galton Watson process

{Y_{n}, n \in Z}

with correlated stationary immigration process

{ϵ_{n}, n \in Z}

. Two situations are presented. The first one is when

Cov (ϵ_{0}, ϵ_{k}) = 0

for

k

larger than some

k_{0}

: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when

{ϵ_{n}, n \in Z}

has general correlation structure: under mixing assumptions, we exhibit an estimator for the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.

查看原文本刊更多论文

具有相关迁移的次临界高尔顿-沃森过程的估计

我们考虑一个观测到的次临界高尔顿沃森过程{Yn，n∈Z}与相关平稳迁移过程{ϵn，n∈Z}。提出了两种情况。第一个是当Cov（ϵ0，ϵk）=0且k大于某个k时，给出了繁殖率和平均迁移率的一致估计，并证明了中心极限定理。第二种是当{ϵn，n∈Z}具有一般相关结构时，在混合假设下，我们给出了繁殖率对数的一个估计量，并证明了它收敛于二次均值并具有显式的速度。此外，当混合系数下降得足够快时，我们给出并证明了估计量的两项展开式。给出了数值说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Stochastic Processes and their Applications 数学-统计学与概率论

CiteScore

2.90

自引率

7.10%

发文量

180

审稿时长

23.6 weeks

期刊介绍： Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.