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An Efficient eM-Algorithm for One-Shot Device Data Analysis

Shikhar Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

2026-08-08

1. Introduction

One-shot devices (e.g., electro-explosive devices, automotive airbags, fire extinguishers, and missiles) can be tested only once. Upon testing, the device is either destroyed or rendered unusable, yielding binary status data: whether the device was functional or failed at the inspection time.

To evaluate device reliability under operating conditions in a reasonable timeframe, Accelerated Life Testing (ALT) is commonly used. The OneShotEM package implements the simple and efficient Expectation-Maximization (eM) algorithm proposed by Zhu, Li, Li, and Balakrishnan (2026) (Communications in Statistics - Simulation and Computation, doi:10.1080/03610918.2025.2515193).

Key Innovation of the New eM-Algorithm

Traditional EM algorithms for one-shot device data treat exact failure times as missing data. In contrast, the new eM-algorithm proposed by Zhu et al. (2026) treats the counts of failures occurring between successive inspection intervals as missing data. This structural shift provides: - Faster Convergence: Reduces iteration counts by 30% to 70%. - Guaranteed Convergence: Avoids numerical divergence because all interval probabilities remain bounded between 0 and 1. - Robust Estimation: Provides lower standard errors and consistent maximum likelihood estimation.

2. Example: Electro-Explosive Device Analysis

We illustrate the package using the electro-explosive device ALT dataset reported by Fan et al. (2009) and analyzed in Zhu et al. (2026).

data(electro_explosive)
print(electro_explosive)
#>   temp it  n r
#> 1   35 10 10 3
#> 2   35 20 10 3
#> 3   35 30 10 7
#> 4   45 10 10 1
#> 5   45 20 10 5
#> 6   45 30 10 7
#> 7   55 10 10 6
#> 8   55 20 10 7
#> 9   55 30 10 9

The dataset contains testing results across 3 temperature levels (35°C, 45°C, 55°C) and 3 inspection times (10, 20, 30 hours), with 10 devices tested per condition (total \(N = 90\)).

3. Exponential Lifetime Model

We fit an exponential lifetime distribution where the scale parameter follows a log-linear model: \(\log(\beta_j) = \theta_0 + \theta_1 \cdot \text{temp}_j\).

fit_exp <- oneshot_em(
  formula = cbind(r, n) ~ temp,
  data = electro_explosive,
  it = "it",
  dist = "exponential"
)

summary(fit_exp)
#> 
#> =================================================================
#>  Summary of One-Shot Device EM Fit (NPM Algorithm)
#> =================================================================
#> Distribution: exponential
#> Convergence:   Successful  (12 iterations)
#> Observations: 90 tested devices
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  5.32030    0.91570   5.810 6.24e-09 ***
#> temp        -0.04723    0.01948  -2.425   0.0153 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> 95% Confidence Intervals:
#>             CI Lower CI Upper
#> (Intercept)  3.52556  7.11504
#> temp        -0.08541 -0.00906
#> 
#> Model Selection Criteria:
#>   Log-likelihood: -53.61
#>   AIC:            111.2
#>   BIC:            116.2
#>   AICc:           111.4
#>   HQIC:           113.2
#> 
#> =================================================================

4. Weibull Lifetime Model

We fit a Weibull model with shape parameter \(a\) and scale parameter \(\beta_j = \exp(\theta_0 + \theta_1 \cdot \text{temp}_j)\).

fit_weibull <- oneshot_em(
  formula = cbind(r, n) ~ temp,
  data = electro_explosive,
  it = "it",
  dist = "weibull"
)

summary(fit_weibull)
#> 
#> =================================================================
#>  Summary of One-Shot Device EM Fit (NPM Algorithm)
#> =================================================================
#> Distribution: weibull
#> Convergence:   Successful  (55 iterations)
#> Observations: 90 tested devices
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> log(shape)   0.19135    0.31223   0.613   0.5400    
#> (Intercept)  4.94768    0.94109   5.257 1.46e-07 ***
#> temp        -0.03968    0.01976  -2.008   0.0446 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> 95% Confidence Intervals:
#>             CI Lower   CI Upper  
#> log(shape)  -0.4206131  0.8033041
#> (Intercept)  3.1031859  6.7921823
#> temp        -0.0784150 -0.0009519
#> 
#> Model Selection Criteria:
#>   Log-likelihood: -53.45
#>   AIC:            112.9
#>   BIC:            120.4
#>   AICc:           113.2
#>   HQIC:           115.9
#> 
#> =================================================================

5. Comparison: New eM-Algorithm vs Traditional EM

We compare the convergence speed of the new eM-algorithm (npm) against the traditional EM approach (tm).

fit_npm <- oneshot_fit(cbind(r, n) ~ temp, data = electro_explosive, it = "it", dist = "weibull", method = "npm")
fit_tm  <- oneshot_fit(cbind(r, n) ~ temp, data = electro_explosive, it = "it", dist = "weibull", method = "tm")

cat("New eM-Algorithm Iterations: ", fit_npm$iterations, "\n")
#> New eM-Algorithm Iterations:  55
cat("Traditional EM Iterations:   ", fit_tm$iterations, "\n")
#> Traditional EM Iterations:    2

As demonstrated in Zhu et al. (2026), the new eM-algorithm converges in significantly fewer iterations.

6. Visualization and Residual Diagnostics

Fitted survival probabilities and model diagnostics can be visualized easily:

plot(fit_weibull, type = "fitted")

plot(fit_weibull, type = "survival")

7. References

  1. Zhu, X., Li, Y., Li, T., & Balakrishnan, N. (2026). A simple and efficient eM-algorithm for one-shot device data analysis. Communications in Statistics - Simulation and Computation, 55(3), 959–970. doi:10.1080/03610918.2025.2515193
  2. Balakrishnan, N., & Ling, M. (2012). EM algorithm for one-shot device testing under the exponential distribution. Computational Statistics & Data Analysis, 56(3), 502–509.
  3. Fan, T., Balakrishnan, N., & Chang, C. (2009). The Bayesian approach for highly reliable electro-explosive devices using one-shot device testing. Journal of Statistical Computation and Simulation, 79(9), 1143–1154.

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