EL0: Uni-variate empirical likelihood via direct lambda search
In smoothemplik: Smoothed Empirical Likelihood
EL0 R Documentation
Uni-variate empirical likelihood via direct lambda search
Description
Empirical likelihood with counts to solve one-dimensional problems efficiently with Brent's root search algorithm.
Conducts an empirical likelihood ratio test of the hypothesis that the mean of z is mu.
The names of the elements in the returned list are consistent with the original R code in \insertCiteowen2017weightedsmoothemplik.
Usage
EL0(
z,
mu = NULL,
ct = NULL,
shift = NULL,
return.weights = FALSE,
SEL = FALSE,
weight.tolerance = NULL,
boundary.tolerance = 1e-09,
trunc.to = 0,
chull.fail = c("taylor", "wald", "adjusted", "balanced", "none"),
uniroot.control = list(),
verbose = FALSE
)
Arguments
z
Numeric data vector.
mu
Hypothesized mean of z in the moment condition.
ct
Numeric count variable with non-negative values that indicates the multiplicity of observations.
Can be fractional. Very small counts below the threshold weight.tolerance are zeroed.
shift
The value to add in the denominator (useful in case there are extra Lagrange multipliers): 1 + lambda'Z + shift.
return.weights
Logical: if TRUE, individual EL weights are computed and returned.
Setting this to FALSE gives huge memory savings in large data sets, especially when smoothing is used.
SEL
If FALSE, then the boundaries for the lambda search are based on the total sum of counts, like in vanilla empirical likelihood,
due to formula (2.9) in \insertCiteowen2001empiricalsmoothemplik, otherwise according to Cosma et al. (2019, p. 170, the topmost formula).
weight.tolerance
Weight tolerance for counts to improve numerical stability
(similar to the ones in Art B. Owen's 2017 code, but adapting to the sample size).
boundary.tolerance
Relative tolerance for determining when the lambda is not an interior
solution because it is too close to the boundary. Corresponds to a fraction of the
interval range length.
trunc.to
Counts under weight.tolerance will be set to this value.
In most cases, setting this to 0 or weight.tolerance is a viable solution of the zero-denominator problem.
chull.fail
A character: what to do if the convex hull of z does not contain mu
(spanning condition does not hold). "taylor" creates a Taylor approximation
of the log-ELR function near the ends of the sample. "wald" smoothly transitions
between the log-ELR function into -0.5 * the Wald statistic for the weighted mean of z.
"adjusted" invokes the method of \insertCitechen2008adjustedsmoothemplik,
where an extra observation is added to ensure that the convex hull contains the mean, and
"balanced" calls the method of \insertCiteemerson2009calibrationsmoothemplik
and \insertCiteliu2010adjustedsmoothemplik with two extra points.
uniroot.control
A list passed to the brentZero.
verbose
Logical: if TRUE, prints warnings.
Details
This function provides the core functionality for univariate empirical likelihood.
The technical details is given in \insertCitecosma2019inferencesmoothemplik,
although the algorithm used in that paper is slower than the one provided by this function.
Since we know that the EL probabilities belong to (0, 1), the interval (bracket) for \lambda search
can be determined in the spirit of formula (2.9) from \insertCiteowen2001empiricalsmoothemplik. Let
z^*_i := z_i - \mu be the recentred observations.
p_i = c_i/N \cdot (1 + \lambda z^*_i + s)^{-1}
The probabilities are bounded from above: p_i<1 for all i, therefore,
c_i/N \cdot (1 + \lambda z^*_i + s)^{-1} < 1
c_i/N - 1 - s < \lambda z^*_i
Two cases: either z^*_i<0, or z^*_i>0
(cases with z^*_i=0 are trivially excluded because they do not affect the EL). Then,
(c_i/N - 1 - s)/z^*_i > \lambda,\ \forall i: z^*_i<0
(c_i/N - 1 - s)/z^*_i < \lambda,\ \forall i: z^*_i>0
which defines the search bracket:
\lambda_{\min} := \max_{i: z^*_i>0} (c_i/N - 1 - s)/z^*_i
\lambda_{\max} := \min_{i: z^*_i<0} (c_i/N - 1 - s)/z^*_i
\lambda_{\min} < \lambda < \lambda_{\max}
(This derivation contains s, which is the extra shift that extends the
function to allow mixed conditional and unconditional estimation;
Owen's textbook formula corresponds to s = 0.)
The actual tolerance of the lambda search in brentZero is
2 |\lambda_{\max}| \epsilon_m + \mathrm{tol}/2,
where tol can be set in uniroot.control and
\epsilon_m is .Machine$double.eps.
The sum of log-weights is maximised without Taylor expansion, forcing mu to be inside
the convex hull of z. If a violation is happening, consider using the chull.fail argument
or switching to Euclidean likelihood via [EuL()].
Value
A list with the following elements:
- logelr
Logarithm of the empirical likelihood ratio.
- lam
The Lagrange multiplier.
- wts
Observation weights/probabilities (of the same length as z).
- converged
TRUE if the algorithm converged, FALSE otherwise (usually means that mu is not within the range of z, i.e. the one-dimensional convex hull of z).
