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Sufficient Statistics Homework Statement and Solution
- Thread starter abeliando
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In summary, the conversation is about a problem with an attachment that involves using the I(.) function and frequentists' approach to estimate the mean of an exponentially distributed variable. One person suggests ignoring the I(.) function and only considering samples greater than the mean, while the other person explains that the problem involves estimating the mean of a variable that follows the exponential distribution and asks for clarification on the "usual" estimate.
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hikaru1221
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I think you're into too much math. The I(.) function, if I get its meaning correctly, is nothing but to indicate that it is impossible to sample [itex]x\leq\mu[/itex]. With frequentists' approach, this means a lot: you can safely ignore the bunch of I(.)'s and consider only those samples greater than [itex]\mu[/itex]. This should solve the problem 
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Ray Vickson
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abeliando said:
Ignoring the fancy notation, what you have for given, fixed μ is that Y = X-μ is exponentially distributed with unknown mean, and you are trying to estimate that mean. You are being asked to show some property of the "usual" estimate.
FAQ: Sufficient Statistics Homework Statement and Solution
What is a sufficient statistic?
A sufficient statistic is a summary statistic that contains all the information about a dataset that is needed to make inferences about a population parameter. In other words, it captures the essential information about the data without losing any important details.
Why are sufficient statistics important?
Sufficient statistics are important because they allow us to reduce the dimensionality of a dataset, making it easier to analyze and draw conclusions. They also help us avoid redundant information and can simplify complex models.
How do you determine if a statistic is sufficient?
To determine if a statistic is sufficient, we can use the Factorization Theorem, which states that a statistic is sufficient if the likelihood function can be written as a product of two functions - one that depends on the data only through the statistic and one that depends on the parameter of interest. Additionally, we can use the Neyman-Fisher factorization criterion to test if a statistic is sufficient.
Can a statistic be both necessary and sufficient?
Yes, a statistic can be both necessary and sufficient. A necessary statistic is one that is required to estimate a population parameter, while a sufficient statistic contains all the necessary information to make inferences about the parameter. Therefore, a statistic that is both necessary and sufficient means that it is the only statistic needed to estimate the parameter and contains all the necessary information.
Are there different types of sufficient statistics?
Yes, there are different types of sufficient statistics. These include minimal sufficient statistics, complete sufficient statistics, and ancillary statistics. Minimal sufficient statistics are the smallest possible subset of the data that is sufficient, while complete sufficient statistics are those that contain all the information about the parameter. Ancillary statistics are those that are not sufficient but can provide additional information about the data.