Questions tagged [parameter-estimation]

Question 1

Let $X_1,...,X_n$ be a random sample with exponential distribution Exp$(\lambda^2+\lambda)$ What is the method of moments estimator of $\lambda$? So PDF is $F(x;\lambda) = (\lambda^2+\lambda)\exp(-(\...

Question 2

Let $X_1, X_2, \dots , X_n$ be an i.i.d. sample from the mixture distribution \begin{equation} \label{eqn:mixture distribution} p_{\epsilon,\theta} = (1 - \epsilon)p_{\theta} + \epsilon \delta, \end{...

Question 3

Theorem (Cramér-Rao inequality). Consider a sample from a parametric model satisfying regularity conditions. Let $\theta^*$ be an unbiased estimator of $\tau(\theta)$. Then for any $\theta \in \Theta$,...

Question 4

Consider a sequence of i.i.d. random variables $X_1,\, \dots,\, X_n$ whose mean is denoted as $x_0$ and variance $\sigma^2 < \infty$. From the Strong Law of Large Numbers, the empirical mean $\bar ...

Question 5

The scalar target $z$ is modeled as $$f(x,y) = \underline c^T \underline b, \qquad \underline b=\begin{bmatrix} 1 \\ x \\ ln(y+d) \\ x \cdot ln(y+d) \end{bmatrix},$$ with unknown parameter $d$ and ...

Question 6

I am trying to learn the theory of estimation, primarily from a mathematical (measure-theoretic/probabilistic) perspective. More specifically, I'm looking for resources that cover one-parameter and ...

Question 7

Question is in the title. Given that $\delta:=\delta(\mathbf X_n)$ is MVUE (minimum variance unbiased estimator) of a scalar parameter $\theta$, we are asked to show that for all natural numbers $k$, $...

Question 8

Summary I am looking for a convex and robust formulation to fit an ellipse to a set of points. Specifically, can handle an extreme condition number of the Scattering Matrix. Full Question The ...

Question 9

Does it make sense to study statistical models in which the maximum likelihood estimator (MLE) $ \hat{\theta}_n $ exists only for infinitely many $ n $, but not necessarily for all $ n $? Suppose, for ...

Question 10

Throughout mathematical statistics, the Fisher information comes up quite frequently as a measure of information. I understand that in the case where you have a single parameter, the Fisher ...

Question 11

Let $X_1, X_2,...,X_n$ be $iid$ continuous uniform $\mathcal{U} (0,\theta)$ and let $T=Max(X_i)$ Show that the family of distributions of T is complete. Step I: Find the CDF (using independence ...

Question 12

In All of Statistics, chapter 6.2, it states "a parametric model takes the form $F=\{f(x; θ) : θ ∈ Θ\}$ ...". Then, in chapter 13.1, it states, "The Simple Linear Regression Model $Y_i =...

Question 13

Let $X_1,X_2,...,X_n (n\geq 2)$ be a random sample from a distribution with probability density function: $$ f(x;\theta) = \begin{cases} \theta x^{\theta-1}, \hspace{1 cm} 0\leq x \leq 1 \\ ...

Question 14

Consider $X_1, X_2, \dots, X_n$ as a random sample from a distribution with the probability density function (pdf): $$ f(x) = \begin{cases} e^{-(x - \theta)}, & \text{for } x < \theta, -\infty&...

Question 15

Let $X_1,X_2,\dots,X_n$ be idd Poisson distributions with unknown mean $\lambda>0$. Find the $100(1-\alpha)\%$ confidence interval for $\lambda$, where $\alpha\in(0,1)$. I think we need to find an ...

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Questions tagged [parameter-estimation]

The method of moments and maximum likelihood estimator for exponential distribution $\lambda^2 + \lambda$

Expectation under true distribution with mixture samples

Uniqueness of the function $\tau(\theta)$ in the Cramér-Rao inequality

Transfer of strong consistency

Identifiability & estimation: $d$ and $\underline c$ from $\lvert S \rvert = 6$

Reference request for theory of estimation

Show that powers of an MVUE is an MVUE

Robust Method to Fit an Ellipse in $\mathbb{R}^{2}$

Does it make sense to consider situations where the MLE exists only for infinitely many $n$?

Intuition behind matrix form of Fisher information

Proving completeness for a statistic [duplicate]

Is a linear model an instance of a parametric model?

Existence of unbiased efficient estimators

Unbiased Estimator

Find a CI for a sample of Poisson distributions

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