Unbiased Estimator for b: - Sum of ln(xi)/n

  • Thread starter DavidLiew
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In summary, an unbiased estimator for b is a statistical method used to estimate the true value of b in a dataset. It is calculated by taking the sum of the natural logarithm of each data point in the dataset and dividing it by the total number of data points (n). This is represented by the formula: (1/n) * SUM(ln(xi)). The sum of ln(xi)/n is used as an estimator for b because it is unbiased, consistent, and easy to calculate and interpret. The purpose of using an unbiased estimator for b is to obtain an accurate estimate of the true value of b without any bias from the data. However, the estimator for b can still be biased if the dataset is not representative or if there are systematic
  • #1
DavidLiew
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If I want shows that [tex]\hat{b}[/tex] is an unbiased estimator for the b
where [tex]\hat{b}[/tex] = - [tex]\sum[/tex] ln xi /n
f(x)= [tex]\frac{1}{b}[/tex] e(1-b/b)
 
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  • #2
DavidLiew said:
If I want shows that [tex]\hat{b}[/tex] is an unbiased estimator for the b
where [tex]\hat{b}[/tex] = - [tex]\sum[/tex] ln xi /n
f(x)= [tex]\frac{1}{b}[/tex] e(1-b/b)

Is that meant to be (1-b)/b? If its not then you'll get 1/b which is basically a uniform distribution,given that the domain is accurate.
 
  • #3
You need to show the expected value of b hat = b.
 

What is an unbiased estimator for b?

An unbiased estimator for b is a statistical method used to estimate the true value of b in a dataset. It is unbiased because, on average, it produces an estimate that is equal to the true value of b.

How is the estimator for b calculated?

The estimator for b is calculated by taking the sum of the natural logarithm of each data point in the dataset and dividing it by the total number of data points (n). This is represented by the formula: (1/n) * SUM(ln(xi)).

Why is the sum of ln(xi)/n used as an estimator for b?

The sum of ln(xi)/n is used as an estimator for b because it has desirable statistical properties, such as being unbiased and consistent. It is also relatively easy to calculate and interpret.

What is the purpose of using an unbiased estimator for b?

The purpose of using an unbiased estimator for b is to obtain an estimate of the true value of b that is not affected by any bias in the data. This ensures that the estimate is as accurate as possible.

Can the estimator for b be biased?

Yes, the estimator for b can be biased if the dataset used to calculate it is not representative of the population or if there are any systematic errors in the data. However, using the sum of ln(xi)/n as an estimator for b helps to minimize this bias.

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