Statistics- unbiased estimator #3

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In summary, the conversation discusses two unbiased estimators for an unknown value, Theta, based on 5 observations from a uniform distribution. The estimators are Theta1=(6/5)*Ymax and Theta2=6*Ymin. To determine which estimator is better, the variances of both estimators are calculated and it is noted that because the distribution is symmetric, the variances of Ymin and Ymax are equal. However, the first estimator is divided by 5, making it 1/25 of the variance of the second estimator. This means that the first estimator has a smaller variance and is therefore a better choice. Additionally, the sample size of n=5 is mentioned, but it is not clear how it
  • #1
Roni1985
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Homework Statement



Suppose that n=5 observations are taken from the uniform pdf, fY=1/[tex]\Theta[/tex] 0<y<[tex]\Theta[/tex]
where [tex]\Theta[/tex] is unknown. Two unbiased estimators for [tex]\Theta[/tex] are

[tex]\Theta[/tex]1= (6/5)*Ymax
[tex]\Theta[/tex]2= (6)*Ymin

which estimator would be better to use? hint: What must be true of Var(Ymax) and Var(Ymin) given that fy is symmetric ? Does your answer as to which estimator is better make sense on intuitive grounds ? Explain.

Homework Equations



Var(Y)=E[Y2] - E[Y]2

Ymin=n*(1-FY(y))n-1fY(y)

Ymax=n*(FY(y))n-1fY(y)

The Attempt at a Solution



I'm trying to calculate the variances of both the estimators and see which one is smaller.
But, there is the extra information like, n=5 and that the distribution is symmetric, and I'm sure I need to use it.
Plus, I don't know how to answer it on intuitive grounds.
Would appreciate any help.
Thanks.

EDIT:

Now, I am still trying to solve it, and if I know that the distribution is symmetric, the variance of Ymin and Ymax are the same.
However, since the first estimator is divided by 5, we know that it's 1/25 of the variance of the second estimator.
Can I say this ? Is this the"on intuitive grounds" answer that they are asking ?
What about the sample size n=5 ? what do I do with it ?

Thanks
 
Last edited:
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  • #2
anybody ?

can't figure it out :\
 

1. What is an unbiased estimator in statistics?

An unbiased estimator is a statistical measurement that accurately estimates the true value of a population parameter without any systematic errors or biases. This means that on average, the estimator will produce a value that is equal to the true value of the parameter being estimated.

2. How is an unbiased estimator different from a biased estimator?

An unbiased estimator produces estimates that are, on average, equal to the true value of the population parameter being estimated. In contrast, a biased estimator consistently produces estimates that are either higher or lower than the true value, resulting in a systematic error.

3. What is the importance of using an unbiased estimator in statistical analysis?

Using an unbiased estimator is crucial in statistical analysis because it ensures that the estimated values accurately represent the true values of the population. This allows for more accurate and reliable conclusions to be drawn from the data.

4. How do you determine if an estimator is unbiased?

An estimator can be determined to be unbiased by comparing its expected value to the true value of the population parameter. If the expected value is equal to the true value, then the estimator is unbiased. Additionally, mathematical proofs and simulation studies can also be used to determine the bias of an estimator.

5. Are all unbiased estimators equally desirable?

No, not all unbiased estimators are equally desirable. Some may have lower variances, meaning they produce less variability in their estimates, while others may have higher efficiency, meaning they produce more precise estimates. The most desirable unbiased estimator is one that has both low variance and high efficiency.

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