getDeltaH_down: Estimation of the Optimal 'H_delta' Value to Guarantee...

In LGCU: Implementation of Learning Gamma CUSUM (Cumulative Sum) Control Charts

Estimation of the Optimal H_delta Value to Guarantee Performance in the Downward CUSUM Control Chart

Description

This function calculates the optimal value of H_delta that guarantees a specific performance in the Gamma CUSUM control chart for downward detection. It employs a Monte Carlo simulation approach and an iterative adjustment process to determine the appropriate value.

Following the methodology presented by Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), this function allows adjusting H_delta for different sample size configurations, ensuring that the control chart maintains the desired performance in terms of expected ARL.

Features:

• Implements Monte Carlo simulations to estimate H_delta.

• Based on parameter estimates obtained in Phase I.

• Iteratively adjusts H_delta until the specified ARL is achieved.

• Displays the total execution time using tictoc.

Recommendations

• This function is useful for estimating H_delta values in scenarios where the sample size differs from the values reported in the reference paper:

  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 International, 40(3), 1279-1301.

• The adjustment process is iterative and computationally demanding, as its execution time depends on the number of iterations (N_init + N_final) and the sample size (n_I).

• It is recommended to establish an appropriate convergence criterion to optimize execution time without compromising the accuracy of H_delta estimation.

• For selecting values of H_minus, a, and b, it is advisable to consult the reference paper, which provides specific calibration strategies and recommendations.

Usage

getDeltaH_down(
  n_I,
  alpha,
  beta,
  beta_ratio,
  H_minus,
  a,
  b,
  ARL_esp,
  m,
  N_init,
  N_final,
  known_alpha
)

Arguments

n_I	Sample size in Phase I.
alpha	Shape parameter of the Gamma distribution.
beta	Scale parameter of the Gamma distribution.
beta_ratio	Ratio between beta and its estimate.
H_minus	Initial lower limit of the CUSUM chart.
a	Tolerance level for the expected ARL (0 <= a < 1).
b	Tolerance level for the expected ARL (0 < b < 1).
ARL_esp	Desired expected ARL value.
m	Number of states in the Markov matrix.
N_init	Number of initial iterations.
N_final	Number of final iterations.
known_alpha	Indicates whether alpha is known (TRUE) or should be estimated (FALSE).

Value

A numerical value corresponding to the optimal H_delta for the downward CUSUM control chart, ensuring the expected performance.

Examples

getDeltaH_down(n_I = 100, alpha = 1, beta = 1, beta_ratio = 1/2,
               H_minus = -4.1497, a = 0.1, b = 0.05, ARL_esp = 370,
               m = 100, N_init = 10, N_final = 1000, known_alpha = TRUE)
               

LGCU documentation built on April 12, 2025, 1:59 a.m.