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Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution

# Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution,10.1214/aoms/1177692707,The Annals of Mathematical Statistics,Mic

Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution
Let $f_\theta(x) = f(x - \theta), \theta, x\in R$, where $f(x) = 0$ for $x \leqq 0$ and let $\hat{\theta}_n$ be the maximum likelihood estimate (MLE) of $\theta$ based on a sample of size $n$. If $\alpha = \lim f'(x)$ exists as $x \rightarrow 0$, and $0 < \alpha < \infty$, then under some regularity conditions, it is shown that $\alpha_n(\hat{\theta}_n - \theta)$ has an asymptotic standard normal distribution where $2\alpha_n^2 = \alpha n \log n$ and that if $\theta$ is regarded as a random variable with a prior density, then the posterior distribution of $\alpha_n(\theta - \hat{\theta}_n)$ converges to normality in probability.
Journal: The Annals of Mathematical Statistics , vol. 43, no. 1972, pp. 113-122, 1972
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## Citation Context (2)

• ...For α = 2 and 1 < α <2 its rate is (nlog n)−1/2 and n −1/α, respectively; see Woodroofe (1972, 1974), and Akahira (1975a); for location models with additional parameters see Smith (1985)...

### Ursula U. Müller, et al. Estimation in Nonparametric Regression with Non-Regular Errors

• ...(n3`logn) 1/2 (0 n - O) -~ n(O, 1), which generalises Woodroofe [20]...

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