| Title: | Maximized Monte Carlo |
|---|---|
| Description: | An implementation of the Monte Carlo techniques described in details by Dufour (2006) <doi:10.1016/j.jeconom.2005.06.007> and Dufour and Khalaf (2007) <doi:10.1002/9780470996249.ch24>. The two main features available are the Monte Carlo method with tie-breaker, mc(), for discrete statistics, and the Maximized Monte Carlo, mmc(), for statistics with nuisance parameters. |
| Authors: | Julien Neves [aut], Jean-Marie Dufour [aut, ths], Gabriel Rodriguez-Rondon [cre] |
| Maintainer: | Gabriel Rodriguez-Rondon <[email protected]> |
| License: | GPL(>= 3) |
| Version: | 0.1.2 |
| Built: | 2025-03-02 04:34:04 UTC |
| Source: | https://github.com/julienneves/maxmc |
Functions that implement the Maximized Monte Carlo technique based on Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.
The main functions of MaxMC are mmc and mc.
Julien Neves, [email protected] (Maintainer)
Jean-Marie Dufour, [email protected]
Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.
Dufour, J.-M. and Khalaf L. (2003), Monte Carlo Test Methods in Econometrics. in Badi H. Baltagi, ed., A Companion to Theoretical Econometrics, Blackwell Publishing Ltd, 494-519.
Find the Monte Carlo (MC) p-value by generating N replications of a statistic.
mc( y, statistic, ..., dgp = function(y) sample(y, replace = TRUE), N = 99, type = c("geq", "leq", "absolute", "two-tailed") )mc( y, statistic, ..., dgp = function(y) sample(y, replace = TRUE), N = 99, type = c("geq", "leq", "absolute", "two-tailed") )
y |
A vector or data frame. |
statistic |
A function or a character string that specifies
how the statistic is computed. The function needs to input the
|
... |
Other named arguments for statistic which are passed unchanged each time it is called |
dgp |
A function. The function inputs the first argument
|
N |
An atomic vector. Number of replications of the test statistic. |
type |
A character string. It specifies the type of test
the p-value function produces. The possible values are
|
The dgp function defined by the user is used to
generate new observations in order to compute the simulated
statistics.
Then pvalue is applied to the statistic and
its simulated values. pvalue computes the
p-value by ranking the statistic compared to its simulated
values. Ties in the ranking are broken according to a uniform
distribution.
We allow for four types of p-value: leq, geq,
absolute and two-tailed. For one-tailed test,
leq returns the proportion of simulated values smaller
than the statistic while geq returns the proportion of
simulated values greater than the statistic. For two-tailed
test with a symmetric statistic, one can use the
absolute value of the statistic and its simulated values to
retrieve a two-tailed test (i.e. type = absolute).
If the statistic is not symmetric, one can specify the p-value
type as two-tailed which is equivalent to twice the minimum
of leq and geq.
Ties in the ranking are broken according to a uniform distribution.
The returned value is an object of class mc
containing the following components:
S0 |
Observed value of |
pval |
Monte Carlo p-value of |
y |
Data specified in call. |
statistic |
|
dgp |
|
N |
Number of replications specified in call. |
type |
|
call |
Original call to |
seed |
Value of |
Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.
Dufour, J.-M. and Khalaf L. (2003), Monte Carlo Test Methods in Econometrics. in Badi H. Baltagi, ed., A Companion to Theoretical Econometrics, Blackwell Publishing Ltd, 494-519.
## Example 1 ## Kolmogorov-Smirnov Test using Monte Carlo # Set seed set.seed(999) # Generate sample data y <- rgamma(8, shape = 2, rate = 1) # Set data generating process function dgp <- function(y) rgamma(length(y), shape = 2, rate = 1) # Set the statistic function to the Kolomogorov-Smirnov test for gamma distribution statistic <- function(y){ out <- ks.test(y, "pgamma", shape = 2, rate = 1) return(out$statistic) } # Apply the Monte Carlo test with tie-breaker mc(y, statistic = statistic, dgp = dgp, N = 999, type = "two-tailed")## Example 1 ## Kolmogorov-Smirnov Test using Monte Carlo # Set seed set.seed(999) # Generate sample data y <- rgamma(8, shape = 2, rate = 1) # Set data generating process function dgp <- function(y) rgamma(length(y), shape = 2, rate = 1) # Set the statistic function to the Kolomogorov-Smirnov test for gamma distribution statistic <- function(y){ out <- ks.test(y, "pgamma", shape = 2, rate = 1) return(out$statistic) } # Apply the Monte Carlo test with tie-breaker mc(y, statistic = statistic, dgp = dgp, N = 999, type = "two-tailed")
The dgp function defined by the user is used to
generate new observations in order to compute the simulated
statistics.
