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Testing for sub-models of the skew t-distribution

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Abstract

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically \(\chi ^2\). The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (J Am Stat Assoc 82:605–610, 1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self–Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.

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Acknowledgements

This research was partially supported by MIUR, Italy, with grant PRIN 2006132978. The numerical work was undertaken on the R package sn written by Azzalini and available at http://cran.r-project.org/src/contrib/PACKAGES.html. The Authors thank an anonymous referee for his constructive comments.

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Correspondence to Anna Clara Monti.

Appendix

Appendix

Let \(S_{(\xi ,\omega ,\alpha ,\nu )}(y)=\big \{ S_\xi (y), S_\omega (y), S_\alpha (y), S_\nu (y) \big \}'\) be the score function of the skew t-model based on a single observation y; thus, \(S_{\xi }(y)=\partial \ln f\big ( y; \xi , \omega , \alpha , \nu \big ) / \partial \xi \), and so forth. The components of S(y) are

$$\begin{aligned} S_\xi (y)= & {} {{ z\tau ^2}\over \omega }-{{\alpha \tau \nu } \over {\omega (\nu +z^2)} } w, \, \, S_\omega (y)=- {1 \over \omega }+{{ z^2\tau ^2}\over \omega }-{{\varsigma \nu } \over {\omega (\nu +z^2)} }w, \, \, S_\alpha (y)= z \tau w ,\\ S_\nu (y)= & {} {1 \over 2}\biggr \{\varPsi \bigg ( {\nu \over 2}+1 \bigg )-\varPsi \bigg ( {\nu \over 2}\bigg ) -{{2 \nu + 1} \over {\nu (\nu +1)}} - \ln \bigg (1+{z^2\over \nu } {\bigg )} + {{z^2\tau ^2} \over {\nu }},\\&+ {{\alpha z \big (z^2-1\big )} \over {\big ( \nu + z^2 \big )^2\tau }} w +{\gamma \over T(\varsigma ; \nu +1)}\biggr \}, \end{aligned}$$

where \(\varsigma =\alpha z \tau \), \(w=t(\varsigma ; \nu +1) / T(\varsigma ;\nu +1)\),

$$\begin{aligned} \gamma =\int _{- \infty }^\varsigma \bigg \{{{ (\nu +2)u^2 } \over { (\nu +1)(\nu +1+u^2)}} - \ln \bigg ( 1+{u^2 \over { \nu +1}}\bigg )\bigg \} t(u;\nu +1)du, \end{aligned}$$

and \(\varPsi (x)=\partial \ln \big \{\varGamma (x)\big \}/\partial x\). The expected information matrix for \(\big (\xi ,\omega ,\alpha ,\nu \big )\) is singular under the skew normal distribution since \(S_\nu (y)\) can be shown to be of order \(O\big (\nu ^{-2}\big )\), and hence, it vanishes as \(\nu \rightarrow \infty \). The situation is further complicated under the normal distribution: when \(\alpha =0\), as \(\nu \rightarrow \infty \), the components of the score vector for the location parameter \(\xi \) and the skewness parameter \(\alpha \) tend to \(S^N_\xi (y)= {{ z}/\omega }\) and \(S^N_\alpha (y)= z(2 / \pi )^{1/2}\), which are linearly dependent.

The problem of the score component \(S_{\nu }(y)\) tending to 0 as \(\nu \rightarrow \infty \) can be remedied by using the inverse degrees of freedom, \(\kappa =1/\nu \), in place of \(\nu \). Thus, the parameter now becomes \((\xi ,\omega ,\alpha ,\kappa )\), and the skew normal distribution corresponds to the boundary case \(\kappa =0\).

The component of the score function corresponding to \(\kappa \) is \(S_{\kappa }(y)=-\kappa ^{-2}S_{\nu }(y)\). As \(\kappa \rightarrow 0\), the components of the score function for \(\big (\xi ,\omega ,\alpha ,\kappa \big )\) tend to

$$\begin{aligned} S^{SN}_\xi (y)= & {} {{ z}\over \omega }-{{\alpha } \over {\omega } } w_\phi , \quad S^{SN}_\omega (y)=- {1 \over \omega }+{{ z^2}\over \omega }-{{{\alpha z} \over {\omega } } } w_\phi , \quad S^{SN}_\alpha (y)= z w_\phi ,\\ S^{SN}_\kappa (y)= & {} {1 \over 4} \big \{z^4-2 z^2-1 - \alpha z \big (2 z^2+\alpha ^2 z^2-1) w_\phi \big \}, \end{aligned}$$

where \(w_\phi =\phi (\alpha z) /\varPhi (\alpha z)\), and \(\phi (z)\) and \(\varPhi (z)\) are the density function and cumulative distribution function, respectively, of the standard normal distribution. In particular, \(S^{SN}_\kappa (y)\) is non-zero, and the components \(S_\xi ^{SN} (y)\), \(S_\omega ^{SN} (y)\), \(S_\alpha ^{SN} (y)\), and \(S^{SN}_\kappa (y)\) are linearly independent provided \(\alpha \ne 0\). Consequently, the information matrix for the parameterization \(\big (\xi ,\omega ,\alpha ,\kappa \big )\) is non-singular under the skew normal distribution when \(\alpha \ne 0\).

