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:

**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`

.**Obtain the constants**for \(\Large Y\). This can be done using`find_constants`

or by simulating the distribution with`nonnormvar1`

.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.**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)`

).**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 \(\Large Y^*\).**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 \(\Large \alpha\).**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`

.

Use these steps to compare a simulated **exponential(mean = 2) variable** to the theoretical exponential(mean = 2) density. (Note that the `printr`

package is invoked to display the tables.)

In R, the exponential parameter is `rate <- 1/mean`

.

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

Note that `calc_theory`

returns the standard deviation, not the variance. The simulation functions require variance as the input.

```
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
```

Look at the power method constants.

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

c0 | c1 | c2 | c3 | c4 | c5 |
---|---|---|---|---|---|

-0.3077396 | 0.8005605 | 0.318764 | 0.0335001 | -0.0036748 | 0.0001587 |

Look at a summary of the target distribution.

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

Distribution | mean | sd | skew | skurtosis | fifth | sixth | |
---|---|---|---|---|---|---|---|

mean | 1 | 2 | 2 | 2 | 6 | 24 | 120 |

Compare to a summary of the simulated distribution.

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

Distribution | mean | sd | skew | skurtosis | fifth | sixth | |
---|---|---|---|---|---|---|---|

X1 | 1 | 1.99987 | 2.0024 | 2.03382 | 6.18067 | 23.74145 | 100.3358 |

`H_exp$valid.pdf`

`## [1] "TRUE"`

Let \(\Large \alpha = 0.05\).

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

`## [1] 5.991465`

Since the exponential(2) distribution has a mean and standard deviation equal to 2, solve \(\Large 2 * p(z') + 2 - y_star = 0\) for \(\Large z'\). Here, \(\Large p(z') = c0 + c1 * z' + c2 * z'^2 + c3 * z'^3 + c4 * z'^4 + c5 * z'^5\).

```
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`

This is approximately equal to the \(\Large \alpha\) value of 0.05, indicating the method provides a **good approximation to the actual distribution.**

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

We can also plot the empirical cdf and show the cumulative probability up to y_star.

```
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])))
```

trimmed_mean | median | mode | max_height |
---|---|---|---|

1.858381 | 1.384521 | 0.104872 | 1.094213 |

Headrick, T C, and R K Kowalchuk. 2007. “The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data.” *Journal of Statistical Computation and Simulation* 77: 229–49. doi:10.1080/10629360600605065.