Let X1, X2, …, Xn be a random sample from a Poisson...

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.

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.

Key Concepts

Minimum Variance Unbiased Estimator (MVUE)

The MVUE is an unbiased estimator that has the lowest variance among all unbiased estimators of a parameter or a function of a parameter. It is often derived by leveraging complete sufficient statistics along with the Rao-Blackwell and Lehmann-Scheffe theorems, ensuring optimality in the sense of the mean squared error for unbiased estimation.

Delta Method

The Delta Method is a statistical technique used to derive the asymptotic distribution of a function of an estimator, particularly when the estimator itself is asymptotically normal. It relies on a first-order Taylor expansion to approximate the transformation, allowing us to determine the asymptotic variance of the function of the estimator. This method is especially useful when transforming parameter estimates, such as converting the MLE of ? into an estimator for functions like P(X ? 1).

Asymptotic Distribution

The asymptotic distribution refers to the probability distribution that an estimator approaches as the sample size becomes very large. For MLEs, under regularity conditions, they are asymptotically normal with a mean equal to the true parameter value and a variance equal to the inverse of the Fisher information. This result provides a basis for constructing confidence intervals and conducting hypothesis tests for large samples.

Maximum Likelihood Estimation (MLE)

Maximum Likelihood Estimation is a method for estimating the parameters of a statistical model by maximizing the likelihood function, which measures the probability of observing the given data. In the context of the Poisson distribution, the MLE for the parameter ? is typically obtained by setting the derivative of the log-likelihood with respect to ? to zero and solving for ?, often yielding ?-hat as the sample mean.

Rao-Blackwell Theorem

The Rao-Blackwell theorem is a method for improving an unbiased estimator by conditioning on a sufficient statistic. It provides a systematic way to derive an estimator with lower variance by taking the conditional expectation of any unbiased estimator with respect to a sufficient statistic, thereby ensuring that the resulting estimator is at least as good, in terms of variance, as the original estimator.

Complete Sufficient Statistic

A statistic is sufficient if it captures all the information in the sample relevant to estimating the parameter, and it is complete if no non?trivial function of it has zero expectation for all values of the parameter. For a Poisson sample, the sum of the observed values serves as a complete sufficient statistic for the parameter ?, meaning that any unbiased estimator that is a function of this sum is, by the Lehmann-Scheffe theorem, the unique MVUE.

Poisson Distribution

A Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, under the assumption that these events occur with a known constant rate and independently of the time since the last event. It is characterized by its parameter ?, which represents the average rate of occurrence, and has the probability mass function P(X = k) = e^(??) ?^k/k! for non-negative integer values of k.

Lehmann-Scheffe Theorem

The Lehmann-Scheffe theorem states that if an unbiased estimator is a function of a complete sufficient statistic, then it is the unique minimum variance unbiased estimator (MVUE) for the parameter. This theorem is crucial in finding optimal unbiased estimators in problems where complete sufficient statistics exist, such as in the estimation problems involving the Poisson distribution.

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