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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