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Questions tagged [parameter-estimation]

Questions about parameter estimation. Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured/empirical data that has a random component. (Def: http://en.m.wikipedia.org/wiki/Estimation_theory)

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Average Rank versus Ranked Average in Parameter Estimation

I have the following problem: In a cricket tournament, the eleven batsmen of a team play 100 matches before the final. The runs scored by each are available. Determine the average rank of the batsmen ...
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Calculating the bias of the inverse of a sample covariance matrix

It's standard in a stats class to calculate the bias of the sample covariance matrix (or lack thereof), but I'm having trouble finding any exact results on how the inverse of the sample covariance ...
mather's user avatar
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Calculating Standard Error (SE) of a Nonlinear Function Using SE of Its Parameters

I'm working with a nonlinear function, specifically a beta weighting scheme, which generates weights that can vary in shape (e.g., decaying, hump-shaped, U-shaped). I have estimates for the parameters ...
mexx's user avatar
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4 votes
3 answers
95 views

Maximum Likelihood Estimation for Poisson Mean with Given Observations

You have a sample of $n$ i.i.d. realizations of the random variable $X$ distributed as a Poisson with parameter $\lambda$. It is known that: $n_1$ values are greater than or equal to $2$; $n_2$ ...
Emalas's user avatar
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Expected value of an estimator of shape parameter of the generalized Pareto distribution

I would like to compute the expected value and variance of the kappa parameter for the generalized Pareto distribution, where $$ \hat{\kappa} = \frac{\hat{\sigma}^2}{{s}} $$ Where $$ s = \frac{1}{n} \...
norh's user avatar
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1 vote
2 answers
43 views

Is $\frac{1}{n}\sum_{i=1}^{n}X_i$ is a sufficient estimator for $\lambda$ in the Poisson distribution?

I know from this question that $\sum_{i=1}^{n}X_i$ is a sufficient estimator for $\lambda$ in the Poisson distribution. However, from looking at the proof I can see that $\frac{1}{n}\sum_{i=1}^{n}X_i$ ...
gbd's user avatar
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5 votes
2 answers
187 views

Finding MLE given dependent observations from uniform distribution $U(0,\theta)$ [closed]

Suppose we are given random variables $X_1,...,X_n$ that are uniformly distributed on the interval $[0,\theta]$, with $\theta >0$ unknown. I know that if the $X_1,...,X_n$ are furthermore ...
user007's user avatar
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1 answer
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A min-max optimization problem for a given probability density function?

I am trying to solve a min-max optimization problem for a given probability density function. The problem is defined as follows: Let the random variable $X_\theta$ have the PDF: $$ f_\theta(x) = 1 - |...
Luna Belle's user avatar
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19 views

Minimum MSE Estimator among all possible estimates. Sheldon Ross Exercise 59 in Chapter 7 [duplicate]

In Sheldon Ross' Introduction to Probability and Statistics for Engineers and Scientists, the problem $59$ in chapter $7$ asks us that if $X_{1},...,X_{n}$ denote a sample from a population whose mean ...
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Non-linear parameter optimization using Python

I have a model that generates the curve represented by the red squares the data represented by the black circles. The model curve (red squares) depends on some parameters to fitting. Is there any ...
Emerson P L's user avatar
4 votes
1 answer
81 views

UMVU estimator of $\lambda^2$ via Rao-Blackwell

I have been working on a problem, which goes as follows: Given the statistical model $(\mathcal{X},\mathcal{B},\mathcal{P})$, where $\mathcal{P}=\{P_{\lambda}^{\otimes}:P_{\lambda}=Pos(\lambda), \...
tychonovs-scholar's user avatar
2 votes
0 answers
99 views

Generalization error bound for Empirical Risk minimizer on Gaussian noisy data

I have datapoints that are sampled from a distribution $\mathbb{D}$. Each datapoint is a tuple $(t,y)$ of a time $t \in [0,T]$ that is sampled uniformly and a value $y(t) \sim u(t) + \mathcal{N}(0, \...
Paul Joh's user avatar
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Predicting simulated data for a known curve

