统计代写|非参数统计代写Nonparametric Statistics代考|Power of Tests for Unordered Alternatives

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据 分析
• Foundations of Data Science 数据科学基础

统计代写|非参数统计代写Nonparametric Statistics代考|Power of Tests for Unordered Alternatives

A much easier path to power involves analogy with (2.22): Approximate the alternative distribution using the null variance structure. This results in the non-central chi-square approximation using (1.4).

Because the Mann-Whitney and Wilcoxon statistics differ only by an additive constant, the Kruskal-Wallis test may be re-expressed as
$$\left{\sum_{k=1}^{K}\left(T_{k}-M_{k}\left(N-M_{k}\right) \kappa^{\circ}\right)^{2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right] .$$
Here $\kappa^{\circ}=1 / 2$; this is the null value of $\kappa_{k l}$, and the null hypothesis specifies that this does not depend on $k$ or $l$, and $\psi=1 / \sqrt{12}$, a multiplicative constant arising in the variance of the Mann-Whitney-Wilcoxon statistic. Many of the following equations follow from (4.25); furthermore, (4.25) also approximately describes other statistics to be considered later, and analogous consequences may be drawn for these statistics as well, with a different value for $\psi$. Hence the additional complication of leaving a variable in (4.25) whose value is known will be justified by using consequences of (4.25) later without deriving them again. Here,
$$T_{k}=\sum_{j=1}^{M_{k}} \sum_{l=1, l \neq k}^{K} \sum_{i=1}^{M_{1}} I\left(X_{k j}>X_{l i}\right),$$
the Mann-Whitney statistic for testing whether group $k$ differs from all of the other groups, with all of the other groups collapsed.

The variance matrix for rank sums comprising $W_{H}$ is singular (that is, it does not have an inverse), and the argument justifying (1.4) relied on the presence of an inverse. The argument of $\S 4.2 .2$ calculated the appropriate quadratic form, dropping one of the categories to obtain an invertible variance matrix, and then showed that this quadratic form is the same as that generating $t$. The same argument shows that the appropriate non-centrality parameter is
$$\xi=\left{\sum_{k=1}^{K}\left(\mathrm{E}{A}\left[T{k}\right]-M_{k}\left(N-M_{k}\right)(1 / 2)\right)^{2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right],$$

统计代写|非参数统计代写Nonparametric Statistics代考|Ordered Alternatives

where $U_{i j}$ is the Mann-Whitney-Wilcoxon statistic for testing groups $i$ vs. $j$. Reject the null hypothesis when $J$ is large. This statistic may be expressed as $\sum_{k=1}^{K} c_{k} \bar{R}{k}$. plus a constant, for some $c{k}$ satisfying (4.5); that is, $J$ may be defined as a contrast of the rank means, and the approach of this subsection may be viewed as the analog of the parametric approach of $\S 4.1 .1$.

Critical values for $J$ can be calibrated using a Gaussian approximation. Under the null hypothesis, the expectation of $J$ is
$$\mathrm{E}{0}[J]=\sum{i<j} M_{i} M_{j} / 2=N^{2} / 4-\sum_{i} M_{i}^{2} / 4$$
and the variance is
$$\operatorname{Var}{0}[J]=\frac{1}{12} \sum{i=2}^{K} \operatorname{Var}{0}\left[U{i}\right]=\frac{1}{12} \sum_{i=2}^{K} M_{i} m_{i-1}\left(m_{i}+1\right)$$
here $U_{i}$ is the Mann-Whitney statistic for testing group $i$ vs. all preceding groups combined, and $m_{i}=\sum_{j=1}^{i} M_{j}$. The second equality in (4.20) follows from independence of the values $U_{i}$ (Terpstra, 1952). A simpler expression for this variance is
$$\operatorname{Var}{0}[J]=\frac{1}{72}\left[N(N+1)(2 N+1)-\sum{i=1}^{K} M_{i}\left(M_{i}+1\right)\left(2 M_{i}+1\right)\right] \text {. }$$
This test might be corrected for ties, and has certain other desirable properties (Terpstra, 1952).

Jonckheere (1954), apparently independently, suggested a statistic that is twice $J$, centered to have zero expectation, and calculated the variance, skewness, and kurtosis. The resulting test is generally called the Jonckheere-Terpstra test.

