In the distance correlation test, we can consider different distances. In particular, we can consider ℓp distance，What p should we use?
Added by Laura G.
Step 1
Distance correlation is a measure of dependence between two random variables or datasets, which can capture both linear and non-linear relationships. Show more…
Show all steps
Your feedback will help us improve your experience
Rachel Gore and 93 other AP CS educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
wald test based on the distance measure in econometrics
Rachel G.
Continuing with Exercise $6.3 .14$, besides estimation there is also a nice geometric interpretation for testing. For the model $(6.3 .26)$, consider the hypotheses $$ H_{0}: \theta=\theta_{0} \text { versus } H_{1}: \theta \neq \theta_{0} $$ where $\theta_{0}$ is specified. Given a norm $\|\cdot\|$ on $R^{n}$, denote by $d(\mathbf{X}, V)$ the distance between $\mathbf{X}$ and the subspace $V$; i.e., $d(\mathbf{X}, V)=\|\mathbf{X}-\widehat{\boldsymbol{\mu}}\|$, where $\hat{\boldsymbol{\mu}}$ is defined in equation $(6.3 .27) .$ If $H_{0}$ is true, then $\hat{\mu}$ should be close to $\mu=\theta_{0} 1$ and, hence, $\left\|\mathbf{X}-\theta_{0} \mathbf{1}\right\|$ should be close to $d(\mathbf{X}, V) .$ Denote the difference by $$ R D=\left\|\mathbf{X}-\theta_{0} \mathbf{1}\right\|-\|\mathbf{X}-\widehat{\boldsymbol{\mu}}\| $$ Small values of $R D$ indicate that the null hypothesis is true, while large values indicate $H_{1}$. So our rejection rule when using $R D$ is Reject $H_{0}$ in favor of $H_{1}$ if $R D>c$. (a) If the error pdf is the Laplace, $(6.1 .6)$, show that expression $(6.3 .31)$ is equivalent to the likelihood ratio test when the norm is given by $(6.3 .28)$. (b) If the error pdf is the $N(0,1)$, show that expression (6.3.31) is equivalent to the likelihood ratio test when the norm is given by the square of the $l_{2}$ norm, $(6.3 .29) .$
Maximum Likelihood Methods
Maximum Likelihood Tests
Suppose data is obtained from 20 pairs of (x, y) and the sample correlation coefficient is 0.81. (a) Test the hypothesis that H0: ρ = 0 against H1: ρ ≠ 0 with α = 0.01. Reject the null hypothesis. (b) Test the hypothesis that H0: ρ = 0.39 against H1: ρ ≠ 0.39 with α = 0.01. Reject the null hypothesis. (c) Construct a 95% two-sided confidence interval for the correlation coefficient. Round your answer to 3 decimal places: 0.659 ≤ ρ ≤ 0.961.
Madhur L.
Recommended Textbooks
Computer Science and Information Technology
Introduction to Programming Using Python
Computer Science - An Overview
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD