Ahmad Bazzi
  • Member for 3 years, 6 months
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Interpreting a matrix calculation
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7 votes

Consider the random variable $$y = c^T x$$Then the mean of $y$ is $$E(y) = c^T E(x) = c^T \mu$$ and variance of $y$ is $$var(y) = E(y - c^T \mu)^2 = E(y^2) - (c^T \mu)^2 = E(y^2) + c^T \mu\mu^T c \...

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Derivation of the distribution of $\hat{\phi}=[\hat{\phi}_1, \cdots, \hat{\phi}_p]$ in AR(p) models
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5 votes

It doesn't have to be thru Yule-Walker. You can use the OLS approach, namely, $$X_t = \pmb X_{t-1}^T \pmb \phi + n_t , \qquad t = p+1,p+2,\ldots$$ where $\pmb X_{t-1} = [X_{t-1} \ldots X_{t-p} ]$and $...

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How to find distribution function of sum of 2 random variables that are uniformly distributed?
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3 votes

PDF We can first derive the PDF using convolution of two PDFs: Case 1: If $0 \leq y \leq 1$, then $f_{X_1}(y - x_2) = 1$ if $ 0\leq x_2 \leq y$, and $f_{X_1}(y-x_2) = 0$ if $x_2 > y$. This means ...

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Why is $y\sim\mathcal{N}(Xβ,\sigma^{2} I_n)$?
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2 votes

Adding a constant $X\beta$ to a normally distributed random variable is again normal. The mean of the $y = X\beta + \epsilon$ is: $$E(y) = E(X\beta + \epsilon) = E(X\beta) + E(\epsilon) = X\beta + 0 ...

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Computing the training and testing error on $k$ nearest neighbors
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2 votes

I will share a picture with you to clear your ambiguities. Assume you've got the training data in 2D space that are labeled either red or green. On the left figure, you've got a test data point (in ...

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Exact Confidence Interval for Poisson using Gamma-Poisson Relationship
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2 votes

Given \begin{equation} X \sim {{\chi}_{2n}}^2 \end{equation} and \begin{equation} Y \sim P(\lambda) \end{equation} Let's prove (the complement of your expression) \begin{equation} P(X < 2\lambda)...

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$SSR = \sum(\hat y_t - \bar y)^2 = \sum\hat y_t^2 - n \bar y^2$
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1 votes

Let's decompose \begin{equation} SSR = \sum\limits_{t=1}^n(\hat y_t - \bar y)^2 = \sum\limits_{t=1}^n \hat y_t^2 - 2\sum\limits_{t=1}^n\hat y_t \bar y + \sum\limits_{t=1}^n \bar y^2 \end{equation} ...

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What are the differences between $y= x\beta+\varepsilon$ and $ y = x\hat{\beta}, r=y-\hat{y}$
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1 votes

$y = X \beta + \epsilon$ is your model. After estimating $\beta$, which is denoted as $\hat{\beta}$, using any estimation method, you could then use it to estimate your $y$, i.e. $\hat{y} = X\hat{\...

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Finding OLS estimator for $\beta$ where $y_i=\beta+ 2 \beta x_i+\epsilon_i$
1 votes

Part $(a)$ seems legit. Computing $E(\hat{\beta})$ Taking it from where you stopped \begin{equation} \hat{\beta}=\frac{\sum y_i+2y_ix_i}{\sum1+4x_i+4x_i^2} \end{equation} Then \begin{equation}...

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Calculate Error Variance without exact information (OLS)
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0 votes

Using the definition of inner product $$y^Ty =\overbrace{\sum_{t=1}^{T} y^2_t}^{\textsf{given}} $$ $$\hat {\beta^T}X^Ty = \hat{\beta}_1\overbrace{\sum_{t=1}^{T} x_{t1} y_t}^{\textsf{given}} + \hat{\...

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Deriving the sampling distribution of MLE for Normal distribution
0 votes

Disclaimer: This is not my proof, but rather an understanding of one of Lukács theorems. First transform $X_k$ using orthogonal matrix $V$ so that it contains entries equal to $\frac{1}{ \sqrt{n}}$ ...

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