Comparison of Simulated Distribution to Theoretical Distribution or Empirical Data

Headrick and Kowalchuk (2007) outlined a general method for comparing a simulated distribution $\Large Y$ to a given theoretical distribution $\Large Y^*$. Note that these could easily be modified for comparison to an empirical vector of data:

1. Obtain the standardized cumulants (skewness, kurtosis, fifth, and sixth) for $\Large Y^*$. This can be done using calc_theory along with either the distribution name (plus up to 4 parameters) or the pdf fx (plus support bounds). In the case of an empirical vector of data, use calc_moments or calc_fisherk.

2. Obtain the constants for $\Large Y$. This can be done using find_constants or by simulating the distribution with nonnormvar1.

3. Determine whether these constants produce a valid power method pdf. The results of find_constants or nonnormvar1 indicate whether the constants yield an invalid or valid pdf. The constants may also be checked using pdf_check. If the constants generate an invalid pdf, the user should check if the kurtosis falls above the lower bound (using calc_lower_skurt). If yes, a vector of sixth cumulant correction values should be used in find_constants or nonnormvar1 to find the smallest correction that produces valid pdf constants.

4. Select a critical value from $\Large Y^*$, i.e. $\Large y^*$ such that $\Large Pr(Y^* \ge y^*) = \alpha$. This can be done using the appropriate quantile function and $\Large 1 - \alpha$ value (i.e. qexp(1 - 0.05)).

5. Solve $\Large m_{2}^{1/2} * p(z') + m_{1} - y^* = 0$ for $\Large z'$, where $\Large m_{1}$ and $\Large m_{2}$ are the 1st and 2nd moments of $\Large Y^*$.

6. Calculate $\Large 1 - \Phi(z')$, the corresponding probability for the approximation $\Large Y$ to $\Large Y^*$ (i.e. $\Large 1 - \Phi(z') = 0.05$) and compare to the target value $\Large \alpha$.

7. Plot a parametric graph of $\Large Y^*$ and $\Large Y$. This can be done with a set of constants using plot_pdf_theory (overlay = TRUE) or with a simulated vector of data using plot_sim_pdf_theory (overlay = TRUE). If comparing to an empirical vector of data, use plot_pdf_ext or plot_sim_pdf_ext.

library("SimMultiCorrData")
library("printr")
stcums <- calc_theory(Dist = "Exponential", params = 0.5)

H_exp <- nonnormvar1("Polynomial", means = stcums[1], vars = stcums[2]^2, 
                     skews = stcums[3], skurts = stcums[4], 
                     fifths = stcums[5], sixths = stcums[6], n = 10000, 
                     seed = 1234)

## Constants: Distribution  1  
## 
## Constants calculation time: 0 minutes 
## Total Simulation time: 0 minutes

as.matrix(H_exp$constants, nrow = 1, ncol = 6, byrow = TRUE)

as.matrix(round(H_exp$summary_targetcont[, c("Distribution", "mean", "sd", 
                                             "skew", "skurtosis", "fifth", 
                                             "sixth")], 5), nrow = 1, ncol = 7,
          byrow = TRUE)

as.matrix(round(H_exp$summary_continuous[, c("Distribution", "mean", "sd", 
                                             "skew", "skurtosis", "fifth", 
                                             "sixth")], 5), nrow = 1, ncol = 7,
          byrow = TRUE)

H_exp$valid.pdf

## [1] "TRUE"

y_star <- qexp(1 - 0.05, rate = 0.5) # note that rate = 1/mean
y_star

## [1] 5.991465

f_exp <- function(z, c, y) {
  return(2 * (c[1] + c[2] * z + c[3] * z^2 + c[4] * z^3 + c[5] * z^4 + 
                c[6] * z^5) + 2 - y)
}

z_prime <- uniroot(f_exp, interval = c(-1e06, 1e06), 
                   c = as.numeric(H_exp$constants), y = y_star)$root
z_prime

## [1] 1.644926

1 - pnorm(z_prime)

## [1] 0.04999249

plot_sim_pdf_theory(sim_y = H_exp$continuous_variable[, 1], 
                    Dist = "Exponential", params = 0.5)

plot_sim_cdf(sim_y = H_exp$continuous_variable[, 1], calc_cprob = TRUE, 
             delta = y_star)

as.matrix(t(stats_pdf(c = H_exp$constants[1, ], method = "Polynomial", 
                    alpha = 0.025, mu = stcums[1], sigma = stcums[2])))