This is some code (it is less than an R package) for estimating a distribution of a latent variable using Kotlarski’s Lemma; in particular, implementing the results in Li and Vuong (1998).

The setup is one where one is interested in the distribution of some latent variable X. One observes two measured with error versions of X; call these X_1 and X_2, and they are given by

X_1 = X + e_1
X_2 = X + e_2

and under the condition that X, e_1, and e_2 are mutually independent. The provided code (two files: kotlarski.R and tuning_parameters.R) will estimate the pdf of X in this case.

To get things working, you to follow the following steps

• Execute the code in tuning_parameters.R – you should set the values of the tuning parameters to be whatever you want them to be (some preliminary values are set there that work in the example below, but are not guaranteed to work across applications)
• Using your data that includes exactly two measurements of the latent variable, save these in variables called X1 and X2
• Once you have completed these two steps, just run cf2dens(kotlarski, tgrid, xgrid) – kotlarski is the name of the function that does most of the work here, tgrid and xgrid are set in tuning_parameters.R

#-----------------------------------------------------------------------------
# Some simulations to check that everything works
#-----------------------------------------------------------------------------

# load the code
source("/path/to/code/kotlarski.R")
source("/path/to/code/tuning_parameters.R")

n <- 5000
X <- rnorm(n)
e1 <- rnorm(n)
e2 <- rnorm(n)
X1 <- X + e1
X2 <- X + e2


# run the code to produce the pdf of x
dd <- cf2dens(kotlarski, tgrid, xgrid)

# plot the estimated pdf
plot(xgrid, dd)

# compare to true pdf
curve(dnorm(x), add=TRUE)