Let X1, X2, …, Xn, n>2, be a random sample from the binomial...

Let $X_{1}, X_{2}, \ldots, X_{n}, n>2$, be a random sample from the binomial distribution $b(1, \theta)$. (a) Show that $Y_{1}=X_{1}+X_{2}+\cdots+X_{n}$ is a complete sufficient statistic for $\theta$. (b) Find the function $\varphi\left(Y_{1}\right)$ that is the MVUE of $\theta$. (c) Let $Y_{2}=\left(X_{1}+X_{2}\right) / 2$ and compute $E\left(Y_{2}\right)$. (d) Determine $E\left(Y_{2} \mid Y_{1}=y_{1}\right)$.

Let $X_{1}, X_{2}, \ldots, X_{n}, n>2$, be a random sample from the binomial distribution $b(1, \theta)$.
(a) Show that $Y_{1}=X_{1}+X_{2}+\cdots+X_{n}$ is a complete sufficient statistic for $\theta$.
(b) Find the function $\varphi\left(Y_{1}\right)$ that is the MVUE of $\theta$.
(c) Let $Y_{2}=\left(X_{1}+X_{2}\right) / 2$ and compute $E\left(Y_{2}\right)$.
(d) Determine $E\left(Y_{2} \mid Y_{1}=y_{1}\right)$.

Key Concepts

Sufficient Statistic

A statistic is sufficient for a parameter if it captures all the information in the data about that parameter, meaning that the conditional distribution of the data given the statistic does not depend on the parameter. In other words, once the sufficient statistic is known, no additional data processing can extract further information about the parameter, which is particularly useful in reducing the dimensionality of the problem.

Complete Statistic

A statistic is complete if any measurable function of the statistic that has an expected value of zero for all parameter values is almost surely zero. Completeness is a strong property that, when combined with sufficiency, guarantees the uniqueness of unbiased estimators that are functions of the statistic, ultimately aiding in the derivation of the minimum variance unbiased estimator.

Minimum Variance Unbiased Estimator (MVUE)

The MVUE is an estimator that is unbiased for the parameter of interest and achieves the lowest possible variance among all unbiased estimators. When a complete sufficient statistic exists, the Lehmann–Scheffé theorem ensures that any unbiased estimator that is a function of the complete sufficient statistic is the unique MVUE.

Rao-Blackwell Theorem

The Rao-Blackwell theorem is a procedure to improve an estimator by conditioning it on a sufficient statistic, resulting in an estimator that is at least as good as the original in terms of mean squared error. This process often leads to finding the MVUE by taking an initial unbiased estimator and refining it via its conditional expectation given the complete sufficient statistic.

Conditional Expectation

Conditional expectation is the expected value of a random variable given that certain conditions or events are known to occur. It is a fundamental concept in probability theory that helps simplify problems, especially in the context of deriving properties of estimators by conditioning on sufficient statistics, which can reduce variance and provide insights into the structure of the estimator.

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