N = 5, 6, 7, 8: Nested hypothesis tests and truncation dependence of $|V_{cb}|$

The determination of $|V_{cb}|$ from exclusive semileptonic $B\to D^*\ell\nu$ decays is sensitive to the choice of form factor parametrization. Larger $|V_{cb}|$ values are obtained fitting the BGL versus the CLN parametrization to recent Belle measurements. For the BGL parametrization, published fits use different numbers of parameters. We propose a method based on nested hypothesis tests to determine the optimal number of BGL parameters to fit the data, and find that six parameters are optimal to fit the Belle tagged and unfolded measurement. We further explore the differences between fits that use different numbers of parameters. The fits which yield $|V_{cb}|$ values in better agreement with determinations from inclusive semileptonic decays, tend to exhibit tensions with heavy quark symmetry expectations. These have to be resolved before the determinations of $|V_{cb}|$ from exclusive and inclusive decays can be considered understood.