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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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Show Consistency for every component

For $j=1,...,k$ let $t_{n,j}:\Omega_n \rightarrow \mathbb{R}$ be an estimator for $h_j(\theta) \in \mathbb{R}$. Show that $t_n(X)=(t_{n,1}(X),...,t_{n,k}(X))$ is a consistent estimator of $h(\theta)=(...
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1answer
36 views

Which estimator would be better in terms of Mean Square Error?

Let $X_1,\ldots,X_n$ be a a sample from a Uniform Distribution $(0,\theta)$ where $\theta > 0$ is an unknown parameter. I have found the estimator based on the sample mean $$\hat{\theta}=2\bar{X}$...
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2answers
28 views

How to calculate the bias of the estimator for variance?

Question: For observations $x_1$, $x_2$, . . . , $x_n$ with sample average $\bar{x}$, we can use an estimator for the population variance: $\hat\sigma^2$ = $\frac1n\cdot $ $\sum\limits_{i=1}^n (x_i - ...
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1answer
51 views

Finding the UMVUE of $\frac{1}{\lambda}$

I have been given the pdf: $$f_X (x; \lambda) = \left(\frac{\lambda}{\pi}\right)^{\frac{1}{2}} x^{-\frac{3}{2}} e^{-\frac{\lambda}{x}} $$ with support $x>0$ and $\lambda>0$. I am asked to ...
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1answer
27 views

Find the time gap between two vehicle based on km/h of the vehicle

I am currently researching on VANET for identifying the road capacity based on vehicle moving speed. Below is my question, If a car moving on 8km/h and distance to the next car is 15 meters, what ...
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1answer
25 views

Finding the estimator of π(1-π) from a random sample of n Bernoulli trials.

A random sample of n independent Bernoulli trials with success probability π results in R successes. Derive an unbiased estimator of π (1 − π). So, from what I understand (correct me if anything I ...
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24 views

Rao Blackwell and sufficient statistics

Suppose that X1, . . . , Xn are independent identically distributed random variables with a B(m, θ) distribution where m is a known positive integer and θ is unknown. I have shown that θ* = X1/m is ...
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1answer
21 views

Rao-Blackwell and Cramer-Rao LB comparison

Let $X_1, X_2, \dots, X_n$ be a random sample following the Geometric distribution. $$ \prod\limits_{i=1}^{n} f(x_i|p) = (1-p)^{\sum\limits_{i=1}^n x_i-n}p^n $$ Since the pmf of the Geometric ...
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28 views

If $X_i \sim U(\theta-\frac{1}{2};\theta+\frac{1}{2})$, show that $[X_{(1)},X_{(n)}]$ is a confidence interval

Let $X_1,...X_n$ random sample from $f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$. a) Show that $[X_{(1)},X_{(n)}]$ is a confidence interval for $\theta$. b) Compute the ...
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1answer
38 views

Estimating mean from a biased sample

Imagine that somebody had chosen $N$ numbers from a normal distribution with mean $\mu$ and variance $1$ ($\mu$ is unknown to you) and only showed you all $n \le N$ numbers which are greater that $\mu$...
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1answer
38 views

Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$

I am working on this problem. Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$ . So far I am thinking of using the invariant property of MLEs, so I let $$\hat{\theta} = \...
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1answer
46 views

Finding a confidence interval for shifted exponential distribution

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ . a) if ...
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1answer
24 views

Finding shortest Confidence Interval for an Exponential Distribution

Let $X$ such that $f_{X}(x\mid\theta) = \theta e^{-\theta x} I_{(0, \infty)}(x)$, where $\theta > 0$. If $[X, 2X]$ is a confidence interval for $\frac{1}{\theta}$: a)Find the confidence ...
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Estimation of noise in a particular random process

I have a problem for which I need help. Hypothesis: Let $m_0 \in \Bbb R$, $\sigma_x \in \Bbb R_+$, $\sigma_m \in \Bbb R_+$ and $n \in \Bbb N$. Let $\Delta M_1$, $\Delta M_2$, ..., $\Delta M_n$ be $n$...
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33 views

Likelihood ratio test for the exponential distribution

Let $Y$ be a random exponentially distributed variable, with mean $\lambda$. That is, its probability density function is $$f(y) = \left\{ \begin{array}{ c c } \frac{1}{\lambda}e^{\frac{...
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1answer
29 views

Finding aconfidence interval for $\theta$ of the uniform distribution on $(0, \frac{1}{\theta})$

Suppose $X_1,\ldots, X_n$ are i.i.d. random variables $Uniform (0, 1/ \theta)$. Find a 95% confidence interval for $\theta$. What I tried: $f_{X} (x) = \frac{1}{1/\theta}=\theta, F_{X} (x) = \frac{...
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1answer
39 views