I am a newbie here seeking advice on a mathematical problem I am currently having in my research. I have a pre-existing curve created by extrapolating known fitted experimental data. As shown below, ...
SSh's user avatar
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1 answer
37 views

Method of Moment for Normal mixtures $p\cdot N(0, 1) + q\cdot N(\eta, 1)$

Setup Let $X_1,\ldots , X_n$ be random variables according to $$ p\cdot N(0, 1) + q\cdot N(\eta, 1),\ p\in (0, 1), q:=1-p. $$ We use method of moments to obtain the needed starting $\sqrt{n}$-...
ytnb's user avatar
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2 votes
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structural identifiability of ordinary differential equations is preserved when adding terms independent of parameters

Assume I have an ordinary differential equation of the form: \begin{equation*} \frac{dx}{dt} = f(x,t,\Theta), \ x(0) = x_0 \end{equation*} with $f:\mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^l \...
Paul Joh's user avatar
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How to derive the sampling distribution of method of moment estimators for uniform distribution

Let $X_1$, ..., $X_n$ be a random sample from $Unif[a,b]$. By simple calculation, the method of moment estimator for $(a,b)$ is $(\hat{a},\hat{b})=$ $$(\frac{1}{n}\sum_{i=1}^n X_i - \sqrt{3(\frac{1}{n}...
jcm22's user avatar
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1 answer
32 views

Expectation of the inverse of an estimator using delta method linearization

I'm working on a research internship about the precision of consumption price indexes and face some probems that leads me to ask a lot of questions about the methods I used and need to be answered. ...
Steve R. NUNES's user avatar
1 vote
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48 views

UMVUE of $\mu^p$ where $X_1,\cdots,X_n\sim\mathcal{N}(\mu,\sigma^2)$

$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\V}{\mathbb{V}}$ Let $p\in\mathbb{N}$. Is there a nice general expression for the UMVUE of $\mu^p$, where $X_1,\cdots,X_n\sim\mathcal{N}(\mu,\sigma^2)$ are i....
harrydiv321's user avatar
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2 answers
85 views

Given single observation, x, from a uniform distribution of 0 to N where 0 < N < 90 and N is uniformly distributed, what is the expected value of N?

In a physical system, an angle, $\theta$, is fixed but unobservable. $\theta$ is restricted to the first quadrant, so $0 \le \theta \le90$. An angle, $\alpha$, lies between $0$ and $\theta$. Only a ...
max's user avatar
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1 vote
0 answers
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Sample mean of Bernoulli trials is admissible under squared loss

Let $X_1,\ldots,X_n$ be i.i.d. Bernoulli trials with probability $\theta\in(0,1)$, and let $L:(0,1)\times[0,1]\to\mathbb{R}$ be the squared loss function, i.e. $L(\theta,a)=(\theta-a)^2$. I am trying ...
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13 views

Calculating the asymptotic distribution of a function of an LLN-like quantity.

I'm preparing for my statistics exam. I'm considering the following question - there is a continuously differentiable strictly monotonic function $\mu : \mathbb{R} \rightarrow \mathbb{R}$ and a ...
Featherball's user avatar
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32 views

Convolution and differentiation - two cascaded transfer functions

I'm just checking if the math I developed for a project is correct. I have two cascaded transfer functions named $H_{inv}(s)$ and $G(s,\hat{\theta})$. An input signal $u$ is convoluted with the ...
Jean-Fr's user avatar
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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves?

!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
Parth's user avatar
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2 votes
1 answer
64 views

Uniform Convergence of integral with parameter

I need to prove that there is no uniform convergence of $I(\alpha) = \int_{0}^{1}{\sin(\frac{1}{x})\frac{1}{x^{\alpha}}dx}$ where $\alpha \in (0,2)$ what I've tried so far: Let's prove pointwise ...
Jane Doe's user avatar
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0 answers
16 views

Estimating the Parameter $\eta$ in a Mixture Model Involving Gaussian Noise

Problem Description: Consider the mixture model defined by the equation: $$ z = x + \eta \cdot (y - x) + k \cdot \sqrt{\eta} \cdot n $$ where: $ x, y, z $ are known D-dimensional vectors. $ n $ is a ...
BinChen's user avatar
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Finding α-Quantiles of χ2 Distribution for Variance Estimation