mmc( y, statistic, ..., dgp = function(y, v) sample(y, replace = TRUE), est = NULL, lower, upper, N = 99, type = c("geq", "leq", "absolute", "two-tailed"), method = c("GenSA", "pso", "GA", "gridSearch"), control = list(), alpha = NULL, monitor = FALSE )mmc( y, statistic, ..., dgp = function(y, v) sample(y, replace = TRUE), est = NULL, lower, upper, N = 99, type = c("geq", "leq", "absolute", "two-tailed"), method = c("GenSA", "pso", "GA", "gridSearch"), control = list(), alpha = NULL, monitor = FALSE )
y |
A vector or data frame. |
statistic |
A function or a character string that specifies
how the statistic is computed. The function needs to input the
|
... |
Other named arguments for statistic which are passed unchanged each time it is called |
dgp |
A function. The function inputs the first argument
|
est |
A vector with length of v. It is the starting
point of the algorithm. If |
lower |
A vector with length of v. Lower bounds for nuisance parameters under the null. See Details. |
upper |
A vector with length of v. Upper bounds for nuisance parameters under the null. See Details. |
N |
An atomic vector. Number of replications of the test statistic. |
type |
A character string. It specifies the type of test
the p-value function produces. The possible values are
|
method |
A character string. Type of algorithm to be
used for global optimization. The four available methods
are simulated annealing ( |
control |
A list. Arguments to be used to control the
behavior of the algorithm chosen in |
alpha |
An atomic vector. If |
monitor |
A logical. If set to TRUE, then the p-values at every iteration and the cumulative maximum p-value are plotted on a graphical device. Default is FALSE. |
Then pvalue is applied to the statistic and
its simulated values.pvalue computes the
p-value by ranking the statistic compared to its simulated
values. Ties in the ranking are broken according to a
uniform distribution.
We allow for four types of p-value: leq, geq,
absolute and two-tailed. For one-tailed test,
leq returns the proportion of simulated values smaller
than the statistic while geq returns the proportion of
simulated values greater than the statistic. For two-tailed
test with a symmetric statistic, one can use the
absolute value of the statistic and its simulated values to
retrieve a two-tailed test (i.e. type = absolute).
If the statistic is not symmetric, one can specify the p-value
type as two-tailed which is equivalent to twice the minimum
of leq and geq.
Ties in the ranking are broken according to a uniform distribution.
Usually, to ensure that the MMC procedure is exact, lower and
upper must be set such that any theoretically possible
values for the nuisance parameters under the null are covered. This
can be computationally expansive.
Alternatively, the consistent set estimate MMC method (CSEMMC)
which is applicable when a consistent set estimator of the nuisance
parameters is available can be used. If such set is available, by setting
lower and upper accordingly, mmc will yield
an asymptotically justified version of the MMC procedure.
One version of this procedure is the Two-stage constrained maximized
Monte Carlo test, where first a confidence set of level
for the nuisance parameters is obtained and then the MMC with
confidence level is taken over this particular set.
This procedure yields a conservative test with level
. Note that we generally advise
against using asymptotic Wald-type confidence intervals based on
their poor performance. Instead, it is simply best to build
confidence set using problem-specific tools.
The returned value is an object of class mmc
containing the following components:
S0 |
Observed value of |
pval |
Maximized Monte Carlo p-value of |
y |
Data specified in call. |
statistic |
|
dgp |
|
est |
|
lower |
|
upper |
|
N |
Number of replications specified in call. |
type |
|
method |
|
call |
Original call to |
seed |
Value of |
lmc |
If |
opt_result |
An object returning the optimization results. |
rejection |
If |
GenSA
Integer. Maximum number of iterations of the algorithm. Defaults to 1000.
Integer. The program will stop when there is no any improvement in nb.stop.improvement steps. Defaults to 25
Logical.TRUE when the objective function is smooth, or differentiable almost everywhere in the region of par, FALSE otherwise. Default value is TRUE.
Integer. Maximum number of call of the objective function. Default is set to 1e7.
Numeric. Maximum running time in seconds.
Numeric. Initial value for temperature.
Numeric. Parameter for visiting distribution.
Numeric. Parameter for acceptance distribution.
Logical. FALSE means that the objective function has only a few local minima. Default is FALSE which means that the objective function is complicated with many local minima.
psoptim
The maximum number of iterations. Defaults to 1000.
