Can I use the chebyshev inequality if my estimator is only asymptotically unbiased? I am trying to show that my estimator converges in probability to $\mu$ if it is asymptotically unbiased and its asymptotical variance is $0$.
Many thanks!
Asked
Active
Viewed 222 times
1 Answers
1
The short answer is yes!
For more detailed answer we denote the estimator of $\mu$ by $\widehat{\mu}$ then in order to show the convergence in probability write
$$ \mathbb{P}(|\widehat{\mu} - \mu| > \varepsilon) \to 0, \text{ as } n \to \infty. $$
Denote the bias term $b_n = \mathbb{E}\widehat{\mu} - \mu$, then using Markov inequality (or Chebyshev, if you wish) we get
\begin{align} \mathbb{P}(|\widehat{\mu} - \mu| > \varepsilon) &\le \frac{\mathbb{E}(\widehat{\mu} - \mu)^2} {\varepsilon^2} = \frac{\mathbb{E}(\widehat{\mu} - (\mu + b_n) + b_n)^2} {\varepsilon^2} \\ &= \frac{\mathbb{E}(\widehat{\mu} - (\mu + b_n))^2 + \mathbb{E} b_n^2} {\varepsilon^2} = \frac{\mathbb{V}ar(\widehat{\mu}) + b_n^2}{\varepsilon^2} \to 0, \end{align} since the variance tends to $0$ and the bias term $b_n$ tends to $0$ as $n \to \infty$.
tortue
- 2,223