This paper describes a novel statistical bandwidth extension (BWE) technique based on a Gaussian mixture model (GMM) and a sub-band basis spectrum model (SBM), in which each dimensional component represents a specific acoustic space in the frequency domain. The proposed method can achieve the BWE from speech data with an arbitrary frequency bandwidth whereas the conventional methods perform the conversion from fixed narrow-band data. In the proposed method, we train a GMM with SBM parameters extracted from full-band spectra in advance. According to the bandwidth of input signal, the trained GMM is reconstructed to the GMM of the joint probability density between low-band SBM and high-band SBM components. Then high-band SBM components are estimated from low-band SBM components of the input signal based on the reconstructed GMM. Finally, BWE is achieved by adding the spectra decoded from estimated high-band SBM components to the ones of the input signal. To construct the full-band signal from the narrow-band one, we apply this method to log-amplitude spectra and aperiodic components. Objective and subjective evaluation results show that the proposed method extends the bandwidth of speech data robustly for the log-amplitude spectra. Experimental results also indicate that the aperiodic component extracted from the upsampled narrow-band signal realizes the same performance as the restored and the full-band aperiodic components in the proposed method.