Issue 58

S. Çal ı ş kan et al.ii, Frattura ed Integrità Strutturale, 25 (2021) 344-364; DOI: 10.3221/IGF-ESIS.58.25

Figure 6: Normal distribution of staircase data based on Bootstrapping method.

Standard error estimation of re-sample distribution for bootstrapping method is originated by Monte-Carlo approach. Approximating population with larger sample size is advanced at least 200 times re-sampling from original data [28]. During analysis, 200 times original data was re-sampled and only 10 of them can be seen on Tab. 1.

b X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(8)

X*

1 2 3 4 5 6 7 8 9

580 580 580 560 560 560 580 600 620

580 620 580 560 600 560 580 580 560 600

600 600 560 580 600 600 580 620 600 580

580 620 620 580 580 560 600 580 600 620

580 600 580 560 560 560 560 560 600 580

580 560 620 560 580 560 560 620 560

600 600 560 580 580 560 580 600 600

600 580 620 560 560 560 580 600 600

587.5

595 590

567.5 577.5

565

577.5

595

592.5 587.5

10 580

620 560 560

Table 1: Bootstrapping Algorithm.

Mean value and estimated standard deviation of bootstrap replications can be seen from following equations:

   B i 1 X 1 X B

* i

(10)

1 2 2

 

  

B

1







 X X *



s

(11)

i

  B

1



i

1

Similarly, standard deviation of Monte Carlo approximation based on bootstrapping replications can be calculated by following equation:

       2 1 n

s

n

(12)

Standard deviation of 200 means is the bootstrapped standard error of the mean. Mean value can be calculated as 585.11 and estimated standard deviation is 7.09 after bootstrapping analysis in Fig. 7. Standard deviation based on Monte Carlo approximation is 9.19 which is close to bootstrapping method. By using bootstrapping re-sampling, confidence intervals

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