GICARL_CUSUM_up {LGCU} | R Documentation |
ARL Calculation for an Upward CUSUM Control Chart with a Gamma Distribution
Description
This function estimates the average run length (ARL) of an upward CUSUM control chart applied to a Gamma distribution with guaranteed performance (GIC), considering both known and estimated parameters.
This approach follows the methodology described in the work of Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), which provides a detailed analysis of the performance of CUSUM control charts for Gamma distributions with guaranteed efficiency. Specifically, the method implemented in this function enables the precise evaluation of ARL under different parameter settings, ensuring appropriate calibration and monitoring of controlled processes.
Recommendations
For further consultation and to review values of H_delta
and H_plus
, it is recommended to refer to the following article:
Madrid‐Alvarez, H. M., García‐Díaz, J. C., & Tercero‐Gómez, V. G. (2024).
A CUSUM control chart for gamma distribution with guaranteed performance. Quality and Reliability Engineering.
Key Considerations:
The control chart is calibrated with
beta = 1
.When
alpha
andbeta
are known, it is recommended to use the same values foralpha_est = alpha
andbeta_est = 1
.Higher values of
m
increase the accuracy of the results (m = {10, 100}
).
Usage
GICARL_CUSUM_up(
alpha,
beta,
alpha_est,
beta_est,
beta_ratio,
H_plus,
H_delta,
m
)
Arguments
alpha |
Shape parameter of the Gamma distribution. |
beta |
Scale parameter of the Gamma distribution. |
alpha_est |
Estimated shape parameter. |
beta_est |
Estimated scale parameter. |
beta_ratio |
Ratio between |
H_plus |
Upper control limit of the upward CUSUM chart. |
H_delta |
Increment of the GIC threshold. |
m |
Number of divisions for the probability matrix. |
Value
A numeric value representing the average run length (ARL) of the upward CUSUM control chart.
Examples
# Example with known parameters
GICARL_CUSUM_up(alpha = 0.9, beta = 2.136, alpha_est = 0.9, beta_est = 1,
beta_ratio = 2.67, H_plus = 25.1592, H_delta = 0, m = 100)
# Example with estimated parameters
GICARL_CUSUM_up(alpha = 1, beta = 1, alpha_est = 1.2, beta_est = 0.8,
beta_ratio = 2, H_plus = 6.5081, H_delta = 2.9693, m = 100)