gpindex (version 0.6.2)

generalized_mean: Generalized mean

Description

Calculate a weighted generalized mean.

Usage

generalized_mean(r)
arithmetic_mean(x, w = NULL, na.rm = FALSE)
geometric_mean(x, w = NULL, na.rm = FALSE)
harmonic_mean(x, w = NULL, na.rm = FALSE)

Value

generalized_mean() returns a function:

function(x, w = NULL, na.rm = FALSE){...}

This computes the generalized mean of order r of x with weights w.

arithmetic_mean(), geometric_mean(), and harmonic_mean() each return a numeric value for the generalized means of order 1, 0, and -1.

Arguments

r

A finite number giving the order of the generalized mean.

x

A strictly positive numeric vector.

w

A strictly positive numeric vector of weights, the same length as x. The default is to equally weight each element of x.

na.rm

Should missing values in x and w be removed? By default missing values in x or w return a missing value.

Details

The function generalized_mean() returns a function to compute the generalized mean of x with weights w and exponent r (i.e., \(\prod_{i = 1}^{n} x_{i}^{w_{i}}\) when \(r = 0\) and \(\left(\sum_{i = 1}^{n} w_{i} x_{i}^{r}\right)^{1 / r}\) otherwise). This is also called the power mean, Hölder mean, or \(l_p\) mean. See Bullen (2003, p. 175) for a definition, or https://en.wikipedia.org/wiki/Generalized_mean. The generalized mean is the solution to the optimal prediction problem: choose \(m\) to minimize \(\sum_{i = 1}^{n} w_{i} \left[\log(x_{i}) - \log(m) \right]^2\) when \(r = 0\), \(\sum_{i = 1}^{n} w_{i} \left[x_{i}^r - m^r \right]^2\) otherwise.

The functions arithmetic_mean(), geometric_mean(), and harmonic_mean() compute the arithmetic, geometric, and harmonic (or subcontrary) means, also known as the Pythagorean means. These are the most useful means for making price indexes, and correspond to setting r = 1, r = 0, and r = -1 in generalized_mean().

Both x and w should be strictly positive (and finite), especially for the purpose of making a price index. This is not enforced, but the results may not make sense if the generalized mean is not defined. There are two exceptions to this.

1. The convention in Hardy et al. (1952, p. 13) is used in cases where x has zeros: the generalized mean is 0 whenever w is strictly positive and r < 0. (The analogous convention holds whenever at least one element of x is Inf: the generalized mean is Inf whenever w is strictly positive and r > 0.)

2. Some authors let w be non-negative and sum to 1 (e.g., Sydsaeter et al., 2005, p. 47). If w has zeros, then the corresponding element of x has no impact on the mean whenever x is strictly positive. Unlike weighted.mean(), however, zeros in w are not strong zeros, so infinite values in x will propagate even if the corresponding elements of w are zero.

The weights are scaled to sum to 1 to satisfy the definition of a generalized mean. There are certain price indexes where the weights should not be scaled (e.g., the Vartia-I index); use sum() for these cases.

The underlying calculation returned by generalized_mean() is mostly identical to weighted.mean(), with one important exception: missing values in the weights are not treated differently than missing values in x. Setting na.rm = TRUE drops missing values in both x and w, not just x. This ensures that certain useful identities are satisfied with missing values in x. In most cases arithmetic_mean() is a drop-in replacement for weighted.mean().

References

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

Fisher, I. (1922). The Making of Index Numbers. Houghton Mifflin Company.

Hardy, G., Littlewood, J. E., and Polya, G. (1952). Inequalities (2nd edition). Cambridge University Press.

IMF, ILO, Eurostat, UNECE, OECD, and World Bank. (2020). Consumer Price Index Manual: Concepts and Methods. International Monetary Fund.

Lord, N. (2002). Does Smaller Spread Always Mean Larger Product? The Mathematical Gazette, 86(506): 273-274.

Sydsaeter, K., Strom, A., and Berck, P. (2005). Economists' Mathematical Manual (4th edition). Springer.

See Also

transmute_weights() transforms the weights to turn a generalized mean of order \(r\) into a generalized mean of order \(s\).

factor_weights() calculates the weights to factor a mean of products into a product of means.

price_indexes and quantity_index() for simple wrappers that use generalized_mean() to calculate common indexes.

back_period()/base_period() for a simple utility function to turn prices in a table into price relatives.

Other means: extended_mean(), lehmer_mean(), nested_mean()

Examples

x <- 1:3
w <- c(0.25, 0.25, 0.5)

#---- Common generalized means ----

# Arithmetic mean

arithmetic_mean(x, w) # same as weighted.mean(x, w)

# Geometric mean

geometric_mean(x, w) # same as prod(x^w)

# Harmonic mean

harmonic_mean(x, w) # same as 1 / weighted.mean(1 / x, w)

# Quadratic mean / root mean square

generalized_mean(2)(x, w)

# Cubic mean
# Notice that this is larger than the other means so far because
# the generalized mean is increasing in r

generalized_mean(3)(x, w)

#---- Comparing the Pythagorean means ----

# The dispersion between the arithmetic, geometric, and harmonic
# mean usually increases as the variance of 'x' increases

x <- c(1, 3, 5)
y <- c(2, 3, 4)

var(x) > var(y)

arithmetic_mean(x) - geometric_mean(x)
arithmetic_mean(y) - geometric_mean(y)

geometric_mean(x) - harmonic_mean(x)
geometric_mean(y) - harmonic_mean(y)

# But the dispersion between these means is only bounded by the
# variance (Bullen, 2003, p. 156)

arithmetic_mean(x) - geometric_mean(x) >= 2 / 3 * var(x) / (2 * max(x))
arithmetic_mean(x) - geometric_mean(x) <= 2 / 3 * var(x) / (2 * min(x))

# Example by Lord (2002) where the dispersion decreases as the variance
# increases, counter to the claims by Fisher (1922, p. 108) and the
# CPI manual (par. 1.14)

x <- (5 + c(sqrt(5), -sqrt(5), -3)) / 4
y <- (16 + c(7 * sqrt(2), -7 * sqrt(2), 0)) / 16

var(x) > var(y)

arithmetic_mean(x) - geometric_mean(x)
arithmetic_mean(y) - geometric_mean(y)

geometric_mean(x) - harmonic_mean(x)
geometric_mean(y) - harmonic_mean(y)

# The "bias" in the arithmetic and harmonic indexes is also smaller in
# this case, counter to the claim by Fisher (1922, p. 108)

arithmetic_mean(x) * arithmetic_mean(1 / x) - 1
arithmetic_mean(y) * arithmetic_mean(1 / y) - 1

harmonic_mean(x) * harmonic_mean(1 / x) - 1
harmonic_mean(y) * harmonic_mean(1 / y) - 1

#---- Missing values ----

w[2] <- NA

arithmetic_mean(x, w, na.rm = TRUE) # drop the second observation
weighted.mean(x, w, na.rm = TRUE) # still returns NA