The maximum number of function evaluations (not considering any performed during numerical gradient computation). Defaults to Inf.
The tolerance for restarting. Once the maximal distance between the best particle and all other particles is less than reltol*d the algorithm restarts. Defaults to 0 which disables the check for restarting.
The swarm size. Defaults to floor(10+2*sqrt(length(par))) unless type is "SPSO2011" in which case the default is 40.
The exponent for calculating number of informants. Defaults to 3.
The average percentage of informants for each particle. A value of 1 implies that all particles are fully informed. Defaults to 1-(1-1/s)^k.
The exploitation constant. A vector of length 1 or 2. If the length is two, the actual constant used is gradially changed from w[1] to w[2] as the number of iterations or function evaluations approach the limit provided. Defaults to 1/(2*log(2)).
The local exploration constant. Defaults to .5+log(2).
The global exploration constant. Defaults to .5+log(2).
The diameter of the search space. Defaults to the euclidean distance between upper and lower.
The maximal (euclidean) length of the velocity vector. Defaults to NA which disables clamping of the velocity. However, if specified the actual clamping of the length is v.max*d.
Logical; if TRUE the particles are processed in random order. If vectorize is TRUE then the value of rand.order does not matter. Defaults to TRUE.
The maximum number of restarts. Defaults to Inf.
The maximum number of iterations without improvement. Defaults to 25
Logical; if TRUE the particles are processed in a vectorized manner. This reduces the overhead associated with iterating over each particle and may be more time efficient for cheap function evaluations. Defaults to TRUE.
Character vector which describes which reference implementation of SPSO is followed. Can take the value of "SPSO2007" or "SPSO2011". Defaults to "SPSO2007".
GA
the population size.
the probability of crossover between pairs of chromosomes. Typically this is a large value and by default is set to 0.8.
the probability of mutation in a parent chromosome. Usually mutation occurs with a small probability, and by default is set to 0.1.
a logical defaulting to FALSE. If set at TRUE the first attribute attached to the value returned by the user-defined fitness function is used to update the population. Be careful though, this is an experimental feature!
a user-defined function which, if provided, receives the current ga-class object as input, performs post fitness-evaluation steps, then returns an updated version of the object which is used to update the GA search. Be careful though, this is an experimental feature!
the maximum number of iterations to run before the GA search is halted.
the number of consecutive generations without any improvement in the best fitness value before the GA is stopped.
a logical defaulting to FALSE determining whether or not a local search using general-purpose optimisation algorithms should be used. See argument optimArgs for further details and finer control.
a list controlling the local search algorithm with the following components:
a string specifying the general-purpose
optimisation method to be used, by default is set to
"L-BFGS-B". Other possible methods are those reported
in optim.
a value in the range [0,1] specifying the probability of performing a local search at each iteration of GA (default 0.1).
a value in the range [0,1] specifying the pressure selection (default 0.5). The local search is started from a random solution selected with probability proportional to fitness. High values of pressel tend to select the solutions with the largest fitness, whereas low values of pressel assign quasi-uniform probabilities to any solution.
a list of control parameters. See 'Details'
section in optim.
a logical argument specifying if best solutions at each iteration should be saved in a slot called bestSol. See ga-class.
a logical argument specifying if parallel computing should be used (TRUE) or not (FALSE, default) for evaluating the fitness function. This argument could also be used to specify the number of cores to employ; by default, this is taken from detectCores. Finally, the functionality of parallelization depends on system OS: on Windows only 'snow' type functionality is available, while on Unix/Linux/Mac OSX both 'snow' and 'multicore' (default) functionalities are available.
gridSearch
the number of levels. Default is 10.
print information on the number of objective function evaluations
can be loop (the default), multicore or snow. See Details.
a list containing settings that will be passed to mclapply if method is multicore. Must be a list of named elements; see the documentation of mclapply in parallel.
default is NULL. If method snow is used, this must be a cluster object or an integer (the number of cores).
logical: should the names of levels be kept?
does fun expect a list? Default is FALSE
Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.
Dufour, J.-M. and Khalaf L. (2003), Monte Carlo Test Methods in Econometrics. in Badi H. Baltagi, ed., A Companion to Theoretical Econometrics, Blackwell Publishing Ltd, 494-519.
Y. Xiang, S. Gubian. B. Suomela, J. Hoeng (2013). Generalized Simulated Annealing for Efficient Global Optimization: the GenSA Package for R. The R Journal, Volume 5/1, June 2013. URL https://journal.r-project.org/.
