confidence_grid

Description

returns the indicator array

Usage

confidence_grid(
  alpha,
  lower_bds,
  upper_bds,
  seeds,
  generating_fun,
  s_obs,
  tol,
  resolution,
  theta_init = NULL,
  T_stat = ma_depth
)

Arguments

Value

A list containing an indicator array (ind_array) representing the confidence set, the confidence set lower bounds (updated_lower_bds), and the confidence set upper bounds (updated_upper_bds).

Examples

### Note that the examples may take a few seconds to run.
### Regular normal
set.seed(123)
n <- 50 # sample size
R <- 50 # Repro sample size (should be at least 200 for accuracy in practice)
alpha <- .05 # significance level
tol <- 0.01 # tolerance for the confidence set (use smaller tolerance in practice)
s_obs <- c(1.12, 0.67) # the observed sample mean
seeds <- matrix(rnorm(R * (n + 2)), nrow = R, ncol = n + 2) # pre-generated seeds

# this function computes the repro statistics given the seeds and the parameter
s_sample <- function(seeds, theta) {
  # generate the raw data points
  raw_data <- theta[1] + sqrt(theta[2]) * seeds[, 1:n]

  # compute the regular statistics
  s_mean <- apply(raw_data, 1, mean)
  s_var <- apply(raw_data, 1, var)

  return(cbind(s_mean, s_var))
}

lower_bds <- c(0.5, 0.3) # lower bounds for the parameter search region
upper_bds <- c(1.5, 1.3) # upper bounds for the parameter search region

resolution = 10  # resolution of the grid
result <- confidence_grid(alpha, lower_bds, upper_bds, seeds, s_sample, s_obs, tol, resolution)
print(result$ind_array)
print(result$search_lower_bds)
print(result$search_upper_bds)

get_CI

Description

Given the observed statistic, this function computes a confidence interval given that the data generating process is known using the supplied random seeds.

Usage

get_CI(
  alpha,
  lower_bds,
  upper_bds,
  parameter_index,
  seeds,
  generating_fun,
  s_obs,
  tol,
  theta_init = NULL,
  T_stat = ma_depth,
  verbose = FALSE,
  check_input = TRUE
)

Arguments

Value

A length-2 vector representing the obtained confidence interval. In the case when no point is accepted in the search space, return NULL.

Examples

### Note that the examples may take a few seconds to run.
### Regular Normal
set.seed(123)
n <- 30 # sample size
R <- 50 # Repro sample size (should be at least 200 for accuracy in practice)
alpha <- .05 # significance level
tol <- 0.01 # tolerance for the confidence set (use smaller tolerance in practice)
s_obs <- c(1.12, 0.67) # the observed sample mean and variance
seeds <- matrix(rnorm(R * (n + 2)), nrow = R, ncol = n + 2) # pre-generated seeds

# this function computes the repro statistics given the seeds and the parameter
s_sample <- function(seeds, theta) {
# generate the raw data points
  raw_data <- theta[1] + sqrt(theta[2]) * seeds[, 1:n]

  # compute the regular statistics
  s_mean <- apply(raw_data, 1, mean)
  s_var <- apply(raw_data, 1, var)

  return(cbind(s_mean, s_var))
}

lower_bds <- c(-5, 0.01) # lower bounds for the parameter search region
upper_bds <- c(5, 5) # upper bounds for the parameter search region

# choose parameter_index = 1 to get the confidence interval for the mean
mean_CI <- get_CI(alpha, lower_bds, upper_bds, 1, seeds, s_sample, s_obs, tol)
print(mean_CI) # estimated confidence interval for mean
var_CI <- get_CI(alpha, lower_bds, upper_bds, 2, seeds, s_sample, s_obs, tol)
print(var_CI) # estimated confidence interval for variance

grid_projection

Description

Projects the multidimensional indicator array generated by confidence_grid down to 2d for visualization

Usage

grid_projection(indicator_array, index_set)

Arguments

Value

A two-dimensional indicator array ready for visualization (indicator_array projected onto the subspace specified by index_set).

