Factor weights to turn the generalized mean of a product into the product of generalized means. Useful for price-updating the weights in a generalized-mean index.
Value
factor_weights() return a function:
function(x, w = NULL){...}update_weights() returns a numeric vector the same length as x.
Details
The function factor_weights(r) returns a function to compute weights
u(x, w) such that
generalized_mean(r)(x * y, w) ==
generalized_mean(r)(x, w) * generalized_mean(r)(y, u(x, w))This generalizes the result in section C.5 of Chapter 9 of the PPI Manual for chaining the Young index, and gives a way to chain generalized-mean price indexes over time.
Factoring weights with r = 1 sometimes gets called price-updating
weights; update_weights() simply calls factor_weights(1)().
Factoring weights return a value that is the same length as x,
so any missing values in x or the weights will return NA.
Unless all values are NA, however, the result for will still satisfy
the above identity when na.rm = TRUE.
References
ILO, IMF, OECD, UNECE, and World Bank. (2004). Producer Price Index Manual: Theory and Practice. International Monetary Fund.
See also
generalized_mean() for the generalized mean.
grouped() to make these functions operate on grouped data.
Other weights functions:
scale_weights(),
transmute_weights()
Examples
x <- 1:3
y <- 4:6
w <- 3:1
# Factor the harmonic mean by chaining the calculation
harmonic_mean(x * y, w)
#> [1] 5.966851
harmonic_mean(x, w) * harmonic_mean(y, factor_weights(-1)(x, w))
#> [1] 5.966851
# The common case of an arithmetic mean
arithmetic_mean(x * y, w)
#> [1] 8.333333
arithmetic_mean(x, w) * arithmetic_mean(y, update_weights(x, w))
#> [1] 8.333333
# In cases where x and y have the same order, Chebyshev's
# inequality implies that the chained calculation is too small
arithmetic_mean(x * y, w) >
arithmetic_mean(x, w) * arithmetic_mean(y, w)
#> [1] TRUE