Question
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from the Poisson distribution with $0<\theta \leq 2$. Show that the mle of $\theta$ is $\widehat{\theta}=\min \{\bar{X}, 2\}$.
Step 1
The probability mass function of a Poisson distribution is given by $P(X = x) = e^{-\theta}\frac{\theta^x}{x!}$ for $x = 0, 1, 2, \ldots$. Show more…
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Key Concepts
Recommended Videos
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a $N\left(\theta, \sigma^{2}\right)$ distribution, where $\sigma^{2}$ is fixed but $-\infty<\theta<\infty$ (a) Show that the mle of $\theta$ is $\bar{X}$. (b) If $\theta$ is restricted by $0 \leq \theta<\infty$, show that the mle of $\theta$ is $\widehat{\theta}=\max \{0, \bar{X}\}$.
Maximum Likelihood Methods
Maximum Likelihood Estimation
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a Poisson distribution with parameter $\theta>0$ (a) Find the MVUE of $P(X \leq 1)=(1+\theta) e^{-\theta}$. Hint: $\quad$ Let $u\left(x_{1}\right)=1, x_{1} \leq 1$, zero elsewhere, and find $E\left[u\left(X_{1}\right) \mid Y=y\right]$, where $Y=\sum_{1}^{n} X_{i}$ (b) Express the MVUE as a function of the mle of $\theta$. (c) Determine the asymptotic distribution of the mle of $\theta$. (d) Obtain the mle of $P(X \leq 1)$. Then use Theorem $5.2 .9$ to determine its asymptotic distribution.
Sufficiency
Functions of a Parameter
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from each of the following distributions involving the parameter $\theta .$ In each case find the mle of $\theta$ and show that it is a sufficient statistic for $\theta$ and hence a minimal sufficient statistic. (a) $b(1, \theta)$, where $0 \leq \theta \leq 1$. (b) Poisson with mean $\theta>0$. (c) Gamma with $\alpha=3$ and $\beta=\theta>0$. (d) $N(\theta, 1)$, where $-\infty<\theta \leq \infty$. (e) $N(0, \theta)$, where $0<\theta<\infty$.
Minimal Sufficiency and Ancillary Statistics
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