I posted this same question yesterday but it got closed because i hadn't met the guidlines for questions, my apologies guys, so i'm going to re-write it better this time. Exercise 17. Let $X_1, \ldots,...
mathmath's user avatar
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0 answers
7 views

Standard practice to show Biased CRBs

I have a problem with four-parameter estimation. I have derived the variances for the estimated parameters using Monte Carlo simulations (numerical ones) and theoretical ones using the inverse of the ...
CfourPiO's user avatar
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1 vote
0 answers
45 views

Finding UMVUE for Correlation Coefficient in Multivariate Normal Distribution

It is described in Wikipedia that: In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance ...
Hank Wang's user avatar
2 votes
1 answer
85 views

Does the kurtosis need to be finite for the sample variance to be consistent?

It is known (see this answer) that if $\mu_4$ is the fourth central moment of a distribution and $\sigma$ is the standard deviation, then we can write $$\operatorname{Var}(S^2_n)=\frac{1}{n}\left[\...
harrydiv321's user avatar
1 vote
1 answer
51 views

UMVUE of $\mathbb{E}[X^2]=\lambda^2 + \lambda$ where $X\sim\mathrm{Pois}(\lambda)$.

This is the same question as this: UMVUE of $E[X^2]$ where $X_i$ is Poisson $(\lambda)$. Here, I restate the problem for completeness: Let $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \mathrm{Pois}...
pbb's user avatar
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2 votes
2 answers
44 views

Determining Parameter Values for a Set of Solutions Involving Absolute Value Equations

The problem in the textbook asks the following: At what values of the parameter "$a$" does the set of solutions to the equation $|x - 1| + |x - a| = 1 - a$ consist of three integers? I ...
curioushuman's user avatar
1 vote
0 answers
30 views

Estimating the parameters of an ellipse (part 2)

This post is a follow up of this previous one. I would like to clarify why the angle estimator works and how to estimate the axes length. Unfortunately, I still have some trouble with this problem. I ...
matteogost's user avatar
0 votes
1 answer
85 views

Show for $\tau (\theta) = \exp(-\theta c), c>0$ the estimator $E(X) = (\max(0, 1-\frac{c}{\sum x_i}))^{n-1}$ is efficiency unbiased. [closed]

TASK: Let $X = x_1, ... x_k$ are i.i.d. with exponential distribution with parametr $\theta$. Show for $\tau (\theta) = \exp(-\theta c), c>0$ the estimator $E(X) = (\max(0, 1-\frac{c}{\sum x_i}))^{...
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0 answers
45 views

Maximum Likelihood Estimation and Unbiased Estimator for drawing balls without replacement

For $N\geq 1$ there is an urn with $N$ balls labeled with the numbers $1,...,N$ and we want to estimate $N$ by randomly choosing $n\leq N$ balls without replacement. Determine the corresponding ...
Lu1998's user avatar
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0 votes
0 answers
12 views

Poisson log-likelihood parameter estimation with uncertainty on model rate

I have data $k_n$ that I expect to follow a Poisson distribution $P_{\lambda}(k)=\frac{\lambda^k e^{-\lambda}}{k!}$, and a model for the event rate $\lambda$ that depends both on a vector of unknown ...
Cullen Abelson's user avatar
2 votes
0 answers
46 views

Minimum variance unbiased estimator for $\mu$ in Normal location model with known but random variance

Consider observing $X \mid \sigma \sim N(\mu, \sigma^2)$ and $\sigma \sim F$ for some known distribution $F$ supported on the positive reals. We observe a single draw $(X, \sigma)$. An estimator $T(X, ...
Kevin's user avatar
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1 vote
1 answer
74 views

How can an estimator be consistent and asymptotically normal at the same time?

