THE EFFICIENCIES OF ALTERNATIVE ESTIMATORS FOR AN ASYMPTOTIC REGRESSION EQUATION
渐近回归方程中不同估计量的效率AI 解读 ↓
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渐近回归方程中不同估计量的效率
- 研究问题
- 对于文中渐近回归方程的参数，在独立同方差正态观测和相关且方差增长的连续自回归观测下，不同估计量如何权衡效率与偏差？
- 方法与思路
- 作者比较观测值线性函数之比与二次函数之比的估计方法，重点分析四个等距观测的情形，并考察相邻观测间的回归估计。
- 材料陈述的贡献
- 摘要称 Patterson 估计量在两种模型中均有较高渐近效率；自回归式估计可能进一步降低渐近方差，但较大观测方差下可能有明显负偏差。
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- 应查阅原文方程及参数定义，区分四个观测下的详细分析与更多观测下的判断，并核对最大似然结论和偏差比较的条件。
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- 仅据 OpenAlex 索引摘要，版本未注明；原方程缺失、参数符号不完整，需查原 PDF，不能重构模型或将有限情形结论普遍化。
原文摘要
Methods for the estimation of the parameter p in equation (1) have been discussed, this equation representing the expectation of observations on a quantity y for a specified value of x. The methods all involve taking as the estimator the ratio of either two linear functions or two quadratic functions of the observed y. Their relative efficiencies and biases are considered under two models for the generation of observations, the y either being independent and normally distributed about their expectations with constant variance or arising from a continuous autoregressive process in which the variance increases and successive values of y for an individual are correlated. Detailed investigation has been possible only for the very simple situation of four observations equally spaced in x. As is well known, the Patterson estimator, a ratio of two linear functions, is of high asymptotic efficiency for the first model, and it proves to be also highly efficient for the second. An interesting alternative is the calculation of a linear regression of yi+1on yi. This is also highly efficient, and has the advantage of simultaneously estimating the parameter a; moreover, under the second model, the estimators of p and α maximize the likelihood for any number of equally spaced observations. However, the estimator of p appears likely to have a considerable negative bias if the variance of observations about their expectation is at all large. Other quadratic estimators have been examined, but showed no special merits. The indications are that the Patterson estimator is always a fairly safe one to use, for any number of equally spaced observations; its efficiency is never low, and it is unlikely to be seriously biased. The estimator calculated autoregressively is likely to be more efficient (in the narrow sense of having a smaller asymptotic variance), especially for the second model and for a larger number of observations, and if the variance per observation is low this advantage will not be offset by its greater expectation of bias.