On bias reduction estimators of skew-normal and skew-t distributions

Abstract

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.

Keywords: 62F15; Bias-corrected estimators; bias prevention; maximum likelihood estimator; skew-normal; skew-t.

Figure 1.

Modifications to the score function for the skew normal.

Figure 2.

Relative bias and MSE of the considered estimate of α.

Figure 3.

Comparing the behaviour of $M^J(\alpha )$, $M_1(\alpha )$ and $M^{\ast }(\alpha )$.

Figure 4.

Example 4.3. The left panel shows $U(\alpha )$, $U^{sf}(\alpha )$ and $U^{\ast }(\alpha )$ while the right panel shows the relative versions of $l(\alpha )$, $l^{sf}(\alpha )$ and $l^{\ast }(\alpha )$ for the data z.

Figure 5.

Example 4.3. The left panel shows $U(\alpha )$, $U^{sf}(\alpha )$ and $U^{\ast }(\alpha )$ while the right panel shows the relative versions of $l(\alpha )$, $l^{sf}(\alpha )$ and $l^{\ast }(\alpha )$ for the data $|\mathsf{z}|$.

Figure 6.

Modification term of the score function for skew t distributions with 3 (left panel) and 5 (right panel) degrees of freedom.

Figure 7.

Relative bias and MSE of the considered estimate of $\alpha$.

Figure 8.

Example 6.2. The left panel shows $U_p(\alpha )$, $U_p^{sf}(\alpha )$ and $U_p^{\ast }(\alpha )$ while the right panel shows the relative versions of $l_p(\alpha )$, $l_p^{sf}(\alpha )$ and $l_p^{\ast }(\alpha )$ for the data y.

Figure 9.

Relative bias and MSE of the considered estimate of $\alpha$.

Figure 10.

Observed coverage of 0.95 confidence intervals based on $W^{\ast }(\alpha )$ with M = 5000 iterations.

Figure 11.

Histograms of the Data set 1 (left panel) and Data set 2 (right panel) and the three fitted pdf curves.