Empirical likelihood test

Tests a linear hypothesis with various calibration options.

# S4 method for EL
elt(
  object,
  rhs = NULL,
  lhs = NULL,
  alpha = 0.05,
  calibrate = "chisq",
  control = NULL
)

Arguments

object: An object that inherits from EL.
rhs: A numeric vector or a column matrix for the right-hand side of hypothesis, with as many entries as the rows in lhs. Defaults to NULL. See ‘Details’.
lhs: A numeric matrix or a vector (treated as a row matrix) for the left-hand side of a hypothesis. Each row gives a linear combination of the parameters in object. The number of columns must be equal to the number of parameters. Or a character vector with a symbolic description of the hypothesis is allowed. Defaults to NULL. See ‘Details’.
alpha: A single numeric for the significance level. Defaults to 0.05.
calibrate: A single character representing the calibration method. It is case-insensitive and must be one of "ael", "boot", "chisq", or "f". Defaults to "chisq". See ‘Details’.
control: An object of class ControlEL constructed by el_control(). Defaults to NULL and inherits the control slot in object.

Value

An object of class of ELT. If lhs is non-NULL, the optim slot corresponds to that of CEL. Otherwise, it corresponds to that of EL.

Details

elt() performs the constrained minimization of $l(\theta)$ described in CEL. rhs and lhs cannot be both NULL. For non-NULL lhs, it is required that lhs have full row rank $q \leq p$ and $p$ be equal to the number of parameters in the object.

Depending on the specification of rhs and lhs, we have the following three cases:

If both rhs and lhs are non-NULL, the constrained minimization is performed with the right-hand side $r$ and the left-hand side $L$ as $$\inf_{\theta: L\theta = r} l(\theta).$$
If rhs is NULL, $r$ is set to the zero vector as $\inf_{\theta: L\theta = 0} l(\theta)$.
If lhs is NULL, $L$ is set to the identity matrix and the problem reduces to evaluating at $r$ as $l(r)$.

calibrate specifies the calibration method used. Four methods are available: "ael" (adjusted empirical likelihood calibration), "boot" (bootstrap calibration), "chisq" (chi-square calibration), and "f" ($F$ calibration). When lhs is not NULL, only "chisq" is available. "f" is applicable only to the mean parameter. The an slot in control applies specifically to "ael", while the nthreads, seed, and B slots apply to "boot".

References

Adimari G, Guolo A (2010). ``A Note on the Asymptotic Behaviour of Empirical Likelihood Statistics.'' Statistical Methods & Applications, 19(4), 463--476. doi:10.1007/s10260-010-0137-9 .

Chen J, Variyath AM, Abraham B (2008). ``Adjusted Empirical Likelihood and Its Properties.'' Journal of Computational and Graphical Statistics, 17(2), 426--443.

Kim E, MacEachern SN, Peruggia M (2024). ``melt: Multiple Empirical Likelihood Tests in R.'' Journal of Statistical Software, 108(5), 1--33. doi:10.18637/jss.v108.i05 .

Qin J, Lawless J (1995). ``Estimating Equations, Empirical Likelihood and Constraints on Parameters.'' Canadian Journal of Statistics, 23(2), 145--159. doi:10.2307/3315441 .

Examples

## Adjusted empirical likelihood calibration
data("precip")
fit <- el_mean(precip, 32)
elt(fit, rhs = 100, calibrate = "ael")
#> 
#> 	Empirical Likelihood Test
#> 
#> Hypothesis:
#> par = 100
#> 
#> Significance level: 0.05, Calibration: Adjusted EL 
#> 
#> Statistic: 45.48, Critical value: 3.841
#> p-value: 1.539e-11 
#> EL evaluation: converged 
#> 

## Bootstrap calibration
elt(fit, rhs = 32, calibrate = "boot")
#> 
#> 	Empirical Likelihood Test
#> 
#> Hypothesis:
#> par = 32
#> 
#> Significance level: 0.05, Calibration: Bootstrap 
#> 
#> Statistic: 2.997, Critical value: 3.928
#> p-value: 0.084 
#> EL evaluation: converged 
#> 

## F calibration
elt(fit, rhs = 32, calibrate = "f")
#> 
#> 	Empirical Likelihood Test
#> 
#> Hypothesis:
#> par = 32
#> 
#> Significance level: 0.05, Calibration: F 
#> 
#> Statistic: 2.997, Critical value: 3.98
#> p-value: 0.0879 
#> EL evaluation: converged 
#> 

## Test of no treatment effect
data("clothianidin")
contrast <- matrix(c(
  1, -1, 0, 0,
  0, 1, -1, 0,
  0, 0, 1, -1
), byrow = TRUE, nrow = 3)
fit2 <- el_lm(clo ~ -1 + trt, clothianidin)
elt(fit2, lhs = contrast)
#> 
#> 	Empirical Likelihood Test
#> 
#> Hypothesis:
#> trtNaked - trtFungicide = 0
#> trtFungicide - trtLow = 0
#> trtLow - trtHigh = 0
#> 
#> Significance level: 0.05, Calibration: Chi-square 
#> 
#> Statistic: 26.6, Critical value: 7.815
#> p-value: 7.148e-06 
#> Constrained EL: converged 
#> 

## A symbolic description of the same hypothesis
elt(fit2, lhs = c(
  "trtNaked - trtFungicide",
  "trtFungicide - trtLow",
  "trtLow - trtHigh"
))
#> 
#> 	Empirical Likelihood Test
#> 
#> Hypothesis:
#> trtNaked - trtFungicide = 0
#> trtFungicide - trtLow = 0
#> trtLow - trtHigh = 0
#> 
#> Significance level: 0.05, Calibration: Chi-square 
#> 
#> Statistic: 26.6, Critical value: 7.815
#> p-value: 7.148e-06 
#> Constrained EL: converged 
#>