gpindex (version 0.6.3)

extended_mean: Extended mean

Description

Calculate a generalized logarithmic mean / extended mean.

Usage

extended_mean(r, s)
generalized_logmean(r)
logmean(a, b, tol = .Machine$double.eps^0.5)

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.

Arguments

r, s

A finite number giving the order of the generalized logarithmic mean / extended mean.

a, b

A strictly positive numeric vector.

tol

The tolerance used to determine if a == 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.

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)
all.equal(generalized_logmean(-1)(x, y), sqrt(x * y))

# 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))

# 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))

# As are heronian and centroidal means.

all.equal(
  extended_mean(0.5, 1.5)(x, y),
  (x + sqrt(x * y) + y) / 3
)
all.equal(
  extended_mean(2, 3)(x, y),
  2 / 3 * (x^2 + x * y + y^2) / (x + y)
)