Finding confidence interval for $\frac{kx^{k-1}}{\theta^k}$

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\theta)=\frac{kx^{k-1}}{\theta^k}, x\in (0,\theta)$$ where $\theta \gt 0 $ and $k$ is a positive integer. Find a $100(1-\alpha)% $% ...
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1answer
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Helping to understand the expected value and variance of an estimator $\hat{X}\left(y\right)=\frac{2}{N}\left(y_1,y_2,…,y_n\right)$

I have found with the following example: Suppose that we have samples y drawn from a uniformly distributed random variable which extends from 0 to x. We would like to estimate x from the data $y_1,......
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1answer
17 views

Construct a symmetric $100(1-a)\%$ confidence interval for $\theta$

Suppose $Y = T - \theta$, where $T$ is a sufficient statistic for $\theta$, with distribution function: $$ F_Y (y) = 1-e^{-yn} $$ To construct a $100(1-a)\%$ confidence interval, we must find two ...
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1answer
42 views

Inequality for Uniform Distribution

Let $X_1,..,X_n$ be a random independent sample $X_1,…,X_n$ from a Uniform$[0,\theta] $ distribution, $\theta \in [0, \infty)$, with probability density function $f(x;\theta) = \begin{cases} 1/\theta,...
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1answer
36 views

Spherical Gaussian MLE

I am having trouble doing a derivation. I want to find the MLE estimate of $\sigma^2$ in a spherical gaussian, i.e when we have set $\Sigma = \sigma^2I$. I have already seen https://stats....
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2answers
44 views

How to find MLE of this piecewise pdf?

Suppose $X_1,\ldots, X_n$ are i.i.d. random variables having pdf $$ f_{\theta}(x)=\left\{\begin{array}{ll}{\theta,} & {0 \leqslant x \leqslant 1} \\ {1-\theta,} & {1<x \leqslant 2}\end{...
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0answers
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Estimating parameter of binomial distribution

We have some solution containing a compound A. We find that when we mix $1$ $\mathrm{mm}^3$ of our solution A with some amount of some compound B, we get a reaction 200 out of 185 times when we check ...
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0answers
43 views

Improve unbiased estimator of $\theta$ with sample from $U(-\theta,\theta)$ using Rao-Blackwell theorem

Let $X_1,X_2,\dots, X_n$ be rvs with pdf: $$f(x\mid \theta)=\frac{1}{2\theta}I(-\theta<x<\theta)$$ I know that $Y=\max|X_{i}|$ is a sufficient statistics for $\theta$ and I found an unbiased ...
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0answers
10 views

Including a delay to the exponentially decaying function

I am trying to fit a function of the form $X(t)=X_0 e^{\lambda t}$ to the following data on time $t$ and $X(t)$ and estimate the parameter value of $\lambda$. $t=[0, 1 ,2 ,3,4,5,6]$ and $X(t)=[...
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1answer
27 views

Uniform Distribution Estimator

Let $X_1,..,X_n$ be For a random sample $X_1,…,X_n$ from a Uniform$[\theta,θ+1] $distribution, with probability density function $f(x;\theta) = \begin{cases} 1, & \theta \le x \le \theta+1 \\ 0, &...
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1answer
52 views

Mean squared error for vectors

I know that when we compare estimators $\hat{b_1}$ and $\hat{b_2}$ to an unknown parameter $\beta$, in classical statistics an estimator $\hat{b_1}$ is said to be "better" than $\hat{b_2}$ if: $$ ...
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1answer
39 views

Find an estimator of $N$ (hypergeometric)

A forest has $N$ (unknown) monkeys. A random sample of $n$ monkeys is selected from the forest, tagged and released back into the forest. After a few days, a random sample of $m$ monkeys is selected ...
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2answers
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MOM for uncommon distribution (using Gamma function)

I'm stuck understanding this question where we are trying to get the MOM for a RV, IID, with a density function as: $$f(x|α) = {Γ(3α)/Γ(α)Γ(2α)} * x^{α-1} * (1-x)^{2α-1}$$ Where alpha is the ...
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46 views

Distribution of random variable $Y = T - \theta$, where $T$ is a sufficient function for $\theta$

Let $X_1, X_2, \dots , X_n$ be a random sample from the distribution with with pdf: $$ f(x;\theta) = e^{\theta-x} I_{[\theta, +\infty)}, \, \theta \in \mathbb{R} $$ For the sample we have: $$ f(x_1, \...
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1answer
42 views

Basu's theorem to show independence

Using Basu's theorem, prove that $\sum\limits_{i = 1 }^n {(X_i - X_{(1)}) }$ and $X_{(1)}$ are independent for any $(\theta, \lambda)$. You may assume that $X_{(1)}$ is complete and sufficient for $θ$ ...
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1answer
59 views

UMVUE of $P(X_1 ≥ t)$ for a two-parameter exponential distribution

I'm attempting to find $(a)$ The UMVUE of $λ$ when $θ$ is known. $(b)$ The UMVUE of $θ$ when $λ$ is known. $(c)$ The UMVUE of $P(X_1 ≥ t)$ for a fixed $t > θ$ when $λ$ is known. I'm new to the ...
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1answer
17 views

