Nested generalized mean

Calculate the (outer) generalized mean of two (inner) generalized means (i.e., crossing generalized means).

Usage

nested_mean(r1, r2, t = c(1, 1))

fisher_mean(x, w1 = NULL, w2 = NULL, na.rm = FALSE)

Arguments

r1: A finite number giving the order of the outer generalized mean.
r2: A pair of finite numbers giving the order of the inner generalized means.
t: A pair of strictly positive weights for the inner generalized means. The default is equal weights.
x: A strictly positive numeric vector.
w1, w2: 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, w1, and w2 be removed? By default missing values in x, w1, or w2 return a missing value.

Value

nested_mean() returns a function:

function(x, w1 = NULL, w2 = NULL, na.rm = FALSE){...}

This computes the generalized mean of order r1 of the generalized mean of order r2[1] of x with weights w1 and the generalized mean of order r2[2] of x with weights w2.

fisher_mean() returns a numeric value for the geometric mean of the arithmetic and harmonic means (i.e., r1 = 0 and r2 = c(1, -1)).

Note

There is some ambiguity about how to remove missing values in w1 or w2 when na.rm = TRUE. The approach here is to remove missing values when calculating each of the inner means individually, rather than removing all missing values prior to any calculations. This means that a different number of data points could be used to calculate the inner means. Use the balanced() operator to balance missing values across w1 and w2 prior to any calculations.

References

Diewert, W. E. (1976). Exact and superlative index numbers. Journal of Econometrics, 4(2): 114–145.

ILO, IMF, OECD, UNECE, and World Bank. (2004). Producer Price Index Manual: Theory and Practice. International Monetary Fund.

Lent, J. and Dorfman, A. H. (2009). Using a weighted average of base period price indexes to approximate a superlative index. Journal of Official Statistics, 25(1):139–149.

Examples

x <- 1:3
w1 <- 4:6
w2 <- 7:9

#---- Making superlative indexes ----

# A function to make the superlative quadratic mean price index by
# Diewert (1976) as a product of generalized means

quadratic_mean_index <- function(r) nested_mean(0, c(r / 2, -r / 2))

quadratic_mean_index(2)(x, w1, w2)
#> [1] 1.912366

# The arithmetic AG mean index by Lent and Dorfman (2009)

agmean_index <- function(tau) nested_mean(1, c(0, 1), c(tau, 1 - tau))

agmean_index(0.25)(x, w1, w1)
#> [1] 2.088801

#---- Walsh index ----

# The (arithmetic) Walsh index is the implicit price index when using a
# superlative quadratic mean quantity index of order 1

p1 <- price6[[2]]
p0 <- price6[[1]]
q1 <- quantity6[[2]]
q0 <- quantity6[[1]]

walsh <- quadratic_mean_index(1)

sum(p1 * q1) / sum(p0 * q0) / walsh(q1 / q0, p0 * q0, p1 * q1)
#> [1] 1.401718

sum(p1 * sqrt(q1 * q0)) / sum(p0 * sqrt(q1 * q0))
#> [1] 1.401718

# Counter to the PPI manual (par. 1.105), it is not a superlative
# quadratic mean price index of order 1

walsh(p1 / p0, p0 * q0, p1 * q1)
#> [1] 1.401534

# That requires using the average value share as weights

walsh_weights <- sqrt(scale_weights(p0 * q0) * scale_weights(p1 * q1))
walsh(p1 / p0, walsh_weights, walsh_weights)
#> [1] 1.401718

#---- Missing values ----

x[1] <- NA
w1[2] <- NA

fisher_mean(x, w1, w2, na.rm = TRUE)
#> [1] 2.699206

# Same as using obs 2 and 3 in an arithmetic mean, and obs 3 in a
# harmonic mean

geometric_mean(c(
  arithmetic_mean(x, w1, na.rm = TRUE),
  harmonic_mean(x, w2, na.rm = TRUE)
))
#> [1] 2.699206

# Use balanced() to use only obs 3 in both inner means

balanced(fisher_mean)(x, w1, w2, na.rm = TRUE)
#> [1] 3