Suppose we have a random sample of size 2 n from a...

Suppose we have a random sample of size $2 n$ from a population denoted by $X,$ and $E(X)=\mu$ and $V(X)=\sigma^{2} .$ Let $$\bar{X}_{1}=\frac{1}{2 n} \sum_{i=1}^{2 n} X_{i} \quad \text { and } \quad \bar{X}_{2}=\frac{1}{n} \sum_{i=1}^{n} X_{i} $$be two estimators of $\mu$. Which is the better estimator of $\mu$ ? Explain your choice.

Suppose we have a random sample of size $2 n$ from a population denoted by $X,$ and $E(X)=\mu$ and $V(X)=\sigma^{2} .$ Let
$$\bar{X}_{1}=\frac{1}{2 n} \sum_{i=1}^{2 n} X_{i} \quad \text { and } \quad \bar{X}_{2}=\frac{1}{n} \sum_{i=1}^{n} X_{i}
$$be two estimators of $\mu$. Which is the better estimator of $\mu$ ? Explain your choice.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Efficiency

Efficiency in estimation is a way to compare unbiased estimators, typically favoring those with lower variance. An efficient estimator is one that achieves the smallest variance among all unbiased estimators, thus providing more precise estimates of the parameter.

Effect of Sample Size

Sample size plays a crucial role in the performance of an estimator. Generally, increasing the sample size leads to a reduction in the variance of the estimator, thereby improving the estimator's precision and making it a better candidate for estimating the parameter.

Unbiasedness

This concept refers to an estimator's property of having an expected value equal to the true parameter being estimated. An unbiased estimator does not systematically overestimate or underestimate the parameter; over many samples, its average will equal the parameter value.

Variance

The variance of an estimator measures the dispersion of its sampling distribution. Lower variance indicates that the estimator tends to be closer to its expected value across repeated samples, making it more stable and reliable in estimation.

Transcript

Please give Ace some feedback