Sto cercando un riferimento in cui è dimostrato che la media armonica
minimizza (in ) la somma degli errori relativi al quadrato
Sto cercando un riferimento in cui è dimostrato che la media armonica
minimizza (in ) la somma degli errori relativi al quadrato
Risposte:
Perché hai bisogno di un riferimento? Questo è un semplice problema di calcolo: perché il problema, come è stato formulato, abbia senso, dobbiamo supporre che tutto . Quindi definire la funzione f ( z ) = n ∑ i = 1 ( x i - z ) 2 Quindi calcola la derivata rispetto az: f′(z)=-2⋅n ∑ i=1(1-z
Come riferimento, forse https://en.wikipedia.org/wiki/Fr%C3%A9chet_mean o https://en.wikipedia.org/wiki/Harmonic_mean o riferimenti in essi.
You could point out that this is a weighted least squares regression with weights .
To make the connection with the references, revert to a standard notation in which you seek to find that minimizes
This is a model with a single constant regressor
I have renamed "" as "" (the "response") and the parameter to be estimated is instead of . The weights are . It is necessary that they all exceed . The solution is
QED.
The same analysis applies to any positive sets of weights, providing a generalization of the harmonic mean and a useful way to characterize it.
When, as in a controlled experiment, the are viewed as fixed (and not random), the machinery of weighted least squares provides confidence intervals and prediction intervals, etc. In other words, casting the problem into this setting automatically gives you a way to assess the precision of the harmonic mean.
Viewing the harmonic mean as the solution to a weighted problem provides insight into its nature and, especially, to its sensitivity to the data. It is now clear that the most important contributors are those with the smallest values of --and their importance has been quantified by the weights matrix .
Douglas C. Montgomery, Elizabeth A. Peck, and G. Geoffrey Vining, Introduction to Linear Regression Analysis. Fifth Edition. J. Wiley, 2012. Section 5.5.2.