LGCU (version 0.1.5)

GICARL_CUSUM_down: ARL Calculation for a Downward CUSUM Control Chart with a Gamma Distribution

Description

This function estimates the average run length (ARL) of a downward 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_minus, 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.

Considerations:

• The control chart is calibrated with beta = 1.

• When alpha and beta are known, it is recommended to use the same values for alpha_est = alpha and beta_est = 1.

• Higher values of m increase the accuracy of the results (m = {10, 100}).

Usage

GICARL_CUSUM_down(
  alpha,
  beta,
  alpha_est,
  beta_est,
  beta_ratio,
  H_minus,
  H_delta,
  m
)

Value

A numeric value representing the average run length (ARL) of the downward CUSUM control chart.

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 beta and its estimation.

H_minus

Lower control limit of the downward CUSUM chart.

H_delta

Increment of the GIC threshold.

m

Number of divisions for the probability matrix.

Examples

# Example with known parameters
GICARL_CUSUM_down(alpha = 1, beta = 1, alpha_est = 1, beta_est = 1,
                  beta_ratio = 1/2.5, H_minus = -2.792, H_delta = 0, m = 100)

# Example with estimated parameters
GICARL_CUSUM_down(alpha = 1, beta = 1, alpha_est = 1, beta_est = 1.1,
                  beta_ratio = 1/2, H_minus = -4.1497, H_delta = 1.5167,
                  m = 100)