I can't work out why the asymptotic distribution of an estimator matters if it is consistent? My understanding is: An estimator, $\hat{\theta}$, is consistent if it converges in probability to the ...
arb6's user avatar
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2 votes
1 answer
64 views

Let $Y_1, \dots, Y_n \sim \; \textrm{iid}$ with pdf $f_Y(y)$. Show that the UMVUE of $\theta$ is given by $\frac{n-1}{\sum_{i=1}^n Y_i}$ [duplicate]

I'm having a difficult time figuring out where to go here. Question: Let $Y_1,\dots, Y_n$ be iid random variables with pdf $f_Y(y) = \theta e^{-\theta y} \;,\; y >0\;,\;\theta >0.$ Show that the ...
Calum's user avatar
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2 votes
2 answers
65 views

Value of the coordinates in parametric form of hyperbola

We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's center. I've learnt that in the parametric form, we take $$x= h + a\sec t$$ and $...
Gwen's user avatar
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0 votes
0 answers
11 views

Minimax Estimation: What's the difference between Minimax, Sharp Minimax, First-Order Sharp Minimax and Second-Order Sharp Minimax Estimator?

I am currently working on my dissertation on Biased Data, and the second chapter focuses on distribution function estimation. Efromovich's work appears to be an outstanding reference on the topic, but ...
Noelia Schz. Mrt's user avatar
2 votes
1 answer
77 views

MLE of $\theta$ from $N(\theta+2, \theta^2)$

Let $X_1, X_2, ..., X_n$ be a random sample from $N(\theta+2, \theta^2)$. Find the MLE of $\theta$. I went through some work and I could not solve for $\theta$. Am I doing anything wrong? $$L( \theta )...
Brian Lam's user avatar
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0 answers
38 views

How to prove $W(T)$ is not a MVB estimator

A random sample $X_1, . . . , X_n$ of size n is taken from the Poisson distribution with parameter $\theta$. Let $X = (X_1, . . . , X_n)^T$ and $x = (x_1, . . . , x_n)^T$. It is proposed now to ...
fa fa's user avatar
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0 votes
0 answers
59 views

Independence of MLEs for subsamples

Suppose I have a sample $\textbf{x} = (x_1, \dots, x_n)$ from stationary and $\beta$-mixing time series. I want to estimate the scalar parameter $\theta$ of the distribution using MLE, i.e., $$ \hat{\...
Grigori's user avatar
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0 answers
14 views

Parameter estimates and uncertainties associated with a Multinomial distribution

I have a process/model the depends on a single parameter $\lambda$ that generates $n+1$ outcomes. From $N$ events I can estimate the probabilities $\hat{p}_k$ from $N_k/N$ using a MLE for a ...
Muzza's user avatar
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0 votes
0 answers
22 views

Fisher Information and Parameter Space

I am reviewing Fisher information and saw that one of the requirements is that the distribution of the data, say $f(x|\theta)$, involves a parameter $\theta$ that is unknown but lies within a given ...
kpr62's user avatar
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1 vote
0 answers
71 views

What does it really mean to take correlation between time series? [closed]

I have a conceptual problem when we extend the correlation to time series. I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from ...
Curious student's user avatar
1 vote
0 answers
49 views

Estimating the parameters of an ellipse

Problem definition Consider a dataset composed by $m$ bivariate measurements \begin{equation*} y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m \end{equation*} uniformly ...
matteogost's user avatar
-1 votes
1 answer
41 views

Check if method of moments estimator is unbiased for $X_1...X_n$ being a random sample from $Uniform[-\theta,\theta]$ [closed]

I am not sure how to do this. To find the method of moments estimator I did: $E[X] = \frac{-\theta + \theta}{2} = 0$ use 2nd moment: $E[X^2] = \frac{(-\theta)^2 + -(\theta^2) + \theta^2}{3} = \frac{\...
autalisk's user avatar
2 votes
0 answers
54 views

How to estimate the best variance-proxy of a sub-Gaussian distribution from data?

Suppose we have $N$ independently identically distributed (i.i.d.) samples $X_1,\cdots,X_N$ generated from a sub-Gaussian random variable $X \sim \mathbb{P}$. Then by definition there exists the ...
Asce's user avatar
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1 vote
1 answer
52 views

Consistency of Biased Estimators

In Statistical Inference, we were taught this theorem, Consider an estimator $T_n$ of population parameter $\theta$, using $n$ samples. $T_n$ is a Consistent Estimator of $\theta$ if $$E[T_n] \to \...
Harry's user avatar
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