# Show that Xbar - Ybar is a consistent estimator

• birdec
In summary, the conversation discusses the consistency of the estimator X bar - Y bar for the difference between the means of two populations. It is stated that for an estimator to be consistent, its bias and variance must equal 0. However, it is pointed out that this is incorrect and the definition of consistency should be looked up. Additionally, it is mentioned that if the variance were zero, the random variable would have only one possible value.
birdec
Suppose that X sub 1, X sub 2,... X sub n and Y sub 1, Y sub 2,... Y sub n are independent random samples from populations with means mu sub x and mu sub y and variances sigma squared sub x and sigma squared sub y , respectively. Show that X bar - Y bar is a consistent estimator of mu sub x - mu sub y.

I know that the Bias and Variance must equal 0 so...

Bias (Xbar - Ybar) =
[E(Xbar) - mu sub x] - [E(Ybar) - mu sub y]
= 0 Variance (Xbar - Ybar)
[sigma squared sub x /n] - [sigma squared sub y /n]
= 0

I'm pretty sure this is incorrect.

birdec said:
I know that the Bias and Variance must equal 0

That isn't what it means for an estimator to be consistent. Why don't you look up the definition?

(If the variance were zero, the random variable would have only one possible value.)

## What is the meaning of "consistent estimator"?

A consistent estimator is a statistical method or formula that produces increasingly accurate results as the sample size increases. In other words, as more data is collected, the value of the estimator will approach the true population value.

## How can we show that Xbar - Ybar is a consistent estimator?

We can show that Xbar - Ybar is a consistent estimator by proving that as the sample size increases, the difference between the sample means (Xbar and Ybar) becomes smaller and closer to the true population mean. This can be demonstrated mathematically using the law of large numbers or by conducting simulations.

## What are the assumptions for using Xbar - Ybar as a consistent estimator?

The assumptions for using Xbar - Ybar as a consistent estimator are that the samples are drawn from independent populations, the populations have the same variance, and the samples are randomly selected from their respective populations.

## What are the advantages of using Xbar - Ybar as a consistent estimator?

The advantages of using Xbar - Ybar as a consistent estimator include its simplicity and ease of calculation, its ability to estimate the difference between two population means, and its reliability as the sample size increases.

## Are there any limitations to using Xbar - Ybar as a consistent estimator?

One limitation of using Xbar - Ybar as a consistent estimator is that it assumes the samples are drawn from normally distributed populations. Additionally, if the samples are biased or not representative of the population, the estimator may not produce accurate results.

• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
282
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
949
• Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
12
Views
3K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
9
Views
985
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K