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Table 1 Bias and root mean squared error on Monte Carlo simulation when \(\alpha =0.5\), \(\beta =0.2\), \(\lambda =0.5\) and \(\theta =8\)

From: The beta Burr type X distribution properties with application

n

Parameter

Bias

RMSE

100

\(\hat{\alpha }\)

3.34 × 10−2

4.73 × 10−4

\(\hat{\beta }\)

−7.47 × 10−3

1.05 × 10−4

\(\hat{\lambda }\)

−1.93 × 10−3

2.73 × 10−5

\(\hat{\theta }\)

2.09 × 10−4

2.95 × 10−6

500

\(\hat{\alpha }\)

1.13 × 10−2

1.60 × 10−4

\(\hat{\beta }\)

−6.52 × 10−4

9.23 × 10−5

\(\hat{\lambda }\)

−3.84 × 10−3

5.43 × 10−5

\(\hat{\theta }\)

−8.10 × 10−7

1.13 × 10−8

1000

\(\hat{\alpha }\)

−1.63 × 10−2

2.31 × 10−4

\(\hat{\beta }\)

−9.48 × 10−3

1.34 × 10−4

\(\hat{\lambda }\)

−2.12 × 10−4

3.01 × 10−6

\(\hat{\theta }\)

4.65 × 10−7

6.57 × 10−9

1500

\(\hat{\alpha }\)

−1.24 × 10−8

1.76 × 10−10

\(\hat{\beta }\)

5.42 × 10−8

7.66 × 10−10

\(\hat{\lambda }\)

6.58 × 10−5

9.31 × 10−6

\(\hat{\theta }\)

−1.13 × 10−9

1.60 × 10−11