In this post we shall give another proof of the famous AM-GM-HM inequality:
If are positive real numbers, then AM GM HM, precisely
I remember in high school the AM-GM inequality is proved using direct induction or calculus. Later on I knew proofs using backward induction, convexity, Jensen’s inequality etc (See the above Wikipedia link for details). The following proof, which I learnt in a statistics class, comes from Stefanski (1996). The only prerequisite is 100-200 level calculus.
We first make the proof mathematics-like. After that, we describe its meaning in statistics.
Proof. Fix . Consider a map given by
By standard differentiation techniques we learnt in calculus,
Using the fact we get the AM-GM inequality.
Next, we consider given by
The fact implies the GM-HM inequality. The proof is completed.
The function is indeed the joint probability density function (also called likelihood) of independent random variables Exponential() with density ( is the indicator function). Then for a fixed ,
is the Likelihood Ratio Test statistic of testing the null hypothesis against the alternative hypothesis are not all equal. Obviously . The test rejects the null hypothesis if and only if is too small. This is because, if the numerator of , which is the likelihood when is true, is small then is less likely to occur.
The function is the likelihood of independent random variables where Exponential(). Then given ,
is the Likelihood Ratio Test statistic of testing the null hypothesis against the alternative hypothesis are not all equal. The test rejects the null hypothesis if and only if is too small.