Sufficient Statistic: T, S Pairing Explained

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Discussion Overview

The discussion revolves around the concept of sufficient statistics in statistics, specifically examining the pairing of a sufficient statistic T with another statistic S. Participants explore the implications of this pairing on the sufficiency of the combined statistic, addressing both theoretical and practical aspects.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that if T(x1...xn) is a sufficient statistic paired with S(x1...xn), then (T,S) remains a sufficient statistic, seeking a formal explanation for this idea.
  • Another participant expresses uncertainty and raises a concern that the method of pairing T and S (e.g., addition or multiplication) could affect the sufficiency, arguing that if the pairing alters the information in T, the combined statistic may not be sufficient.
  • A participant reiterates the idea that if T is sufficient for a parameter 'theta', then the vector (T,S) could also be sufficient, suggesting that ignoring S would not affect the sufficiency derived from T alone.
  • Further clarification is sought regarding whether a vector valued statistic (T,S) can be sufficient for a scalar valued parameter 'theta', indicating that this concept is not fully understood by all participants.

Areas of Agreement / Disagreement

Participants express differing views on the implications of pairing sufficient statistics, with some supporting the idea that (T,S) remains sufficient while others question the conditions under which this holds true. The discussion remains unresolved regarding the formal explanation and the effects of different pairing methods.

Contextual Notes

Participants highlight the importance of the method of pairing in determining sufficiency, indicating that assumptions about the nature of T and S and their relationship are crucial to the discussion.

EvLer
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If T(x1...xn) is a sufficient statistic and is paired up with another statistic S(x1...xn) then the (T,S) is still a sufficient statistic i think, correct?
But how would I "formally" explain that?
thanks.
 
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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.
 
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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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EnumaElish said:
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 scalar then it can be sufficient for a parameter 'theta' which is also scalar valued. Then you mean that, a vector valued statistic (T,S) can be sufficient for a scalar valued parameter 'theta'?... The idea is unknown to me.
 
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