Non-Central Beta Distribution
X ~ Beta(a, b, λ)
Noncentral / Ratio-of-Chi-Squares Family • Published July 22, 2026
The exact distribution of sample R² under a nonzero population R², built the same way the already-published non-central F extends the already-published F: replacing the ratio's numerator Chi-Square with its non-central counterpart, giving a Poisson-mixture-of-Beta density with no scipy equivalent, verified against a truncated-mixture quadrature and a 3-million-draw simulation, no closed-form MLE for any of the three parameters, and simulation via the ratio-of-(noncentral, central)-Chi-Squares construction.
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About This Reference Sheet
The exact distribution of sample R² under a nonzero population R², built the same way the already-published non-central F extends the already-published F: replacing the ratio's numerator Chi-Square with its non-central counterpart, giving a Poisson-mixture-of-Beta density with no scipy equivalent, verified against a truncated-mixture quadrature and a 3-million-draw simulation, no closed-form MLE for any of the three parameters, and simulation via the ratio-of-(noncentral, central)-Chi-Squares construction.
Support
x ∈ [0, 1]
Parameters
a > 0, b > 0 (shape parameters), λ ≥ 0 (noncentrality)