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通过清空盒子构造 πPS 抽样设计AI 解读

We present a simple but universal technique for the construction of $\pi PS$ sampling designs. A tool that is used in the construction consists of playing a game in which objects are removed from $N$ boxes, $n$ at a time, and at most one from each box at a time. Necessary and sufficient conditions on $N, n$ and the contents of the boxes are established such that all boxes can be emptied by this process. It is shown that every $\pi PS$ design can be derived from such a game.…

Hadamard 矩阵及其应用AI 解读

An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the…

拟合不完整模型所致混杂的一种度量AI 解读

This paper proposes and develops a method for selecting a design to estimate a set of linear parametric functions in cases wherein the adequacy of the preliminary linear model is in doubt. The proposed method relies on the norm of the aliasing matrix. If the expected value of the estimator $\hat{\psi}$ of a set of linear functions $\psi = L_1\theta_1$ using a design $\Gamma$, under the true model is $E\lbrack\hat{\psi}\rbrack = \psi + A_\Gamma \theta_2$, then the norm $\|A_\Gamma\| =$…

用于同方差检验的独立逐步残差AI 解读

Regression models which specify independent, homoscedastic and normally distributed errors may be analyzed in a stepwise manner to produce calculated residuals having this same property. If the nth residual is calculated as the deviation of the nth observation from its predicted value based on a least squares fit to only the first n observations then the resulting sequence of residuals, appropriately normalized, are not only mutually independent and homoscedastic but are…