Limit theorems for ratios of order statistics from uniform distributions
Abstract Let {Xni,1≤i≤mn,n≥1} $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}$ be an array of independent random variables with uniform distribution on [0,θn] $[0, \theta _{n}]$, and {Xn(k),k=1,2,…,mn} $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}$ be the kth order statistics of the random variables {Xni,1≤i≤mn}...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2019-11-01
|
| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2256-7 |
| _version_ | 1818619470878343168 |
|---|---|
| author | Shoufang Xu Changlin Mei Yu Miao |
| author_facet | Shoufang Xu Changlin Mei Yu Miao |
| author_sort | Shoufang Xu |
| collection | DOAJ |
| description | Abstract Let {Xni,1≤i≤mn,n≥1} $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}$ be an array of independent random variables with uniform distribution on [0,θn] $[0, \theta _{n}]$, and {Xn(k),k=1,2,…,mn} $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}$ be the kth order statistics of the random variables {Xni,1≤i≤mn} $\{X_{ni}, 1 \leq i \leq m_{n}\}$. We study the limit properties of ratios {Rnij=Xn(j)/Xn(i),1≤i<j≤mn} $\{R_{nij}=X_{n(j)}/X_{n(i)}, 1\leq i < j \leq m_{n}\}$ for fixed sample size mn=m $m_{n}=m$ based on their moment conditions. For 1=i<j≤m $1=i < j \leq m$, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for 2=i<j≤m $2=i < j \leq m$, we obtain some classical limit theorems and self-normalized limit theorems. |
| first_indexed | 2024-12-16T17:38:00Z |
| format | Article |
| id | doaj.art-642032c401474a2790ecb1837fef49b8 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-16T17:38:00Z |
| publishDate | 2019-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-642032c401474a2790ecb1837fef49b82022-12-21T22:22:40ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-11-012019111410.1186/s13660-019-2256-7Limit theorems for ratios of order statistics from uniform distributionsShoufang Xu0Changlin Mei1Yu Miao2Department of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong UniversityDepartment of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong UniversityCollege of Mathematics and Information Science, Henan Normal UniversityAbstract Let {Xni,1≤i≤mn,n≥1} $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}$ be an array of independent random variables with uniform distribution on [0,θn] $[0, \theta _{n}]$, and {Xn(k),k=1,2,…,mn} $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}$ be the kth order statistics of the random variables {Xni,1≤i≤mn} $\{X_{ni}, 1 \leq i \leq m_{n}\}$. We study the limit properties of ratios {Rnij=Xn(j)/Xn(i),1≤i<j≤mn} $\{R_{nij}=X_{n(j)}/X_{n(i)}, 1\leq i < j \leq m_{n}\}$ for fixed sample size mn=m $m_{n}=m$ based on their moment conditions. For 1=i<j≤m $1=i < j \leq m$, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for 2=i<j≤m $2=i < j \leq m$, we obtain some classical limit theorems and self-normalized limit theorems.http://link.springer.com/article/10.1186/s13660-019-2256-7Uniform distributionOrder statisticsLaw of large numbersComplete convergenceLarge deviation principle |
| spellingShingle | Shoufang Xu Changlin Mei Yu Miao Limit theorems for ratios of order statistics from uniform distributions Journal of Inequalities and Applications Uniform distribution Order statistics Law of large numbers Complete convergence Large deviation principle |
| title | Limit theorems for ratios of order statistics from uniform distributions |
| title_full | Limit theorems for ratios of order statistics from uniform distributions |
| title_fullStr | Limit theorems for ratios of order statistics from uniform distributions |
| title_full_unstemmed | Limit theorems for ratios of order statistics from uniform distributions |
| title_short | Limit theorems for ratios of order statistics from uniform distributions |
| title_sort | limit theorems for ratios of order statistics from uniform distributions |
| topic | Uniform distribution Order statistics Law of large numbers Complete convergence Large deviation principle |
| url | http://link.springer.com/article/10.1186/s13660-019-2256-7 |
| work_keys_str_mv | AT shoufangxu limittheoremsforratiosoforderstatisticsfromuniformdistributions AT changlinmei limittheoremsforratiosoforderstatisticsfromuniformdistributions AT yumiao limittheoremsforratiosoforderstatisticsfromuniformdistributions |