Intereting Posts

Books that every student “needs” to go through
Prove divergence and conclude that there is no universal “smallest” comparison series to test divergence
How does one calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?
How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)
Not sure how to go about solving this integral
Matrix graph and irreducibility
$\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
Edge coloring of the cube
(Why) is topology nonfirstorderizable?
Embedding $S_n$ into $A_{n+2}$
Is 2201 really the only non-palindromic number whose cube is palindromic?
A series of Lemmas about $C^{\infty}$ vector fields
How to differentiate product of vectors (that gives scalar) by vector?
$p(x)$ irreducible polynomial $\iff J=\langle p(x)\rangle$ is a maximal ideal in $K$ $\iff K/J$ is a field
Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$

Assume $\boldsymbol{y}=\boldsymbol{\iota}\beta_1+\boldsymbol{x}\beta_2+\boldsymbol{u}$ where $\boldsymbol\iota$ is the n-vector of ones and $\{u_i\}$ are i.i.d. with $E(u_i)=0$ and $E(u_i^2)=\sigma^2$. Now, assume that $\boldsymbol{x’x}/n\to c>0$ and $\boldsymbol{\iota’x}/n\to 0$ as $n\to\infty.$ Suppose there is an estimator $\hat\gamma$ (independent from $\boldsymbol{u}$) for the ratio of the coefficients $\gamma=\beta_1/\beta_2$, and it follows that

$$

\sqrt{n}(\hat\gamma-\gamma)\overset{A}{\sim}\mathcal{N}(0,\lambda^2).

$$

Define

$$

\hat\beta_2=\frac{(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)’\boldsymbol{y}}{(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)'(\boldsymbol{x}+\boldsymbol{\iota}\hat\gamma)}.

$$

What is the asymptotic distribution of this estimator? I’ve tried to decompose $\hat\beta_2$ but however I do that I end up with correlation problem, i.e. I cannot derive the asymptotic distribution of the sum of all parts from decomposition.

Any suggestion is appreciated.

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- $T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer

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- Upper bound for the strict partition on K summands
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- Solving the recurrence $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{2}\log(n)$
- Giving an asymptotically tight bound on sum $\sum_{k=1}^n (\log_2 k)^2$
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- The Dido problem with an arclength constraint
- Integration by parts in Riemann-Stieltjes integral from Rudin
- 1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?
- computing the orbits for a group action
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- Number of 5 letter words with at least two consecutive letters same
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- Measurable Maps and Continuous Functions
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- How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$
- Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?
- How are mathematicians taught to write with such an expository style?