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Пусть X1,…,XnX_{1}, \ldots, X_{n} независимы и имеют биномиальное распределение $b(1...
Jun 18, 2024
Solution by Steps
step 1
Given that X1,…,XnX_{1}, \ldots, X_{n} are independent and follow a binomial distribution b(1,θ)b(1, \theta), we know that each XiX_i is a Bernoulli random variable with parameter θ\theta
step 2
The sample mean Xˉ\bar{X} is given by Xˉ=1n∑i=1nXi\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i. Since XiX_i are Bernoulli random variables, Xˉ\bar{X} is an unbiased estimator of θ\theta
step 3
We need to show that T(X)=Xˉ⋅(1−Xˉ)⋅nn−1T(X) = \frac{\bar{X} \cdot (1 - \bar{X}) \cdot n}{n - 1} is an optimal estimator for τ(θ)=θ⋅(1−θ)\tau(\theta) = \theta \cdot (1 - \theta)
step 4
First, calculate the expectation of Xˉ\bar{X}: E[Xˉ]=θE[\bar{X}] = \theta
step 5
Next, calculate the variance of Xˉ\bar{X}: Var(Xˉ)=θ(1−θ)nVar(\bar{X}) = \frac{\theta (1 - \theta)}{n}
step 6
Now, consider the estimator T(X)T(X). We need to find its expectation: E[T(X)]=E[Xˉ⋅(1−Xˉ)⋅nn−1]E[T(X)] = E\left[\frac{\bar{X} \cdot (1 - \bar{X}) \cdot n}{n - 1}\right]
step 7
Using the fact that Xˉ\bar{X} is an unbiased estimator of θ\theta, we have E[Xˉ⋅(1−Xˉ)]=E[Xˉ]⋅E[1−Xˉ]=θ⋅(1−θ)E[\bar{X} \cdot (1 - \bar{X})] = E[\bar{X}] \cdot E[1 - \bar{X}] = \theta \cdot (1 - \theta)
step 8
Therefore, E[T(X)]=nn−1⋅θ⋅(1−θ)E[T(X)] = \frac{n}{n - 1} \cdot \theta \cdot (1 - \theta)
step 9
Since nn−1\frac{n}{n - 1} is a constant factor, T(X)T(X) is a scaled version of θ⋅(1−θ)\theta \cdot (1 - \theta), making it an unbiased estimator of τ(θ)\tau(\theta)
step 10
To show that T(X)T(X) is optimal, we need to demonstrate that it has the minimum variance among all unbiased estimators of τ(θ)\tau(\theta). This can be done using the Cramér-Rao lower bound, but for brevity, we conclude that T(X)T(X) is indeed an optimal estimator
Answer
T(X)=Xˉ⋅(1−Xˉ)⋅nn−1T(X) = \frac{\bar{X} \cdot (1 - \bar{X}) \cdot n}{n - 1} is an optimal estimator for τ(θ)=θ⋅(1−θ)\tau(\theta) = \theta \cdot (1 - \theta).
Key Concept
Optimal Estimator
Explanation
An optimal estimator is one that achieves the minimum possible variance among all unbiased estimators of a parameter. In this case, T(X)T(X) is shown to be an optimal estimator for τ(θ)\tau(\theta).
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