Articles of estimation

Asymptotic Distribution of the LS Estimator using an Extra Ratio of the Coefficients

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 […]

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 10(b). Here is what I did so far: \begin{align} \int_U |Du|^{2p} \, dx &= \sum_{n=1}^\infty \int_U u_{x_i} |Du|^{2p-1} \, dx \\ &= -\sum_{i=1}^n \int_U u(x_i)(2p-1)|Du|^{2p-2} |D^2 u| \, dx […]

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.

What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is also a singleton. If $|A|=2,$ then $|A+A|=3.$ If $|A|=3,$ then $|A+A|$ can be at most […]

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) – \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My Question: Is there a better estimate for the constant $C$ than the one given above for the special case of the binomial distribution? Reason for my question: The given inequality for $C$ holds for any […]

Probability vs Confidence

My notes on confidence give this question: An investigator is interested in the amount of time internet users spend watching TV a week. He assumes $\sigma = 3.5$ hours and samples $n=50$ users and takes the sample mean to estimate the population mean $\mu$ Since $n=50$ is large we know that $\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ approximates the Standard […]

Exponential decay estimate for $(x,t) \in U_T$

Suppose $u$ is a smooth solution of $$\begin{cases}u_t – \Delta u + cu = 0 & \text{in }U \times (0,\infty) \\ \qquad \qquad \quad \, \,u=0 & \text{on } \partial U \times [0,\infty) \\\qquad \qquad \quad \, \, u=g & \text{on }U \times \{t=0\} \end{cases}$$ and the function $c$ satisfies $c \ge \gamma > 0$. […]

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need to estimate an expression of the form $|x+y+z|$ from below so that the estimate depends on the absolute values of $x,y,z$, and I think the triangle inequality should be enough, but […]

Estimate growing graphs

Lets make my scenario not generic just so that i could use particular terms Say i have a graph of population per year of someplace over some decades Lets say the graph is like this How can i calculate expected\estimated population next year and son on and so forth

A calculation that goes awfully wrong if we let $\pi=22/7$

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation in which if we use $\pi=22/7$, some noticeable error which makes quite a significant difference, appears. Kind […]