# Tagged Questions

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component.

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### Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$\int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{\star}$$ (The limits of the integrals are all ...
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### Monotonicity of Sample Mean

$X_1,X_2,\ldots$ are drawn i.i.d. from a distribution with mean $\mu$. Define $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ Prove that $\forall t \quad E[|\bar{X}_t - \mu|] \geq E[|\bar{X}_{t+1} - \mu|]$
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### Definition of Bias for Baysian Estimation

I know for parameter estimation the estimator $\hat{\theta}(X)$ of $\theta$ based on observation $\theta$ is said to be unbiased if \begin{align} E[ \hat{\theta}(X)]=\theta, \ \forall \theta. \end{...
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### Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$\hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x)$$ where $1_A(x)$ is the ...
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### Show that estimates are unbiased

The following is a problem in my book that I don't really understand: We take a random sample: $x_1,x_2,\ldots,x_n$ from a population that is $N(μ,σ)$ where $\mu$ and $\sigma$ are unknown. We build ...
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$X_1,X_2,\ldots,X_n$ are iid random variables $B(1,\theta)$ where $0< \theta<1$. Let $w = 1$ if $\sum_i X_i = n$ and $0$ otherwise. What is the best unbiased estimator of $\theta^2$. Attempt: ...
Suppose to have a set of data $\{y_i, u_i\}_{i=1}^m$, where $y_i \in \mathbb{R}$ and $u_i \in \mathbb{R}^n$. The claim is that $$y_i = u_i^\top \theta + \varepsilon$$ where $\theta \in \mathbb{R}^n$...
Assume there is a Set of data which follows a known distribution (e.g. normal distribution). $$S = \left\{ a_0,a_1 ... a_n \right\}$$ When taking a subset from S S_k = \left\{ a_0,a_1 ... a_k \...