lehmer_mean {gpindex} | R Documentation |
Lehmer mean
Description
Calculate a weighted Lehmer mean.
Usage
lehmer_mean(r)
contraharmonic_mean(x, w = NULL, na.rm = FALSE)
Arguments
r |
A finite number giving the order of the Lehmer mean. |
x |
A strictly positive numeric vector. |
w |
A strictly positive numeric vector of weights, the same length as
|
na.rm |
Should missing values in |
Details
The function lehmer_mean()
returns a function to compute the Lehmer
mean of order r
of x
with weights w
, which is
calculated as the arithmetic mean of x
with weights .
This is also called the counter-harmonic mean or generalized anti-harmonic
mean. See Bullen (2003, p. 245) for a definition, or
https://en.wikipedia.org/wiki/Lehmer_mean.
The Lehmer mean of order 2 is sometimes called the contraharmonic (or
anti-harmonic) mean. The function contraharmonic_mean()
simply calls
lehmer_mean(2)()
. Like the generalized mean, the contraharmonic mean
is the solution to an optimal prediction problem: choose to minimize
. The Lehmer mean of order -1 has a similar interpretation,
replacing
with
,
and together these bound the harmonic and arithmetic means.
The Lehmer mean is an alternative to the generalized mean that generalizes
the Pythagorean means. The function lehmer_mean(1)()
is identical to
arithmetic_mean()
, lehmer_mean(0)()
is identical to
harmonic_mean()
, and lehmer_mean(0.5)()
is identical to
geometric_mean()
with two values and no weights. See von der Lippe
(2015) for more details on the use of these means for making price indexes.
Value
lehmer_mean()
returns a function:
function(x, w = NULL, na.rm = FALSE){...}
This computes the Lehmer mean of order r
of x
with weights
w
.
contraharmonic_mean()
returns a numeric value for the Lehmer mean of
order 2.
Note
lehmer_mean()
can be defined on the extended real line, so that
r = -Inf / Inf
returns min()
/max()
, to agree with the
definition in, e.g., Bullen (2003). This is not implemented, and r
must be finite.
References
Bullen, P. S. (2003). Handbook of Means and Their Inequalities. Springer Science+Business Media.
Lehmer, D. H. (1971). On the Compounding of Certain Means. Journal of Mathematical Analysis and Applications, 36(1): 183-200.
von der Lippe, P. (2015). Generalized Statistical Means and New Price Index Formulas, Notes on some unexplored index formulas, their interpretations and generalizations. Munich Personal RePEc Archive paper no. 64952.
See Also
Other means:
extended_mean()
,
generalized_mean()
,
nested_mean()
Examples
x <- 2:3
w <- c(0.25, 0.75)
#---- The Pythagorean means are special cases of the Lehmer mean ----
all.equal(lehmer_mean(1)(x, w), arithmetic_mean(x, w))
all.equal(lehmer_mean(0)(x, w), harmonic_mean(x, w))
all.equal(lehmer_mean(0.5)(x), geometric_mean(x))
#---- Comparing Lehmer means and generalized means ----
# When r < 1, the generalized mean is larger than the corresponding
# Lehmer mean
lehmer_mean(-1)(x, w) < generalized_mean(-1)(x, w)
# The reverse is true when r > 1
lehmer_mean(3)(x, w) > generalized_mean(3)(x, w)
# This implies the contraharmonic mean is larger than the quadratic
# mean, and therefore the Pythagorean means
contraharmonic_mean(x, w) > arithmetic_mean(x, w)
contraharmonic_mean(x, w) > geometric_mean(x, w)
contraharmonic_mean(x, w) > harmonic_mean(x, w)
# ... and the logarithmic mean
contraharmonic_mean(2:3) > logmean(2, 3)
# The difference between the arithmetic mean and contraharmonic mean
# is proportional to the variance of x
weighted_var <- function(x, w) {
arithmetic_mean((x - arithmetic_mean(x, w))^2, w)
}
arithmetic_mean(x, w) + weighted_var(x, w) / arithmetic_mean(x, w)
contraharmonic_mean(x, w)
#---- Changing the order of the mean ----
# It is easy to modify the weights to turn a Lehmer mean of order r
# into a Lehmer mean of order s because the Lehmer mean can be
# expressed as an arithmetic mean
r <- 2
s <- -3
lehmer_mean(r)(x, w)
lehmer_mean(s)(x, w * x^(r - 1) / x^(s - 1))
# The weights can also be modified to turn a Lehmer mean of order r
# into a generalized mean of order s
lehmer_mean(r)(x, w)
generalized_mean(s)(x, transmute_weights(1, s)(x, w * x^(r - 1)))
# ... and vice versa
lehmer_mean(r)(x, transmute_weights(s, 1)(x, w) / x^(r - 1))
generalized_mean(s)(x, w)
#---- Percent-change contributions ----
# Percent-change contributions for a price index based on the Lehmer
# mean are easy to calculate
scale_weights(w * x^(r - 1)) * (x - 1)