Articles of estimation

Estimating sum of n elements by throwing away half of elements

I’ve got a task where i need to proove the asymptotic big-Theta equation: $$ \log n! = \Theta(n \, \log n) $$ $ \ $ Since $f(\mathit{n}) \in \Theta(g(\mathit{n}))$ means that $g(n)\cdot k{_1} \leq f(n) \leq g(n)\cdot k{_2}$, one way i can proove the equation above is to find the coefficients $k{_1}$ and $k{_2}$, considering […]

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z \right| $$ tends to zero as $R \to \infty$. By using the ML-inequality one has $$ \left| \int_{\gamma_2} e^{iz^2}\mathrm{d}z […]

Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. I am stuck on calculating the Fisher Information, which is given by $-nE_{\theta}\left(\dfrac{d^{2}}{d\theta^{2}}\log f(X\mid\theta)\right)$. So far, I have the second derivative of the log likelihood as $\dfrac{-n}{\theta^{2}}+\dfrac{\theta(n-\sum […]

Proving an integral is finite

I have the following integral: $$\displaystyle \int_{\mathbb{R}^2} \left( \int_{\mathbb{R}^2} \frac{J_{1}(|\alpha|)J_{1}(|k- \alpha|)}{|\alpha||k-\alpha|} \ \mathrm{d}\alpha \right)^2 \ \mathrm{d}k,$$ where both $\alpha$ and $k$ are vectors in $\mathbb{R}^2$, with $k \neq 0$, and $J_{\nu}$ denotes the Bessel function of the first kind. I’m having some trouble with the best way to approach this integral. If we focus on […]

Continuously sampled event: Estimating the value of a future data point, based on past measurements and their tendency

Problem I’d appreciate some ideas on how to define a formula to estimate the value of a future data point for a continuously sampled event, based on past measurements and their tendency. At any given time, I have exactly 15 past measurements of the event. Let’s assume that what I’m trying to predict is the […]

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 […]