- #1

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But how would I "formally" explain that?

thanks.

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- Thread starter EvLer
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- #1

- 458

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But how would I "formally" explain that?

thanks.

- #2

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I am not sure... just giving my thoughts..

Let, T(x) be sufficient for some parameter 'theta' say. Then T(x) contains all information about 'theta'. Now the question is how are you 'pairing up' S(x) with T(x) .... by addition or multiplication or through some other relation? If the process of 'pairing' alters the information in T(x) then the combined statistic may not be sufficient.

As for example, consider the normal distribution N(theta,1). Let x1,x2,...,xn be a random sample from it drawn independently of each other. Then T(x)= (x1+x2....+xn)/n is sufficient (minimal) for theta. Consider

S(x)= (x1-x2-x3...-xn)/n. Then if you pair up by addition, ie, form a new statistic T(x)+S(x), it is not sufficient for theta.

Let, T(x) be sufficient for some parameter 'theta' say. Then T(x) contains all information about 'theta'. Now the question is how are you 'pairing up' S(x) with T(x) .... by addition or multiplication or through some other relation? If the process of 'pairing' alters the information in T(x) then the combined statistic may not be sufficient.

As for example, consider the normal distribution N(theta,1). Let x1,x2,...,xn be a random sample from it drawn independently of each other. Then T(x)= (x1+x2....+xn)/n is sufficient (minimal) for theta. Consider

S(x)= (x1-x2-x3...-xn)/n. Then if you pair up by addition, ie, form a new statistic T(x)+S(x), it is not sufficient for theta.

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- #3

EnumaElish

Science Advisor

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I think what EvLer means is "if a scalar statistic T is sufficient then a vector (pair) of statistics (T,S) is also sufficient." The heuristic answer is "because even if one were to ignore S, they would still have sufficiency by virtue of having T."

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- #4

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I think what EvLer means is "if a scalar statistic T is sufficient then a vector (pair) of statistics (T,S) is also sufficient." The heuristic answer is "because even if one were to ignore S, they would still have sufficiency by virtue of having T."

If T is

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