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框架，但其经典幂基依赖于有限高阶整数阶矩。本文提出参数自适应过渡多项式（PATP），这是一种由连续参数α∈[0,1]控制的有符号奇偶分数阶幂族。二次指数映射p_i(α)连接分形域p_i(0)=1/i、退化线性点p_i(1/2)=1以及有符号奇偶整数幂域p_i(1)=i。针对S=2阶情形，我们推导出闭式方差缩减系数g_2(α)（以有符号与绝对分数阶矩表示），识别出α=1/2处的奇异行为，并给出该公式有意义所需的矩条件与正则性条件。该构造应理解为Kunchenko广义框架内的B型PATP类比，而非对α=1处经典偶幂PMM基的精确复现。基于典型分布的数值实验用于检验有符号奇偶估计量的有限样本性质，并标定柯西分布等极重尾情形的适用边界。