Afrika Statistika

A simple note on some empirical stochastic process as a tool in uniform L-statistics weak laws

Gane Samb Lô

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Abstract

In this paper, we are concerned with the stochastic process $$\beta_n (q_t, t) = \beta_n(t) = \frac{1}{\sqrt{n}}\sum_{j=1}^n \{G_{t,n}(Y(t)) - G_t(Y_j(t))\} q_t(Y_j(t)), \tag{A}$$ where for $n \geq 1$ and $T > 0$, the sequences $\{Y_1(t), Y_2(t), \cdots, Y_n(t), t \in [0, T]\}$ are independent observations of some real stochastic process $Y(t), t \in [0,T]$, for each $t \in [0,T], G_t$ is the distribution function of $Y(t)$ and $G_{t,n}$ is the empirical distribution function based on $Y_1(t), Y_2(t), \cdots, Y_n(t)$ and finally $q_t$ is a bounded real function defined on $\mathbb{R}$. This process appears when investigating some time-dependent L-Statistics which are expressed as a function of some functional empirical process and the process (A). Since the functional empirical process is widely investigated in the literature, the process reveals itself as an important key for L-Statistics laws. In this paper, we state an extended study of this process, give complete calculations of the first moments, the covariance function and find conditions for asymptotic tightness.

Article information

Source
Afr. Stat., Volume 5, Number 1 (2010), 245-251.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.as/1388545346

Mathematical Reviews number (MathSciNet)
MR2920301

Zentralblatt MATH identifier
1328.62288

Subjects
Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties G2G05

Keywords
Empirical processes Order Statistics L-statistics

Citation

Samb Lô, Gane. A simple note on some empirical stochastic process as a tool in uniform L-statistics weak laws. Afr. Stat. 5 (2010), no. 1, 245--251. https://projecteuclid.org/euclid.as/1388545346


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