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Empirical likelihood for general estimating functions

Source: R/el_eval.R
el_eval.Rd

Computes empirical likelihood with general estimating functions.

Usage

el_eval(g, weights = NULL, control = el_control())

Arguments

g

A numeric matrix, or an object that can be coerced to a numeric matrix. Each row corresponds to an observation of an estimating function. The number of rows must be greater than the number of columns.

weights

An optional numeric vector of weights to be used in the fitting process. The length of the vector must be the same as the number of rows in g. Defaults to NULL, corresponding to identical weights. If non-NULL, weighted empirical likelihood is computed.

control

An object of class ControlEL constructed by el_control().

Value

A list of the following optimization results:

  • optim A list with the following optimization results:

    • lambda A numeric vector of the Lagrange multipliers of the dual problem.

    • iterations A single integer for the number of iterations performed.

    • convergence A single logical for the convergence status.

  • logp A numeric vector of the log probabilities of the empirical likelihood.

  • logl A single numeric of the empirical log-likelihood.

  • loglr A single numeric of the empirical log-likelihood ratio.

  • statistic A single numeric of minus twice the empirical log-likelihood ratio with an asymptotic chi-square distribution.

  • df A single integer for the degrees of freedom of the statistic.

  • pval A single numeric for the \(p\)-value of the statistic.

  • nobs A single integer for the number of observations.

  • npar A single integer for the number of parameters.

  • weights A numeric vector of the re-scaled weights used for the model fitting.

Details

Let \(X_i\) be independent and identically distributed \(p\)-dimensional random variable from an unknown distribution \(P\) for \(i = 1, \dots, n\). We assume that \(P\) has a positive definite covariance matrix. For a parameter of interest \(\theta(F) \in {\rm{I\!R}}^p\), consider a \(p\)-dimensional smooth estimating function \(g(X_i, \theta)\) with a moment condition $$\textrm{E}[g(X_i, \theta)] = 0.$$ We assume that there exists an unique \(\theta_0\) that solves the above equation. Given a value of \(\theta\), the (profile) empirical likelihood ratio is defined by $$R(\theta) = \max_{p_i}\left\{\prod_{i = 1}^n np_i : \sum_{i = 1}^n p_i g(X_i, \theta) = 0, p_i \geq 0, \sum_{i = 1}^n p_i = 1 \right\}.$$ el_mean() computes the empirical log-likelihood ratio statistic \(-2\log R(\theta)\) with the \(n\) by \(p\) numeric matrix g, whose \(i\)th row is \(g(X_i, \theta)\). Since the estimating function can be arbitrary, el_eval() does not return an object of class EL, and the associated generics and methods are not applicable.

References

Qin J, Lawless J (1994). “Empirical Likelihood and General Estimating Equations.” The Annals of Statistics, 22(1), 300–325. doi:10.1214/aos/1176325370 .

See also

EL, el_control()

Examples

set.seed(123526)
mu <- 0
sigma <- 1
x <- rnorm(100)
g <- matrix(c(x - mu, (x - mu)^2 - sigma^2), ncol = 2)
el_eval(g, weights = rep(c(1, 2), each = 50))
#> $optim
#> $optim$lambda
#> [1]  0.07496852 -0.13181246
#> 
#> $optim$iterations
#> [1] 5
#> 
#> $optim$convergence
#> [1] TRUE
#> 
#> 
#> $logp
#>   [1] -4.769471 -4.990644 -4.870218 -5.091557 -5.100253 -4.930272 -5.062251
#>   [8] -4.776230 -4.686532 -5.140843 -5.128973 -5.142095 -5.113834 -4.725774
#>  [15] -5.036367 -5.067660 -5.067015 -5.119565 -5.093684 -5.098870 -4.875005
#>  [22] -5.078325 -5.023702 -5.131392 -5.090640 -5.123581 -5.084509 -5.124924
#>  [29] -5.126164 -5.116142 -4.858517 -4.609268 -4.397857 -5.128368 -5.085189
#>  [36] -5.028656 -4.858664 -5.096852 -5.133205 -4.281854 -5.140959 -4.618976
#>  [43] -5.024714 -5.101541 -4.946874 -5.143709 -5.063363 -5.131133 -5.118396
#>  [50] -5.114350 -4.450663 -4.389785 -3.925409 -4.416857 -4.447601 -4.371702
#>  [57] -4.442178 -4.450460 -4.438083 -4.379111 -4.449239 -4.105659 -4.330148
#>  [64] -4.447834 -4.438277 -4.235448 -4.383028 -4.312055 -4.180299 -4.417466
#>  [71] -4.432212 -4.396910 -4.081117 -4.330231 -4.394711 -4.127873 -4.432886
#>  [78] -4.433487 -4.258212 -4.363482 -4.236043 -4.426048 -4.434566 -4.422859
#>  [85] -4.415198 -4.302786 -3.988760 -4.300249 -4.429157 -4.320378 -4.330964
#>  [92] -4.447207 -4.435400 -4.448065 -4.450586 -4.450673 -4.437071 -4.429319
#>  [99] -4.423978 -3.975990
#> 
#> $logl
#> [1] -456.2696
#> 
#> $loglr
#> [1] -1.415871
#> 
#> $statistic
#> [1] 2.831742
#> 
#> $df
#> [1] 2
#> 
#> $pval
#> [1] 0.2427141
#> 
#> $nobs
#> [1] 100
#> 
#> $npar
#> [1] 2
#> 
#> $weights
#>   [1] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>   [8] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [15] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [22] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [29] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [36] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [43] 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667 0.6666667
#>  [50] 0.6666667 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [57] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [64] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [71] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [78] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [85] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [92] 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333 1.3333333
#>  [99] 1.3333333 1.3333333
#>