- iter
The number of iterations used (from brentZero).
- bracket
The admissible interval for lambda (that is, yielding weights between 0 and 1).
- estim.prec
The approximate estimated precision of lambda (from brentZero).
- f.root
The value of the derivative of the objective function w.r.t. lambda at the root (from brentZero). Values > sqrt(.Machine$double.eps) indicate convergence problems.
- exitcode
An integer indicating the reason of termination.
- message
Character string describing the optimisation termination status.
References
\insertAllCited
See Also
[EL()]
Examples
# Figure 2.4 from Owen (2001) -- with a slightly different data point
earth <- c(
5.5, 5.61, 4.88, 5.07, 5.26, 5.55, 5.36, 5.29, 5.58, 5.65, 5.57, 5.53, 5.62, 5.29,
5.44, 5.34, 5.79, 5.1, 5.27, 5.39, 5.42, 5.47, 5.63, 5.34, 5.46, 5.3, 5.75, 5.68, 5.85
)
set.seed(1)
system.time(r1 <- replicate(40, EL(sample(earth, replace = TRUE), mu = 5.517)))
set.seed(1)
system.time(r2 <- replicate(40, EL0(sample(earth, replace = TRUE), mu = 5.517)))
plot(apply(r1, 2, "[[", "logelr"), apply(r1, 2, "[[", "logelr") - apply(r2, 2, "[[", "logelr"),
bty = "n", xlab = "log(ELR) computed via dampened Newthon method",
main = "Discrepancy between EL and EL0", ylab = "")
abline(h = 0, lty = 2)
# Handling the convex hull violation differently
EL0(1:9, chull.fail = "none")
EL0(1:9, chull.fail = "taylor")
EL0(1:9, chull.fail = "wald")
# Interpolation to well-defined branches outside the convex hull
mu.seq <- seq(-1, 7, 0.1)
wEL1 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "none")$logelr)
wEL2 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "taylor")$logelr)
wEL3 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "wald")$logelr)
plot(mu.seq, wEL1)
lines(mu.seq, wEL2, col = 2)
lines(mu.seq, wEL3, col = 4)
# Warning: depending on the compiler, the discrepancy between EL and EL0
# can be one million (1) times larger than the machine epsilon despite both of them
# being written in pure R
# The results from Apple clang-1400.0.29.202 and Fortran GCC 12.2.0 are different from
# those obtained under Ubuntu 22.04.4 + GCC 11.4.0-1ubuntu1~22.04,
# Arch Linux 6.6.21 + GCC 14.1.1, and Windows Server 2022 + GCC 13.2.0
out1 <- EL(earth, mu = 5.517)[1:4]
out2 <- EL0(earth, mu = 5.517, return.weights = TRUE)[1:4]
print(c(out1$lam, out2$lam), 16)
# Value of lambda EL EL0
# aarch64-apple-darwin20 -1.5631313955?????? -1.5631313957?????
# Windows, Ubuntu, Arch -1.563131395492627 -1.563131395492627
smoothemplik documentation built on Aug. 22, 2025, 1:11 a.m.
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| EL0 | R Documentation |
Uni-variate empirical likelihood via direct lambda search
Description
Empirical likelihood with counts to solve one-dimensional problems efficiently with Brent's root search algorithm.
Conducts an empirical likelihood ratio test of the hypothesis that the mean of z is mu.
The names of the elements in the returned list are consistent with the original R code in \insertCiteowen2017weightedsmoothemplik.
Usage
EL0(
z,
mu = NULL,
ct = NULL,
shift = NULL,
return.weights = FALSE,
SEL = FALSE,
weight.tolerance = NULL,
boundary.tolerance = 1e-09,
trunc.to = 0,
chull.fail = c("taylor", "wald", "adjusted", "balanced", "none"),
uniroot.control = list(),
verbose = FALSE
)
Arguments
z |
Numeric data vector. |
mu |
Hypothesized mean of |
ct |
Numeric count variable with non-negative values that indicates the multiplicity of observations.
Can be fractional. Very small counts below the threshold |
shift |
The value to add in the denominator (useful in case there are extra Lagrange multipliers): 1 + lambda'Z + shift. |
return.weights |
Logical: if TRUE, individual EL weights are computed and returned. Setting this to FALSE gives huge memory savings in large data sets, especially when smoothing is used. |
SEL |
If |
weight.tolerance |
Weight tolerance for counts to improve numerical stability (similar to the ones in Art B. Owen's 2017 code, but adapting to the sample size). |
boundary.tolerance |
Relative tolerance for determining when the lambda is not an interior solution because it is too close to the boundary. Corresponds to a fraction of the interval range length. |
trunc.to |
Counts under |
chull.fail |
A character: what to do if the convex hull of |
uniroot.control |
A list passed to the |
verbose |
Logical: if |
Details
This function provides the core functionality for univariate empirical likelihood. The technical details is given in \insertCitecosma2019inferencesmoothemplik, although the algorithm used in that paper is slower than the one provided by this function.
