Recent content by Usagi

The question: Suppose $Y$ is discrete and only takes on non-negative integers and that the conditional distribution of $Y$ given $X=x$ is Poisson, that is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$$ where $y = 0, 1, 2, \cdots$. First compute $E(Y|X=x)$ and $Var(Y|X=x)$, does this...

Let $f(x|\theta)$ be a family of densities where the parameter space is finite, i.e., $\theta \in \Theta = \{\theta_1, \cdots, \theta_p\}$. Now consider the likelihood function statistic, defined to be $T(\mathbf{X}) = (f_{\theta}(\mathbf{X}))_{\theta \in \Theta} = (T_1(\mathbf{X}), \cdots...

The context of the following identity is in the Classical Normal Linear Regression Model, ie, $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+ \boldsymbol{u}$ where $\boldsymbol{u}$ is a $n \times 1$ matrix and $u_i \sim iid.N(0, \sigma^2)$ for $i = 1, 2, \cdots, n$ Show that...

Question: By mapping the OLS regression into the GMM framework, write the formula for the standard error of the OLS regression coefficients that corrects for autocorrelation but *not* heteroskedasticity. Furthermore, show that in this case, the conventional standard errors are OK if the $x$'s...

Show, using the $\epsilon-\delta$ definition of continuity, that the modified Dirichlet function, i.e., $f(x) = x$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational, is discontinuous at all points $c \neq 0$ My attempt: Is the following argument right (using the sequential definition of...

Ah yup, makes perfect sense, thank you!

I understand this intuitively, but how can I make it rigorous using $\epsilon-\delta$ arguments?

Thanks! However, how do you that this neighbourhood is contained in (a,b)?

Let $a$, $b$ be reals and $f: (a,b) \rightarrow \mathbb{R}$ be twice continuously differentiable. Assume that there exists $c \in (a,b)$ such that $f(c) = 0$ and that for any $x \in (a,b)$, $f'(x) \neq 0$. Define $g: (a,b) \rightarrow \mathbb{R}$ by $\displaystyle g(x) = x -\frac{f(x)}{f'(x)}$...

Yup that is one way, however I was wondering whether my proof using the sequential characterisation is correct?

Show that $\lim_{x \rightarrow 0} \frac{1}{x}$ does not exist. Is my following argument correct? I will show there exists a sequence $(x_n) \subset \mathbb{R} \backslash \{0\}$ satisfying $x_n \neq 0$ and $(x_n) \rightarrow 0$, but $\lim_{n \rightarrow \infty} f(x_n)$ does not exist. Consider...

I can confirm I didn't miss anything. I've copied the q exactly as it is. Also i think part a) is harder than it seems. This is my working so far: So, define $U_k$ as the random variable that denotes the amount of petrol that car $k$ fills, $k = 1, 2, 3, \cdots$. Thus, $U_k, k = 1, 2, 3...

Thanks, how about part (a)?

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is 100 litres. Suppose cars arrive to the station according to a Poisson process with rate \lambda, and that each car fills independently of all other cars and of the arrival...

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...