Claus Bendtsen. (2012). pso: Particle Swarm Optimization. R package version 1.0.3. https://CRAN.R-project.org/package=pso
Luca Scrucca (2013). GA: A Package for Genetic Algorithms in R. Journal of Statistical Software, 53(4), 1-37. URL https://www.jstatsoft.org/article/view/v053i04.
Luca Scrucca (2016). On some extensions to GA package: hybrid optimisation, parallelisation and islands evolution. Submitted to R Journal. Pre-print available at arXiv URL http://arxiv.org/abs/1605.01931.
Manfred Gilli (2011), Dietmar Maringer and Enrico Schumann. Numerical Methods and Optimization in Finance. Academic Press.
## Example 1 ## Exact Unit Root Test library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) ## Example 2 ## Behrens-Fisher Problem library(MASS) # Set seed set.seed(123) # Generate sample x1 ~ N(0,1) and x2 ~ N(0,4) x1 <- rnorm(15, mean = 0, sd = 1) x2 <- rnorm(25, mean = 0, sd = 2) data <- list(x1 = x1, x2 = x2) # Fit a normal distribution on x1 and x2 using maximum likelihood fit1 <- fitdistr(x1, "normal") fit2 <- fitdistr(x2, "normal") # Extract the estimate for the nuisance parameters v = (sd_1, sd_2) est <- c(fit1$estimate["sd"], fit2$estimate["sd"]) # Set the bounds of the nuisance parameters equal to the 99% CI lower <- est - 2.577 * c(fit2$sd["sd"], fit1$sd["sd"]) upper <- est + 2.577 * c(fit2$sd["sd"], fit1$sd["sd"]) # Set the function for the DGP under the null (i.e. two population means are equal) dgp <- function(data, v) { x1 <- rnorm(length(data$x1), mean = 0, sd = v[1]) x2 <- rnorm(length(data$x2), mean = 0, sd = v[2]) return(list(x1 = x1, x2 = x2)) } # Set the statistic function to Welch's t-test welch <- function(data) { test <- t.test(data$x2, data$x1) return(test$statistic) } # Apply Welch's t-test t.test(data$x2, data$x1) # Apply the mmc procedure mmc(y = data, statistic = welch, dgp = dgp, est = est, lower = lower, upper = upper, N = 99, type = "absolute", method = "pso")## Example 1 ## Exact Unit Root Test library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) ## Example 2 ## Behrens-Fisher Problem library(MASS) # Set seed set.seed(123) # Generate sample x1 ~ N(0,1) and x2 ~ N(0,4) x1 <- rnorm(15, mean = 0, sd = 1) x2 <- rnorm(25, mean = 0, sd = 2) data <- list(x1 = x1, x2 = x2) # Fit a normal distribution on x1 and x2 using maximum likelihood fit1 <- fitdistr(x1, "normal") fit2 <- fitdistr(x2, "normal") # Extract the estimate for the nuisance parameters v = (sd_1, sd_2) est <- c(fit1$estimate["sd"], fit2$estimate["sd"]) # Set the bounds of the nuisance parameters equal to the 99% CI lower <- est - 2.577 * c(fit2$sd["sd"], fit1$sd["sd"]) upper <- est + 2.577 * c(fit2$sd["sd"], fit1$sd["sd"]) # Set the function for the DGP under the null (i.e. two population means are equal) dgp <- function(data, v) { x1 <- rnorm(length(data$x1), mean = 0, sd = v[1]) x2 <- rnorm(length(data$x2), mean = 0, sd = v[2]) return(list(x1 = x1, x2 = x2)) } # Set the statistic function to Welch's t-test welch <- function(data) { test <- t.test(data$x2, data$x1) return(test$statistic) } # Apply Welch's t-test t.test(data$x2, data$x1) # Apply the mmc procedure mmc(y = data, statistic = welch, dgp = dgp, est = est, lower = lower, upper = upper, N = 99, type = "absolute", method = "pso")
mmc ObjectThe plot() method for objects of the
class mmc gives a plot of the best and current p-value found during the iterations of mmc.
## S3 method for class 'mmc' plot(x, ...)## S3 method for class 'mmc' plot(x, ...)
x |
An object of class |
... |
Arguments to be passed to methods, such as graphical parameters (see |
The mmc object is returned invisibly.
## Example library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure est <- mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) # Plot result of object of class 'mmc' plot(est)## Example library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure est <- mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) # Plot result of object of class 'mmc' plot(est)
mc ObjectThis is a method for the function print() for objects of the
class mc.