Examples

### simple projection
ind_arr <- array(c(1, 0, 0, 0, 0, 1, 0, 1), dim = rep(2, 3))
print(ind_arr)
# project this indicator array onto a 2d subspace by first and second dimension
ind_arr_12 <- grid_projection(ind_arr, c(1,2))
print(ind_arr_12)
ind_arr_13 <- grid_projection(ind_arr, c(1,3))
print(ind_arr_13)
ind_arr_23 <- grid_projection(ind_arr, c(2,3))
print(ind_arr_23)

p_value

Description

Given the observed statistic and the given seeds, this function finds the p-value. The method uses simulation-based inference, where having fixed seeds, the parameter is searched which makes the observed statistics most "plausible". In particular, the T_stat function measures the "plausibility" of any data point and the procedure maximizes the rank of the observed T_stat value relative to the "repro" 'T_stat' values. The p-value is determined from the maximum rank and the corresponding parameter is returned.

Usage

p_value(
  lower_bds,
  upper_bds,
  seeds,
  generating_fun,
  s_obs,
  theta_init = NULL,
  T_stat = ma_depth,
  verbose = FALSE,
  check_input = TRUE
)

Arguments

Value

A list containing the most likely parameter in the search region (theta_hat) and its corresponding p-value (p_val).

Examples

### Regular Normal
set.seed(123)
n <- 50 # sample size
R <- 50 # Repro sample size (should be at least 200 for accuracy in practice)
s_obs <- c(1.12, 0.67) # the observed sample mean and variance
seeds <- matrix(rnorm(R * (n + 2)), nrow = R, ncol = n + 2) # pre-generated seeds

# this function computes the repro statistics given the seeds and the parameter
s_sample <- function(seeds, theta) {
  # generate the raw data points
  raw_data <- theta[1] + sqrt(theta[2]) * seeds[, 1:n]

  # compute the regular statistics
  s_mean <- apply(raw_data, 1, mean)
  s_var <- apply(raw_data, 1, var)

  return(cbind(s_mean, s_var))
}

lower_bds <- c(-5, 0.01) # lower bounds for null hypothesis region
upper_bds <- c(5, 5) # upper bounds for null hypothesis region

result <- p_value(lower_bds, upper_bds, seeds, s_sample, s_obs)
print(result$p_val) # the largest p_value found
print(result$theta_hat) # the parameter corresponding to the largest p value

plot_grid

Description

projects the indicator array generated by confidence_grid down to 2d for visualization

Usage

plot_grid(indicator_array, lower_bds, upper_bds, parameter_names = NULL)

Arguments

Value

A grid plot showing the confidence regions.

Examples

### Note that the examples may take a few seconds to run.
### Regular normal
set.seed(123)
n <- 50 # sample size
R <- 50 # Repro sample size (should be at least 200 for accuracy in practice)
alpha <- .05 # significance level
tol <- 1e-2 # tolerance for the confidence set
s_obs <- c(1.12, 0.67) # the observed sample mean
seeds <- matrix(rnorm(R * (n + 2)), nrow = R, ncol = n + 2) # pre-generated seeds

# this function computes the repro statistics given the seeds and the parameter
s_sample <- function(seeds, theta) {
  # generate the raw data points
  raw_data <- theta[1] + sqrt(theta[2]) * seeds[, 1:n]

  # compute the regular statistics
  s_mean <- apply(raw_data, 1, mean)
  s_var <- apply(raw_data, 1, var)

  return(cbind(s_mean, s_var))
}

lower_bds <- c(0.5, 0.4) # lower bounds for the parameter search region
upper_bds <- c(1.5, 1.4) # upper bounds for the parameter search region

resolution = 10  # resolution of the grid

result <- confidence_grid(alpha, lower_bds, upper_bds, seeds, s_sample, s_obs, tol, resolution)
ind_arr <- result$ind_array
parameter_names <- c("mean", "variance") # specifying the names of each parameter
plot_grid(ind_arr, lower_bds, upper_bds, parameter_names)