Calculate a generalized logarithmic mean / extended mean.
Value
generalized_logmean() and extended_mean() return a
function:
function(a, b, tol = .Machine$double.eps^0.5){...}logmean() returns a numeric vector, the same length as
max(length(a), length(b)), giving the component-wise logarithmic mean
of a and b.
Details
The function extended_mean() returns a function to compute the
component-wise extended mean of a and b of orders r and
s. See Bullen (2003, p. 393) for a definition. This is also called
the difference mean, Stolarsky mean, or extended mean-value mean.
The function generalized_logmean() returns a function to compute the
component-wise generalized logarithmic mean of a and b of
order r. See Bullen (2003, p. 385) for a definition. The generalized
logarithmic mean is a special case of the extended mean, corresponding to
extended_mean(r, 1)(), but is more commonly used for price indexes.
The function logmean() returns the ordinary component-wise
logarithmic mean of a and b, and corresponds to
generalized_logmean(1)().
Both a and b should be strictly positive. This is not
enforced, but the results may not make sense when the generalized
logarithmic mean / extended mean is not defined. The usual recycling rules
apply when a and b are not the same length.
By definition, the generalized logarithmic mean / extended mean of a
and b is a when a == b. The tol argument is used
to test equality by checking if abs(a - b) <= tol. The default value
is the same as all.equal(). In some cases it's useful to multiply
tol by a scale factor, such as max(abs(a), abs(b)). This often
doesn't matter when making price indexes, however, as a and b
are usually around 1.
Note
generalized_logmean() can be defined on the extended real line,
so that r = -Inf / Inf returns pmin()/pmax(), 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.
See also
transmute_weights() uses the extended mean to turn a generalized
mean of order \(r\) into a generalized mean of order \(s\).
Other means:
generalized_mean(),
lehmer_mean(),
nested_mean()
Examples
x <- 8:5
y <- 1:4
# The arithmetic and geometric means are special cases of the
# generalized logarithmic mean.
all.equal(generalized_logmean(2)(x, y), (x + y) / 2)
#> [1] TRUE
all.equal(generalized_logmean(-1)(x, y), sqrt(x * y))
#> [1] TRUE
# The harmonic mean cannot be expressed as a logarithmic mean, but can
# be expressed as an extended mean.
all.equal(extended_mean(-2, -1)(x, y), 2 / (1 / x + 1 / y))
#> [1] TRUE
# The quadratic mean is also a type of extended mean.
all.equal(extended_mean(2, 4)(x, y), sqrt(x^2 / 2 + y^2 / 2))
#> [1] TRUE
# As are heronian and centroidal means.
all.equal(
extended_mean(0.5, 1.5)(x, y),
(x + sqrt(x * y) + y) / 3
)
#> [1] TRUE
all.equal(
extended_mean(2, 3)(x, y),
2 / 3 * (x^2 + x * y + y^2) / (x + y)
)
#> [1] TRUE