## S3 method for class 'mc' print(x, digits = getOption("digits"), ...)## S3 method for class 'mc' print(x, digits = getOption("digits"), ...)
x |
an object used to select a method. |
digits |
minimal number of significant digits, see
|
... |
further arguments passed to or from other methods. |
The mc object is returned invisibly.
## Example # Set seed set.seed(999) # Generate sample data y <- rgamma(8, shape = 2, rate = 1) # Set data generating process function dgp <- function(y) rgamma(length(y), shape = 2, rate = 1) # Set the statistic function to the Kolomogorov-Smirnov test for gamma distribution statistic <- function(y){ out <- ks.test(y, "pgamma", shape = 2, rate = 1) return(out$statistic) } # Apply the Monte Carlo test with tie-breaker est <- mc(y, statistic = statistic, dgp = dgp, N = 999, type = "two-tailed") # Print result of object of class 'mc' print(est)## Example # Set seed set.seed(999) # Generate sample data y <- rgamma(8, shape = 2, rate = 1) # Set data generating process function dgp <- function(y) rgamma(length(y), shape = 2, rate = 1) # Set the statistic function to the Kolomogorov-Smirnov test for gamma distribution statistic <- function(y){ out <- ks.test(y, "pgamma", shape = 2, rate = 1) return(out$statistic) } # Apply the Monte Carlo test with tie-breaker est <- mc(y, statistic = statistic, dgp = dgp, N = 999, type = "two-tailed") # Print result of object of class 'mc' print(est)
mmc ObjectThis is a method for the function print() for objects of the
class mmc.
## S3 method for class 'mmc' print(x, digits = getOption("digits"), ...)## S3 method for class 'mmc' print(x, digits = getOption("digits"), ...)
x |
an object used to select a method. |
digits |
minimal number of significant digits, see
|
... |
further arguments passed to or from other methods. |
The mmc object is returned invisibly.
## Example library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure est <- mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) # Print result of object of class 'mmc' print(est)## Example library(fUnitRoots) # Set seed set.seed(123) # Generate an AR(2) process with phi = (-1.5,0.5), and n = 25 y <- filter(rnorm(25), c(-1.5, 0.5), method = "recursive") # Set bounds for the nuisance parameter v lower <- -1 upper <- 1 # Set the function to generate an AR(2) integrated process dgp <- function(y, v) { ran.y <- filter(rnorm(length(y)), c(1-v,v), method = "recursive") } # Set the Augmented-Dicky Fuller statistic statistic <- function(y){ out <- suppressWarnings(adfTest(y, lags = 2, type = "nc")) return(out@test$statistic) } # Apply the mmc procedure est <- mmc(y, statistic = statistic , dgp = dgp, lower = lower, upper = upper, N = 99, type = "leq", method = "GenSA", control = list(max.time = 2)) # Print result of object of class 'mmc' print(est)
Computes the p-value of the statistic by computing its rank compared to its simulated values.
pvalue(S0, S, type = c("geq", "leq", "absolute", "two-tailed"))pvalue(S0, S, type = c("geq", "leq", "absolute", "two-tailed"))
S0 |
An atomic vector. Value of the test statistic applied to the data. |
S |
A vector. It consists of replications of the test statistic.
|
type |
A character string. It specifies the type of test
the p-value function produces. The possible values are
|
We allow for four types of p-value: leq, geq,
absolute and two-tailed. For one-tailed test,
leq returns the proportion of simulated values smaller
than the statistic while geq returns the proportion of
simulated values greater than the statistic. For two-tailed
test with a symmetric satistic, one can use the
absolute value of the statistic and its simulated values to
retrieve a two-tailed test (i.e. type = absolute).
If the statistic is not symmetric, one can specify the p-value
type as two-tailed which is equivalent to twice the minimum
of leq and geq.
Ties in the ranking are broken according to a uniform distribution.
The p-value of the statistic S0 given a vector of replications S.
Dufour, J.-M. (2006), Monte Carlo Tests with nuisance parameters: A general approach to finite sample inference and nonstandard asymptotics in econometrics. Journal of Econometrics, 133(2), 443-447.
Dufour, J.-M. and Khalaf L. (2003), Monte Carlo Test Methods in Econometrics. in Badi H. Baltagi, ed., A Companion to Theoretical Econometrics, Blackwell Publishing Ltd, 494-519.
# Generate sample S0 and simulate statistics S0 = 0 S = rnorm(99) # Compute p-value pvalue(S0, S, type = "geq")# Generate sample S0 and simulate statistics S0 = 0 S = rnorm(99) # Compute p-value pvalue(S0, S, type = "geq")