|WikiProject Statistics||(Rated Start-class, Low-importance)|
|WikiProject Mathematics||(Rated Start-class, Low-importance)|
As stated, the inequality is not true (this is easy to see for n=1). There must be some other condition. coco
- I noticed that you added the condition , which is indeed required. Is there another condition we're missing? --MarkSweep 21:17, 21 August 2005 (UTC)
Hoeffding's theorem 2
I just reverted an edit which (re)introduced a mistake in the presentation of the inequality. As stated here, the inequality involves the probability
Note that S is the sum of n independent random variables. This probability could also be written as
which is how it appears in Hoeffding's paper (Theorem 2, p. 16, using slightly different notation). In other words, Hoeffding's formulation is in terms of the mean of n independent RVs, whereas the formulation used here is in terms of their sum. A recent edit changed this to
Why do we need that X_i's have finite first and second moments? It is not stated in the Hoefdding paper and after all, I think it follows from that X_i lies in [a_i, b_i] i.e. bounded interval.
The article has no mistakes - the condition t>0 is not nessecary and the last comment is obvious. Nevertheless, by a Hoeffding type inequality is meant an inequality, which uses the transform f(t)=exp(h(t-x)) to derive bounds for tail probabilities for sums of independent r.v. and martingales. Therefore, the written inequality is only one of the inequalities Hoeffding introduced, but, regarding it from a statistical point of view, that is not the most important result as it doesn't control the variance of the variables. I would suggest to add the other inequality and explain what is meant by saying "Hoeffding inequality", as it is not one thing.
Special case of Bernstein's inequality
Can someone point out which Bernstein inequality Hoeffding's inequality is a special case of?--Steve Kroon 14:10, 22 May 2007 (UTC)
The "which inequality is a special case of the other" is inconsistent in wikipeida: the article on Hoeffding ineq. says it's more general than Bernstein ineq. While the article on Bernstein ineq. says "special cases of Bernstein ineq. is ... Hoeffding ineq". Someone, please, fix this? -- 184.108.40.206 (talk) 13:11, 14 June 2012 (UTC)
The bounds for variables X_i seem misleading
Older revisions of this article (pre 2009), as well as the original paper state the bounds as
- It does not make any difference (I mean the first and the third version above, the second one is marginally stronger). What is misleading about it?—Emil J. 13:52, 13 December 2010 (UTC)
- I think that the third version deals with centered versions of variable? 220.127.116.11 (talk) 13:43, 16 March 2015 (UTC) Sergey
Problem with bounds
To apply Hoeffding Lemma to the zero-mean variables ,we must have almost surely.
But, and are not equivalent.