Let $X_1,~X_2,\cdots$ be a sequence of independent and identically distributed random variables which are supposed to be observed in sequence. The $n$th value in the sequence is a $k$-record value if exactly $k$ of the first $n$ values (including $X_n$) are at least as large as it. Let ${\bf R}_k$ denote the ordered set of $k$-record values. The famous Ignatov's Theorem states that the random sets ${\bf R}_k(k=1,2,\cdots)$ are independent with common distribution. We introduce one new record named recent-$k$-record in this paper: $X_n$ is a $j$-recent-k-record if there are exactly $j$ values at least as large as $X_n$ in $X_{n-k},~X_{n-k+1},\cdots,~X_{n-1}$. It turns out that recent-k-record brings many interesting problems and some novel properties such as prediction rule and Poisson approximation are proved in this paper. One application named "No Good Record" via the Lov{\'a}sz Local Lemma is also provided. We conclude this paper with some possible extensions for future work.
Ignatov’s theorem recent-k-record Poisson approximation the Lovász Local Lemma
Recent-$k$-record Poisson Approximation Ignatov's theorem the Lovàsz Local Lemma
Birincil Dil | İngilizce |
---|---|
Konular | İstatistik |
Bölüm | İstatistik |
Yazarlar | |
Erken Görünüm Tarihi | 2 Ekim 2024 |
Yayımlanma Tarihi | 15 Ekim 2024 |
Yayımlandığı Sayı | Yıl 2024 Cilt: 53 Sayı: 5 |