Kunchenko's method of polynomial maximization provides a semiparametric apparatus for parameter estimation under non-Gaussian errors, but its classical power basis relies on finite higher-order integer moments. This paper introduces the Parametrically Adaptive Transition Polynomial (PATP), a signed-parity fractional-power family controlled by a continuous parameter alpha in [0,1]. The quadratic exponent map p_i(alpha) connects the fractal regime p_i(0)=1/i, the degenerate linear point p_i(1/2)=1, and the signed-parity integer-power regime p_i(1)=i. For the degree-S=2 case we derive a closed-form variance-reduction coefficient g_2(alpha) in terms of signed and absolute fractional moments, identify the singular behavior at alpha=1/2, and state the moment and regularity conditions under which the formula is meaningful. The construction should be read as a Form-B PATP analogue within Kunchenko's generalized apparatus, not as an exact recovery of the canonical even-power PMM basis at alpha=1. Numerical illustrations on canonical distributions are used to examine the finite-sample behavior of the signed-parity estimator and to mark the boundary of applicability for extremely heavy-tailed cases such as Cauchy.

翻译：Kunchenko多项式最大化方法为非高斯误差下的参数估计提供了半参数工具，但其经典幂基依赖于有限高阶整数矩。本文提出参数自适应过渡多项式（PATP），这是一个由[0,1]区间内连续参数α控制的有符号奇偶分数阶幂族。二次指数映射p_i(α)连接了分形区域p_i(0)=1/i、退化线性点p_i(1/2)=1和有符号奇偶整数幂区域p_i(1)=i。针对阶数S=2的情况，我们推导出以有符号和绝对分数阶矩表示的闭式方差缩减系数g_2(α)，识别了α=1/2处的奇异性，并阐述了该公式成立所需的矩条件与正则性条件。该构造应理解为Kunchenko广义框架下的形式B PATP类比，而非在α=1处对典型偶次幂PMM基的精确恢复。通过对典型分布的数值实验，检验了有符号奇偶估计量的有限样本行为，并标注了柯西分布等极重尾情形的适用边界。