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Modified BIC

Zhang and Siegmund (2007, Biometrics) discussed that in the context of change-point problems, the traditional BIC does not satisfy the technical assumptions of Schwarz (1978, Annals of Statistics) and proposed a modification to improve its performance. The MBIC in Joinpoint regression is derived as an asymptotic approximation of the Bayes factor and is of the form:

\(MBIC \left( k \right) = BIC \left( k \right) +\ \frac{\ln \| X^\prime_k \left( \hat{\tau} \right) X_k \left( \hat{\tau} \right) \|}{n}\ -\ \frac{2}{n} \ln \Gamma \left( \frac{n-k-3}{2} \right)\ -\ \frac{k+3}{n} \ln \left( SSE \left( k \right) \right),\)

where n is the number of observations, Γ(z) is the gamma function:

\(\Gamma \left( z \right) = \int_0^\infty t^{z-1} e^{-t} dt\)

and

\(X_k(\hat{\tau}) \ = \ \left( \stackrel{\stackrel{1 \hspace{2mm} x_1 \hspace{2mm} (x_1 \ - \ \hat{\tau_1})^+ \hspace{2mm} \ldots \hspace{2mm} (x_1 \ - \ \hat{\tau_k})^+}{\hspace{1mm} \vdots \hspace{3mm} \vdots \hspace{5mm} \vdots \hspace{4mm} \backslash \hspace{5mm} \vdots \hspace{3mm}}}{\stackrel{1 \hspace{2mm} x_n \hspace{2mm} (x_n \ - \ \hat{\tau_1})^+ \hspace{2mm} \ldots \hspace{2mm} (x_n \ - \ \hat{\tau_k})^+}{ \ }} \right)\)

for a k-joinpoint model with the values of the independent variable \(\left( x_1, \ldots, x_n \right)\)

and the joinpoints estimated as \(\left( \hat{\tau_1}, \ldots, \hat{\tau_k} \right)\). Note that a+ = max (a,0).

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