Problem

Suppose $x_{1}, x_{2}, \ldots, x_{n}$ represent a random sample from a population with mean $\mu$. We want to determine the conditions that the constants $a_{1}, a_{2}, \ldots, a_{n}$ must satisfy for the estimator:

$$\hat{\mu} = a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n}x_{n}$$

to be an unbiased estimator of $\mu$.

1. What condition must the values $a_{1}, a_{2}, \ldots, a_{n}$ satisfy to ensure that the estimator $\hat{\mu}$ is unbiased?
2. Explain why this condition guarantees the unbiasedness of the estimator.
3. Determine if the sample mean is an unbiased estimator according to this criterion.

Summary

In this exercise, we will solve how to determine the necessary conditions for a linear combination of random variables to be an unbiased estimator. The key lies in understanding the concept of expected value and working with the properties of linearity.

Key Concepts

• Estimator: A function of the sample used to approximate an unknown population parameter.
• Unbiased: An estimator is said to be unbiased when its expected value equals the parameter it estimates.
• Expected Value: The weighted average of all possible values a random variable can take.
• Linear Combination: An expression formed by the sum of terms, each with a constant coefficient.
• Random Sample: A set of independent and identically distributed observations from a population.

Practical Applications

1. The concept of unbiasedness is fundamental in statistical inference to obtain reliable estimates of population parameters.
2. Unbiased estimators are especially important in scientific research where systematic errors must be avoided.
3. In financial data analysis, unbiased estimators are used to evaluate risks and expected returns.
4. In industrial quality control, they are employed to estimate production characteristics without systematic underestimation or overestimation.

Solution

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Step 1: Definition of an unbiased estimator

An estimator $\hat{\mu}$ is considered unbiased if its expected value equals the parameter it aims to estimate:

$$\mathbb{E}[\hat{\mu}] = \mu$$

This is the fundamental condition we need to verify.

Step 2: Expected value of the proposed estimator

The proposed estimator is a linear combination of random variables:

$$\hat{\mu} = a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n}x_{n}$$

Using the linearity property of the expected value, we have:

$$\mathbb{E}[\hat{\mu}] = \mathbb{E}[a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n}x_{n}]$$ $$\mathbb{E}[\hat{\mu}] = a_{1}\mathbb{E}[x_{1}] + a_{2}\mathbb{E}[x_{2}] + \cdots + a_{n}\mathbb{E}[x_{n}]$$

Step 3: Using information about the random sample

Since $x_{1}, x_{2}, \ldots, x_{n}$ are random samples from a population with mean $\mu$, they all have the same expected value:

$$\mathbb{E}[x_{i}] = \mu \quad \text{for all} \quad i = 1, 2, \ldots, n$$

Substituting these values into the previous expression:

$$\mathbb{E}[\hat{\mu}] = a_{1}\mu + a_{2}\mu + \cdots + a_{n}\mu$$

Factoring out $\mu$:

$$\mathbb{E}[\hat{\mu}] = \mu(a_{1} + a_{2} + \cdots + a_{n})$$

Step 4: Determining the condition for unbiasedness

For $\hat{\mu}$ to be unbiased, we need:

$$\mathbb{E}[\hat{\mu}] = \mu$$

Substituting what we obtained in Step 3:

$$\mu(a_{1} + a_{2} + \cdots + a_{n}) = \mu$$

Dividing both sides by $\mu$ (assuming $\mu \neq 0$):

$$a_{1} + a_{2} + \cdots + a_{n} = 1$$

Step 5: Verification with the sample mean

The sample mean is defined as:

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_{i} = \frac{1}{n}x_{1} + \frac{1}{n}x_{2} + \cdots + \frac{1}{n}x_{n}$$

In this case, each coefficient $a_i = \frac{1}{n}$. Verifying the condition:

$$\sum_{i=1}^{n}a_i = \sum_{i=1}^{n}\frac{1}{n} = \frac{n}{n} = 1$$

Thus, the sample mean is an unbiased estimator of the population mean.

Conclusion

The necessary and sufficient condition for the estimator $\hat{\mu}$ to be unbiased is:

$$\boxed{\sum_{i=1}^{n}a_i = 1}$$

This condition ensures that the estimator does not systematically underestimate or overestimate the population parameter $\mu$.