The problem that \(S_\xi (y)\) and \(S_\alpha (y)\) tend to linear dependency as \(\kappa \rightarrow 0\) when \(\alpha =0\) persists for the \((\xi ,\omega ,\alpha ,\kappa )\) parameterization; this parameterization does, however, serve as a useful intermediate step for defining the centered parameterization \(\big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )\), introduced in Sect. 2.1. The expected information matrix for the parameter \(\big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )\) is non-singular as \(\kappa \rightarrow 0\) for all \(\alpha \).

An expression for the Jacobian \(\partial \big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )/\partial \big (\xi ,\omega ,\alpha ,\nu \big )\) given by Azzalini (personal communication, see also Arellano-Valle and Azzalini 2013) yields

$$\begin{aligned} {{\partial \big (\xi ,\omega ,\alpha ,\kappa \big )}\over {\partial \big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )}}\bigg |_{\kappa =0}=D', \end{aligned}$$

where

\(b=(2/\pi )^{1/2}\), \(\delta =\alpha /(1+\alpha ^2)^{1/2}\), \(\lambda _2=1-b^2\delta ^2\), \(\delta '=\big (1+\alpha ^2\big )^{-3/2}\), \(a_1=4 \pi - 12 \delta ^2- \delta ^2 \pi + 4 \delta ^4 \), \(a_2=18 \delta ^2 \pi - 36 \delta ^4- 2 \delta ^4 \pi + 12 \delta ^6 -3\pi ^2\), \(a_3=6\delta ^4+8 \delta ^2 \pi - 36 \delta ^2+3\pi \), and \(c=12-3\pi -4\delta ^2+2\delta ^4\).

Let \(S_{(\mu ,\sigma ^2,\gamma _1,\gamma _2)}(y)=\big \{ S_{\mu }(y), S_{\sigma ^2}(y), S_{\gamma _1}(y), S_{\gamma _2}(y) \big \}'\) be the score function of the skew t-model for the centered parameterization \(\big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )\) based on a single observation y. As \(\kappa \rightarrow 0\), the score function tends to

$$\begin{aligned} \big \{ S^{SN}_{\mu }(y), S^{SN}_{\sigma ^2}(y), S^{SN}_{\gamma _1}(y), S^{SN}_{\gamma _2}(y) \big \}'=D\big \{ S^{SN}_\xi (y), S^{SN}_\omega (y), S^{SN}_\alpha (y), S^{SN}_\nu (y) \big \}', \end{aligned}$$

and hence, it follows from the preceding calculations that the components of the score vector tend to

$$\begin{aligned} S^{SN}_\mu (y)= & {} {z\over \omega }-{\alpha w\over \omega }, \qquad S^{SN}_{\sigma ^2}(y)={1 \over 2 \omega ^2 \lambda _2}\big \{(z-\alpha w) (z-b\delta )-1\big \},\\ S^{SN}_{\gamma _1}(y)= & {} {\lambda _2^{3/2} \over c}\bigg [ -(z-\alpha w) {2\delta ^4-4\delta ^2+\pi \over \delta ^2} +(z^2-1-\alpha z w){b a_3 \over 3 \delta } - zw { b a_2 \over 6 \delta ^2 \delta '}\\&-{2\over 3}\big \{z^4-2z^2-1-\alpha z(2 z^2+\alpha ^2 z^2-1)w\big \} {b \delta (\pi -3) \over 3 }\bigg ], \\ S^{SN}_{\gamma _2}(y)= & {} {\lambda _2^2\over c}\bigg [ -{1\over 4 }b \pi (z-\alpha w){\delta ^2-2 \over \delta } +{1\over 2} \big (z^2-1-\alpha z w\big )\big (\delta ^2-6+\pi \big ) -{1\over 8} zw { a_1 \over \delta \delta ' }\\&-{1\over 8}\big \{z^4-2z^2-1-\alpha z(2 z^2+\alpha ^2 z^2-1)w\big \} (\pi -4) \bigg ]. \end{aligned}$$

As \(\alpha \rightarrow 0\), these expressions tend to

$$\begin{aligned} S_\mu ^N(y)={z\over \sigma }, \qquad S_{\sigma ^2}^{N}(y)={z^2-1 \over 2 \sigma ^2}, \end{aligned}$$
$$\begin{aligned} S_{\gamma _1}^{N}(y)={z^3-3z\over 6}, \qquad S_{\gamma _2}^{N}(y)={1\over 24}\big (z^4-6z^2+3\big ), \end{aligned}$$
(A.1)

which are the components of the score function for \(\big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )\) under the normal model. In particular, under the normal model, the expected information matrix of the skew t-model for \(\big (\mu ,\sigma ^2,\gamma _1,\gamma _2\big )\) is

$$\begin{aligned} \mathrm{diag}\Big ({1\over \sigma ^2}, \quad {1\over 2 \sigma ^2}, \quad {1\over 6}, \quad {1\over 24} \Big ), \end{aligned}$$
(A.2)

which is invertible. Thus, as \(\kappa \rightarrow 0\), the expected information matrix for the centered parameterization is non-singular, even for the case \(\alpha =0\).

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DiCiccio, T.J., Monti, A.C. Testing for sub-models of the skew t-distribution. Stat Methods Appl 27, 25–44 (2018). https://doi.org/10.1007/s10260-017-0387-x

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