Estimating the Logarithm of Characteristic Function and Stability Parameter for Symmetric Stable Laws

Jüri Lember () and Annika Krutto ()
Additional contact information
Jüri Lember: University of Tartu
Annika Krutto: University of Oslo

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 3, 2149-2167

Abstract: Abstract Let $$X_1,\ldots ,X_n$$ X 1 , … , X n be an i.i.d. sample from symmetric stable distribution with stability parameter $$\alpha$$ α and scale parameter $$\gamma$$ γ . Let $$\varphi _n$$ φ n be the empirical characteristic function. We prove a uniform large deviation inequality: given preciseness $$\epsilon >0$$ ϵ > 0 and probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , there exists universal (depending on $$\epsilon$$ ϵ and p but not depending on $$\alpha$$ α and $$\gamma$$ γ ) constant $$\bar{r}>0$$ r ¯ > 0 so that $$P\big (\sup _{u>0:r(u)\le \bar{r}}|r(u)-\hat{r}(u)|\ge \epsilon \big )\le p,$$ P ( sup u > 0 : r ( u ) ≤ r ¯ | r ( u ) - r ^ ( u ) | ≥ ϵ ) ≤ p , where $$r(u)=(u\gamma )^{\alpha }$$ r ( u ) = ( u γ ) α and $$\hat{r}(u)=-\ln |\varphi _n(u)|$$ r ^ ( u ) = - ln | φ n ( u ) | . As an applications of the result, we show how it can be used in estimation the unknown stability parameter $$\alpha$$ α .

Keywords: Stable laws; Large deviation inequalities; Parameter estimation; Stable laws 60E07; Confidence regions 62F25; Point estimation 62F10 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s11009-021-09908-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09908-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/11009

DOI: 10.1007/s11009-021-09908-z

Access Statistics for this article

Methodology and Computing in Applied Probability is currently edited by Joseph Glaz

More articles in Methodology and Computing in Applied Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().