Generalized Binomial Moment Transformations Across Arbitrary Origins • convmoment

convmoment implements the generalized binomial framework for transforming statistical moments of any order across arbitrary origins. One unified rule replaces separate formulas for raw-to-raw, raw-to-central, and central-to-raw conversions.

The method uses a symbolic binomial operator:

\[\mu^\prime_r(k) \equiv (a + b)^r \quad \text{with} \quad a^j \mapsto \mu^\prime_j(a), \quad b = a - k\]

This makes transformations exact (machine precision), extensible to any order, and data-free — convert pre-computed moments in O(r²) instead of O(n·r).

Installation

You can install the development version of convmoment from GitHub with:

# install.packages("pak")
pak::pak("mahmudstat/convmoment")

Key Functions

Function	Description
`conv_moment(x, a, k, r)`	Convert single moment of order `r` from origin `a` to `k`
`conv_moment_all(x, a, k)`	Convert vector of raw moments (orders 1..K) from origin `a` to `k`
`raw2central(raw_moments, origin)`	Raw moments → central moments (about mean)
`central2raw(central_moments, mean_val)`	Central moments → raw moments about 0
`get_moments(x, a, r, na_rm, decimal)`	Compute moments directly from sample data

Quick Examples

Raw-to-Raw

library(convmoment)
x <- c(-1, 7, 39)  # moments about a = 2
conv_moment_all(x, a = 2, k = 5)
#> [1]  -4  22 -78

Single Moment (conv_moment)

x <- c(-1, 7, 39)  # raw moments about a = 2
conv_moment(x, a = 2, k = 5, r = 2)  # 2nd moment about k = 5
#> [1] 22

Raw-to-Central (raw2central)

x <- c(-1, 7, 39)  # raw moments about a = 2
mean_x <- -1 + 2  # mean = raw_moment_1 + origin = 1
central <- raw2central(x, origin = 2)
central
#> [1]  0 22 -78

Central-to-Raw (central2raw)

central <- c(0, 22, -78)  # central moments
raw_back <- central2raw(central, mean_val = 1)
raw_back
#> [1] -1  7 39

Raw-to-Central (using conv_moment_all)

x <- c(-1, 7, 39)  # raw moments about a = 2
mean_x <- -1 + 2  # = 1
central <- conv_moment_all(x, a = 2, k = mean_x)
central
#> [1]  0 22 -78
# Same as raw2central(x, origin = 2)

Central-to-Raw (using conv_moment_all)

central <- c(0, 22, -78)  # central moments
raw_back <- conv_moment_all(central, a = 1, k = 2)
raw_back
#> [1] -1  7 39
# Same as central2raw(central, mean_val = 1)

Round-Trip (exact to machine precision)

x <- c(-1, 7, 39)  # raw moments about a = 2
mean_x <- -1 + 2  # = 1
central <- conv_moment_all(x, a = 2, k = mean_x)
raw_back <- conv_moment_all(central, a = mean_x, k = 2)
all.equal(x, raw_back)  # TRUE

From Data

set.seed(123)
y <- rnorm(100, 5, 2)
get_moments(y, a = 2, r = 4, decimal = 4)
#>   Order Moment
#> 1     1 3.1808
#> 2     2 13.4172
#> 3     3 64.0309
#> 4     4 338.1910

convmoment