统计代写|非参数统计代写Nonparametric Statistics代考|Ordered Alternatives

Consider first the one-sided Jonckheere-Terpstra test of level $\alpha$. Let $T_{J}=$ $J / N^{2}$. In this case, the subscript $J$ represents a label, and not an index.
Denote the critical value by $t_{J}^{\circ}$, satisfying $\mathrm{P}{\theta^{\circ}}\left[T{J} \geq t_{J}^{\circ}\right]=1-\alpha$.
As in (4.23), reduce the alternative hypothesis to a single dimension by letting the alternative parameter vector be a fixed vector times a multiplier $\Delta$. The power function $\varpi_{J, n}(\Delta)=\mathrm{P}{\theta^{A}}\left[T{J} \geq t_{J}^{0}\right]$ satisfies $(2.15),(2.19)$, and (2.21), and hence the efficiency tools for one-dimensional hypotheses developed in $\S 2.4 .2$ may be used. Expressing $\mu_{J}(\Delta)$ as a Taylor series with constant and linear terms,
$$\mu_{J}(\Delta) \approx \sum_{i=1}^{K-1} \sum_{j=i+1}^{K} \lambda_{i} \lambda_{j}\left(\kappa^{\circ}+\kappa^{\prime}\left[\theta_{j}^{A}-\theta_{i}^{A}\right]\right)$$
for $\lambda_{i}=M_{i} / N$, where again $\kappa^{\circ}$ is the common value of $\kappa_{j k}$ under the null hypothesis, and $\kappa^{\prime}$ is the derivative of the probability in (3.25), as a function of the location shift between two groups, evaluated at the null hypothesis, and calculated for various examples in $\S 3.8$.2. Hence
$$\mu_{J}^{\prime}(0)=\sum_{i=1}^{K-1} \sum_{j=i+1}^{K} \lambda_{i} \lambda_{j} \kappa^{\prime}\left[\theta_{j}^{\dagger}-\theta_{i}^{\dagger}\right]$$
Recall that $\kappa_{i j}$ depended on two group indices $i$ and $j$ only because the locations were potentially shifted relative to one another; the value $\kappa^{\circ}$ and its derivative $\kappa^{\prime}$ are evaluated at the null hypothesis of equality of distributions, and hence do not depend on the indices. Furthermore, from (4.21), $\operatorname{Var}\left[T_{J}\right] \approx \frac{1}{36}\left[1-\sum_{k=1}^{K} \lambda_{k}^{3}\right] / N$. Consider the simple case in which $\lambda_{k}=1 / K$ for all $k$, and in which $\theta_{j}^{\dagger}-\theta_{i}^{\dagger}=(j-i)$. Then $\mu_{J}^{\prime}(0)=\kappa^{\prime}\left(K^{2}-1\right) / 6 K$, $\operatorname{Var}\left[T_{J}\right] \approx \frac{1}{36}\left[1-1 / K^{2}\right] / N$, and
$$e_{J}=\kappa^{\prime} \frac{\left(K^{2}-1\right) / 6 K}{\sqrt{\frac{1}{36}\left[1-1 / K^{2}\right]}}=\frac{\kappa^{\prime}\left(K^{2}-1\right)}{\sqrt{K^{2}-1}}=\kappa^{\prime} \sqrt{K^{2}-1}$$

统计代写|非参数统计代写Nonparametric Statistics代考|Power of Tests for Unordered Alternatives

\left{\sum_{k=1}^{K}\left(T_{k}-M_{k}\left(N-M_{k}\right) \kappa^{\circ}\right)^{ 2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right] 。\left{\sum_{k=1}^{K}\left(T_{k}-M_{k}\left(N-M_{k}\right) \kappa^{\circ}\right)^{ 2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right] 。

\xi=\left{\sum_{k=1}^{K}\left(\mathrm{E}{A}\left[T{k}\right]-M_{k}\left(N-M_{ k}\right)(1 / 2)\right)^{2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right],\xi=\left{\sum_{k=1}^{K}\left(\mathrm{E}{A}\left[T{k}\right]-M_{k}\left(N-M_{ k}\right)(1 / 2)\right)^{2} / M_{k}\right} /\left[\psi^{2}(N+1) N\right],

统计代写|非参数统计代写Nonparametric Statistics代考|Ordered Alternatives

Jonckheere (1954) 显然独立地提出了一个统计量是两倍Ĵ，以零期望为中心，并计算方差、偏度和峰度。结果测试通常称为 Jonckheere-Terpstra 测试。

统计代写|非参数统计代写Nonparametric Statistics代考|Ordered Alternatives

μĴ(Δ)≈∑一世=1到−1∑j=一世+1到λ一世λj(ķ∘+ķ′[θj一种−θ一世一种])

μĴ′(0)=∑一世=1到−1∑j=一世+1到λ一世λjķ′[θj†−θ一世†]

广义线性模型代考

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MATLAB代写

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