IID Unbiased Estimator

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In summary, the conversation discusses two different estimators of mu from a population with mean mu and variance sigma^2, Y^bar and W. Y^bar is the average of four IID RVs while W is a weighted average of four Y_i. The expected value and variance of Y^bar can be expressed in terms of mu and sigma^2, and the same can be shown for W. The preferred estimator of mu may depend on the specific values of the weights used in W.
  • #1
joemama69
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Homework Statement



Let Y_1, Y_2, Y_3, Y_4 be IID RV from a population with mean mu and variance sigma^2. Let Y^bar = .25(Y_1+Y_2+Y_3+Y_4) denote the average of these four RV's.

1)What are the expected value and variance of Y^bar in terms of mu and sigma^2

2)Consider a different estimator of mu... W = (1/8)Y_1 + (1/8)Y_2 + (1/4)Y_3 + (1/2)Y_4
This is an example of a weighted average of Y_i. Show that W is also an unbiased estimator of mu and find V(W)

3)Based on your answer, which estimator of mu do you prefer, Y^bar or W.

Homework Equations





The Attempt at a Solution



Welp I'm not really sure where to start. I know that Y^bar is a sample mean from 4 samples while mu is the population mean based on every sample, but I do not know how to represent one in terms of the other
 
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  • #2
joemama69 said:

Homework Statement



Let Y_1, Y_2, Y_3, Y_4 be IID RV from a population with mean mu and variance sigma^2. Let Y^bar = .25(Y_1+Y_2+Y_3+Y_4) denote the average of these four RV's.

1)What are the expected value and variance of Y^bar in terms of mu and sigma^2

2)Consider a different estimator of mu... W = (1/8)Y_1 + (1/8)Y_2 + (1/4)Y_3 + (1/2)Y_4
This is an example of a weighted average of Y_i. Show that W is also an unbiased estimator of mu and find V(W)

3)Based on your answer, which estimator of mu do you prefer, Y^bar or W.

Homework Equations





The Attempt at a Solution



Welp I'm not really sure where to start. I know that Y^bar is a sample mean from 4 samples while mu is the population mean based on every sample, but I do not know how to represent one in terms of the other

Doesn't your text have formulas for the expected value and variance of a sum (linear combination) of IID random variables?
 
  • #3
LCKurtz said:
Doesn't your text have formulas for the expected value and variance of a sum (linear combination) of IID random variables?

Sure does, but how to I express mean and variance IN TERMS of mu and sigma^2

Heres what I got outa the book...

so Y^bar = Sum(Y_i/n)

E(Y^bar) = E(Sum(Y_i/n)) = (1/4)Sum(E(Y_i)) = (1/4)Sum(mu)

V(Y^bar) = V(Sum(Y_i/n)) = (1/16)V(Sum(Y_i)) = (1/16)(Sum(Y_i) + 2 SumSumcov(Y_i, Y_j)) = (1/16)Sum(V(Y_i)) = (1/16)Sum(sigma^2)

So that's part one. Part two I am lossed with the weighted average parts. How would I go about that
 
  • #4
joemama69 said:
Sure does, but how to I express mean and variance IN TERMS of mu and sigma^2

Heres what I got outa the book...

so Y^bar = Sum(Y_i/n)

E(Y^bar) = E(Sum(Y_i/n)) = (1/4)Sum(E(Y_i)) = (1/4)Sum(mu)

Hopefully you are using proper notation on your worksheet. When you write (1/4)sum(mu) what you really mean is$$
\frac 1 4\sum_{i=1}^4\mu$$What does that equal when you write it out?
V(Y^bar) = V(Sum(Y_i/n)) = (1/16)V(Sum(Y_i)) = (1/16)(Sum(Y_i) + 2 SumSumcov(Y_i, Y_j)) = (1/16)Sum(V(Y_i)) = (1/16)Sum(sigma^2)

So that's part one. Part two I am lossed with the weighted average parts. How would I go about that

Same comment about the variance sums. And do you need the covariance for IID random variables?
 
  • #5
Ya sorry about the notation, I'm not up to speed with this syntax

So now my question is regarding part two... How would I find E(W) and V(W) when W = the sum of the weighted averages of Y (as apposed to part 1 where Y^bar was just the sum of the Y's divided by n = 4). Basically I'm confused what to do with the weights i.e. 1/8, 1/8, 1/4, 1/2?
 
  • #6
joemama69 said:
Ya sorry about the notation, I'm not up to speed with this syntax

So now my question is regarding part two... How would I find E(W) and V(W) when W = the sum of the weighted averages of Y (as apposed to part 1 where Y^bar was just the sum of the Y's divided by n = 4). Basically I'm confused what to do with the weights i.e. 1/8, 1/8, 1/4, 1/2?
You have ##W=\sum c_iY_i## with ##Y_i## IID. Look in your text for the mean and variance formulas for such sums.
 

1. What is an IID unbiased estimator?

An IID unbiased estimator is a statistical tool used to estimate the value of a population parameter, such as mean or variance, based on a sample from the population. It is called "IID" because it assumes that the sample observations are independent and identically distributed (IID) from the underlying population. An unbiased estimator is one that, on average, produces estimates that are equal to the true population parameter.

2. How is an IID unbiased estimator calculated?

An IID unbiased estimator is calculated by taking the average of the sample observations. For example, to estimate the population mean, the sample mean is calculated by summing all the observations and dividing by the sample size. This estimate is unbiased because, on average, it will be equal to the true population mean.

3. What is the importance of an IID unbiased estimator?

The importance of an IID unbiased estimator lies in its ability to provide accurate estimates of population parameters. By assuming that the sample is representative of the population and using unbiased estimation techniques, we can gain valuable insights into the population without having to collect data from every single individual.

4. Can an IID unbiased estimator have a large variance?

Yes, an IID unbiased estimator can have a large variance. The variance of an estimator measures how much the estimates vary from sample to sample. It is possible for an unbiased estimator to have a large variance, which means that the estimates may deviate significantly from the true population parameter in some samples. This can happen when the sample size is very small or when the underlying population is highly variable.

5. How can one assess the performance of an IID unbiased estimator?

The performance of an IID unbiased estimator can be assessed in several ways. One way is to compare the estimator's estimates to the known true values in simulated datasets. Another way is to calculate the mean squared error, which measures the average squared difference between the estimates and the true values. Additionally, confidence intervals can be constructed to assess the precision of the estimates. Overall, the goal is to have an estimator with low bias and low variance for accurate and precise estimates.

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