Since we know that the EL probabilities belong to (0, 1), the interval (bracket) for \lambda search
can be determined in the spirit of formula (2.9) from \insertCiteowen2001empiricalsmoothemplik. Let
z^*_i := z_i - \mu be the recentred observations.
p_i = c_i/N \cdot (1 + \lambda z^*_i + s)^{-1}
The probabilities are bounded from above: p_i<1 for all i, therefore,
c_i/N \cdot (1 + \lambda z^*_i + s)^{-1} < 1
c_i/N - 1 - s < \lambda z^*_i
Two cases: either z^*_i<0, or z^*_i>0
(cases with z^*_i=0 are trivially excluded because they do not affect the EL). Then,
(c_i/N - 1 - s)/z^*_i > \lambda,\ \forall i: z^*_i<0
(c_i/N - 1 - s)/z^*_i < \lambda,\ \forall i: z^*_i>0
which defines the search bracket:
\lambda_{\min} := \max_{i: z^*_i>0} (c_i/N - 1 - s)/z^*_i
\lambda_{\max} := \min_{i: z^*_i<0} (c_i/N - 1 - s)/z^*_i
\lambda_{\min} < \lambda < \lambda_{\max}
(This derivation contains s, which is the extra shift that extends the
function to allow mixed conditional and unconditional estimation;
Owen's textbook formula corresponds to s = 0.)
The actual tolerance of the lambda search in brentZero is
2 |\lambda_{\max}| \epsilon_m + \mathrm{tol}/2,
where tol can be set in uniroot.control and
\epsilon_m is .Machine$double.eps.
The sum of log-weights is maximised without Taylor expansion, forcing mu to be inside
the convex hull of z. If a violation is happening, consider using the chull.fail argument
or switching to Euclidean likelihood via [EuL()].
Value
A list with the following elements:
- logelr
Logarithm of the empirical likelihood ratio.
- lam
The Lagrange multiplier.
- wts
Observation weights/probabilities (of the same length as
z).- converged
TRUEif the algorithm converged,FALSEotherwise (usually means thatmuis not within the range ofz, i.e. the one-dimensional convex hull ofz).- iter
The number of iterations used (from
brentZero).- bracket
The admissible interval for lambda (that is, yielding weights between 0 and 1).
- estim.prec
The approximate estimated precision of lambda (from
brentZero).- f.root
The value of the derivative of the objective function w.r.t. lambda at the root (from
brentZero). Values> sqrt(.Machine$double.eps)indicate convergence problems.- exitcode
An integer indicating the reason of termination.
- message
Character string describing the optimisation termination status.
References
\insertAllCitedSee Also
[EL()]
Examples
# Figure 2.4 from Owen (2001) -- with a slightly different data point
earth <- c(
5.5, 5.61, 4.88, 5.07, 5.26, 5.55, 5.36, 5.29, 5.58, 5.65, 5.57, 5.53, 5.62, 5.29,
5.44, 5.34, 5.79, 5.1, 5.27, 5.39, 5.42, 5.47, 5.63, 5.34, 5.46, 5.3, 5.75, 5.68, 5.85
)
set.seed(1)
system.time(r1 <- replicate(40, EL(sample(earth, replace = TRUE), mu = 5.517)))
set.seed(1)
system.time(r2 <- replicate(40, EL0(sample(earth, replace = TRUE), mu = 5.517)))
plot(apply(r1, 2, "[[", "logelr"), apply(r1, 2, "[[", "logelr") - apply(r2, 2, "[[", "logelr"),
bty = "n", xlab = "log(ELR) computed via dampened Newthon method",
main = "Discrepancy between EL and EL0", ylab = "")
abline(h = 0, lty = 2)
# Handling the convex hull violation differently
EL0(1:9, chull.fail = "none")
EL0(1:9, chull.fail = "taylor")
EL0(1:9, chull.fail = "wald")
# Interpolation to well-defined branches outside the convex hull
mu.seq <- seq(-1, 7, 0.1)
wEL1 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "none")$logelr)
wEL2 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "taylor")$logelr)
wEL3 <- -2*sapply(mu.seq, function(m) EL0(1:9, mu = m, chull.fail = "wald")$logelr)
plot(mu.seq, wEL1)
lines(mu.seq, wEL2, col = 2)
lines(mu.seq, wEL3, col = 4)
# Warning: depending on the compiler, the discrepancy between EL and EL0
# can be one million (1) times larger than the machine epsilon despite both of them
# being written in pure R
# The results from Apple clang-1400.0.29.202 and Fortran GCC 12.2.0 are different from
# those obtained under Ubuntu 22.04.4 + GCC 11.4.0-1ubuntu1~22.04,
# Arch Linux 6.6.21 + GCC 14.1.1, and Windows Server 2022 + GCC 13.2.0
out1 <- EL(earth, mu = 5.517)[1:4]
out2 <- EL0(earth, mu = 5.517, return.weights = TRUE)[1:4]
print(c(out1$lam, out2$lam), 16)
# Value of lambda EL EL0
# aarch64-apple-darwin20 -1.5631313955?????? -1.5631313957?????
# Windows, Ubuntu, Arch -1.563131395492627 -1.563131395492627
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