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Under the null hypothesis, if we have $$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$ the test statistic can be construct as: $$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$ …

For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct \begin{equation*} \Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n \end{equation*} for testi …

Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel fu …

Suppose $$X_{ij} = \mu_j + \varepsilon_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N_j$$ ANOVA can allow us to test whether $\mu_1 = \cdots = \mu_J$. In traditional ANOVA, however, the number o …

Now I have a statistic $T_n$ for testing $H_0 \leftrightarrow H_1$, and I have proved that: $$n T_n \rightarrow_d \chi_K^2$$ under $H_0$. Then an asymptotic $\chi^2$ test can be used, an asymptotic le …

Let $Y_t \in \mathbb{R}$ be a response variable and $X_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X_1, Y_1), \cdots, (X_n, Y_n)$. \begin{equation} Y_{t} = \mu( …

The asymptotic expression for $(\hat{m} - m)(x)$ is presented in the Appendix of Fan and Qao. So this question is meaningless.