Using log data in parameter estimation

Say I have data as $x_1,x_2,..x_n$ and I am fitting a model to these data for example, $Y(t)=Y_0 e^{-\lambda t}$. I am trying to estimate the parameter $\lambda$ by fitting to these data. In these ...
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1answer
36 views

Cannot Find Parameter Estimation using Maximum Likelihood Estimation [duplicate]

Given probability density function $$p(x|\phi)=\begin{cases}\dfrac{\phi}{x^2}&\phi<x<\infty\\0&\text{others}\end{cases}.$$ I want to find an estimation for the $\phi$ parameter with ...
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Decomposition of posterior porbability

So, I'm currently working through "Data Analysis: A Bayesian Tutorial 2nd Ed." by D.S. Sivia and I am stuck an equation, which I am not able to reproduce (p.40 - 42 for those who might have a copy). ...
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35 views

estimate Markov chain mean transition time

Assume a continuous time Markov chain which is run through in one direction and finally absorbed at the last state $1 \rightarrow 2 \rightarrow 3 \rightarrow ... \rightarrow n $ The transition ...
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0answers
36 views

Computing Maximum Likelihood and Least Squares Estimators for Parameter Estimation of Gaussian Model

I am having trouble with the following question, particularly the first part. Doesn't least squares require that errors be the same across the RV drawn from a distribution. However the variance ...
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29 views

Does a Best Unbiased Estimator for parameter of exponential distribution exist?

For an exponential distribution $$X \sim \exp(\lambda) = \lambda \ \exp(-\lambda\ x),\ x>0$$ Does there exist an Best Unbiased Estimator (BUE) for $\lambda$ i.e. can it achieve the lower bound ...
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Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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2answers
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How to find a confidence interval for difference in response rates?

I found this question on a past exam: I know how to find a confidence interval with one mean: $\bar x \pm z_{\alpha/2} \frac{\sigma}{\sqrt n}$ and I also know how to do it with two means however ...
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0answers
40 views

Show that the only unbiased estimator for the zero-truncated Poisson distribution is absurd

Consider the zero-truncated Poisson distribution on the striclty positive integers, i.e. \begin{align} \mathbb{P}_{\theta}(X=k) = \frac{\theta^k}{k!(e^{\theta}-1)}\, \, \, , \, \, k=1, 2, ... \end{...
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0answers
20 views

Finding a 2-3 parameter decay function to fit experimental data

I have a series of concentration measurements, that are normalized such that they attain their maximum concentration $y_{max} = 1$ at the location $x_{ \text{max}} = 1$. For $x > 1$ the ...
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1answer
40 views

Finding confidence level

I am studying for my upcoming stats final exam and I come across this question: Let X be a single observation from the uniform distribution: $$f(x) = \begin{cases} 0 & \text{otherwise} \\ ...
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1answer
51 views

Moment estimator $\hat{\theta}$ of $\mathrm{Beta}(\theta,1)$ and bias of $\hat{\theta}$

I'm trying to find the moment estimator for the density function $$f(y)=\theta y^{\theta-1}$$ and check whether this is biased. I know this is a $Beta(\theta,1)$ distribution and it looks like I only ...
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0answers
25 views

Trouble replicating a population abundance model of endangered macaws

I have replicated a study of estimating macaw population size, whereby several point counts were made. I am now looking at the original study and trying to work out how to calculate the estimated ...
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1answer
14 views

How to find the estimator of a signal with additive white noise

We have to derivate the MAP estimation of a signal $x(m)$ observed in AWGN $n(m)$, resulting $y(m) = x(m) + n(m)$, supposing no zero mean in any process. I have done this: To find the MAP estimator, ...
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0answers
8 views

Parameters estimation (fit) sensitivity

I have a linear fit estimation of the following type: y=ax+b I would like to test how much a mismatch in a and b affects the accuracy of y? should I use partial derivative or some other method?
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25 views

Can anyone help with the inverse problem and tuning parameters

For my final year project i want to model the population of London using the Verhulst logistic model. However, to gain more marks i wish to use the inverse problem to tune the parameters to make the ...
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1answer
47 views

Finding an unbiased estimator of $(1+\lambda)e^{-\lambda}$ for Poisson distribution

If $X_1,X_2,\ldots,X_n\sim \mathrm{Pois}(\lambda)$, find an unbiased estimator of $(1+\lambda)e^{-\lambda}$. I am actually supposed to find the UMVUE of $(1+\lambda)e^{-\lambda}$. but I first have ...
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1answer
55 views

finding the maximum likelihood and probability distribution function to fit to data

There are data on a pathogen population $N(t)$ measured over time (t). I am using the ODE of, ${dN\over dt}=-\lambda N$ to model its decline over time. So, $ N(t)=N_0 e^{- \lambda t}$. However, ...