piar (version 0.10.1)

emean: Extended mean

Description

Calculate the component-wise extended mean.

Usage

emean(x, y, order = c(0, 1), tol = .Machine$double.eps^0.5)

Value

A numeric vector, the same length as max(length(x), length(y)), giving the component-wise extended mean of x and y.

Arguments

x, y

[numeric > 0] A strictly positive numeric vector.

order

[numeric(2)] A pair of finite numbers giving the order of the extended mean. The default calculates the ordinary logarithmic mean. Setting either the first or second element to 1 gives the generalized logarithmic mean.

tol

[numeric > 0] The tolerance used to determine if x == y. The default value is the same as all.equal().

Details

The extended mean is also called the difference mean, Stolarsky mean, or extended mean-value mean; see Bullen (2003, p. 393) for details.

Both x and y should be strictly positive. This is not enforced, but the results may not make sense when the extended mean is not defined. The usual recycling rules apply when x and y are not the same length.

By definition, the extended mean of x and y is x when x == y. The tol argument is used to test equality by checking if abs(x - y) <= tol. In some cases it's useful to multiply tol by a scale factor, such as max(abs(x), abs(y)). This often doesn't matter when making price indexes, however, as x and y are usually around 1.

References

Bullen, P. S. (2003). Handbook of Means and Their Inequalities. Springer Science+Business Media.

See Also

Other math functions: gmean(), nested_gmean(), scale_weights(), transmute_weights(), transmute_weights2(), update_weights()

Examples

x <- 8:5
y <- 1:4

# The arithmetic and geometric means are special cases of the
# generalized logarithmic mean.
all.equal(emean(x, y, c(2, 1)), (x + y) / 2)
all.equal(emean(x, y, c(-1, 1)), sqrt(x * y))

# The harmonic mean cannot be expressed as a logarithmic mean, but can
# be expressed as an extended mean.
all.equal(emean(x, y, c(-2, -1)), 2 / (1 / x + 1 / y))

# The quadratic mean is also a type of extended mean.
all.equal(emean(x, y, c(2, 4)), sqrt(x^2 / 2 + y^2 / 2))

# As are heronian and centroidal means.
all.equal(
  emean(x, y, c(0.5, 1.5)),
  (x + sqrt(x * y) + y) / 3
)
all.equal(
  emean(x, y, c(2, 3)),
  2 / 3 * (x^2 + x * y + y^2) / (x + y)
)