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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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Estimator example cube

We make subsequent throws of a fake cubic cube for which the probability of falling out six is $\frac{1}{6}$ - $\epsilon$, the probability of falling out of one is $\frac{1}{6}$ + $\epsilon$ and the ...
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2answers
63 views

Poisson Conditional Expectation ( searching best estimator for h(λ) )

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) (t = a value) E $[$$X_2$ - I{$x_1$=1}|$\sum_{i=1}^n X_i=t$$]$ =E $...
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38 views

Find the Unbiased Estimator (Poisson)

Suppose $x_1$,$x_2$,$x_3$,.....,$x_n$ are i.i.d. random variables with a common density poisson(λ) (I is an indicator function) Find an unbiased estimator for $λ^2$ E $[$$\left(\frac{2 }{e^{-λ}}\...
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Finding the bias of an estimator

Consider the following model: $$ y_i = a + b x_i + c z_i + w_i, $$ where $a,b,c$ are unobserved fixed parameters, $x_i$ and $z_i$ are fixed in repeated samples. Assume also $\mathrm{E}[w_i] = 0$ for ...
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1answer
38 views

Consistent estimator for the variance of a normal distribution

So I have to show that $\hat{\sigma}_n^2=\frac{1}{n}\cdot (\sum_{i=1}^n(X_i-\bar{X})^2)$ is a consistent estimator for the variance $\sigma^2$ when $X_1,X_2,...,X$ are i.i.d. from a normal ...
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1answer
18 views

MVUE for Bernoulli Random Variable

Let $X_1, X_2$ be a random sample from a Bernoulli distribution with $P(X = 1) = p$ and $P(X = 0) = 1-p$. I want to find a MVUE for $p$. $E[X_1]=p$ and $(X_1,X_2)$ is complete and sufficient for $p$. ...
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1answer
29 views

MLE of $p$ for a sequence of RVs with distribution $ P(X_i = k) = (k+1) p^2 (1-p)^k $

Let $ X_i $ be a sequence of iid rv´s with distribution $ P(X_i = k) = (k+1) p^2 (1-p)^k $ with $ k \in \mathbb{N}_0 $ where we set $ \theta := p \in (0,1] $ and $ p $ is unknown. Compute the MLE $ \...
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21 views

Finding UMVUE, if it exists [duplicate]

I'm having trouble finding the uniform minimum variance unbiased estimator (UMVUE) of non-standard distributions/functions. I understand there are few general approaches that work pretty well, but I'...
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Prove that the estimator $\delta(X) = X$ is not an admissible estimator when estimating the mean of a Gaussian

Let us observe a data point $X$ sampled from $N(\theta, 1)$ where $\theta \in \mathbb{R}^+$. We will consider squared/quadratic loss, and let our estimator be $\delta(X) = X$. How can I show that this ...
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1answer
34 views

Consistency of Maximum Likelihood Estimator for Gaussian R.V with Equal Mean and Variance

tl;dr: What is wrong with this MLE estimator? Does it not satisfy the conditions for consistency or did I make a mistake in the calculation. I am trying to compute the MLE estimator for parameter ...
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1answer
21 views

Method of Moments estimators of $\alpha$ and $\beta$

Let 5 numbers 2, 3, 5, 9 and 10 come from a uniform distribution on the interval $[\alpha,\beta]$. Find the method of moments estimators of $\alpha$ and $\beta$. Any help would be appreciated, thank ...
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Consistency and bias of estimators

I've been given a number of examples of estimators and I've been asked whether each one is consistent and/or unbiased. The situation is calculating the arithmetic mean of student's height. Add all ...
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Stochastic approximation of a Cauchy principle value integral.

Suppose I have a random variable $X\sim f_X(x|\boldsymbol\theta)$ with a well-defined expected value. The usual integral for an analytic solution of this expected value is $$\operatorname EX=\int_{\...
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1answer
31 views

Find $c \geq 0$ so that $c\hat{\vartheta}$ is unbiased

I have found the following statistical model. Say $\Omega:=[0,\infty)^{n}$ and $P_{m}$~$[\mathcal{U}(0,m)]^{\otimes n}$ where $m$ is the parameter $\in [0,\infty)$ and $n \in \mathbb N$ Define the ...
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Equivalence of two expected value terms and the distribution of its parameters

Problem: "The number of breakdowns $Y$ per day for a certain machine is a Poisson random variable with mean $\lambda$. The daily cost of repairing these breakdowns is given by $C=3Y^2$. If $Y_1,...,...
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How do I derive a Least Square Estimator of a linear combination of two variables?

I am working on a problem where I have the following model: lm(Y ~ x1 x2) If I have the output of this general model in R, is it possible to derive the LSE of: <...
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2answers
60 views

Counting coupons in a box

I have a question with regards to counting. I come from a background in quantum computing, so I'm not familiar with the relevant literature here. For the interested reader, I provide below more ...
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28 views

Calculating the Mean Square Error (MSE) in Wavelet Denoising

I´m currently reading the paper (to be more precise: it´s a chapter from the book "Shearlets, Multiscale Analysis of Multivariate Data" by Kutyniok and Labate) "Image Processing Using Shearlets" by G....
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Method of moments when the first moment is $0$

I have a quick question regarding the method of moments estimator. Generally, when you have $k$ parameters you want to estimate, it suffices to find $k$ equations using $k$ moments. If you are ...
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1answer
179 views

Method of moments with a Gamma distribution

I'm more so confused on a specific step in obtaining the MOM than completely obtaining the MOM: Given a random sample of $ Y_1 , Y_2,..., Y_i$ ~ $ Gamma (\alpha , \beta)$ find the MOM So I found the ...
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Let $X_1,X_2…X_n $ be a random sample from a $N(\theta ,\theta^2 )$ distribution, where $\theta >0$ is unknown.

Let $X_1,X_2...X_n $ be a random sample from a $N(\theta ,\theta^2 )$ distribution, where $\theta >0$ is unknown.Let $T_1=\sum_{i=1}^{n}X_i$ and $T_2=\sum_{i=1}^{n}X_i^2$ Which of the following ...
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Regression Unbiased Estimator

Suppose $Y_1,Y_2...,Y_n$ are random variables $ε_1,ε_2...,ε_n$ are i.i.d with mean 0 and unknown variance $\hatβ_0 , $$\hatβ_1 $$,\hatσ^2$ are maximum likelihood estimators Given $Y_i=β_0+β_1*x_i + ...
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Examples where estimating the derivative is easier than estimating the actual quantity.

I recently came across a paper in which they author estimated the deriviative of the quanity he was interested in rather than the quantity itself. This was done because apparntly the derivative ...
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1answer
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Based on the ideas of Parameter Estimation and Fitting Probability Distributions, what stops us from making any function be a PDF(PMF)?

Currently I am doing an introduction to parameter estimation and fitting probability distributions to sets of data. So in a small synopsis what I understand the whole process to be like is the ...
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2answers
32 views

What is an intuitive application of estimators?

So we're currently studying Estimators and we just proved Cramér-Rao's inequality and that when it is an equality, then whatever estimator we have is a unique MVUE. All of this to me just sounds like ...
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1answer
22 views

What is the maximum likelihood of a binomial distribution?

i've looked everywhere I could for an answer to this question but no luck ! If I have $X_1 .... X_n$ random variables that are independent and identically distributed such as ∀ $1 <$ $i$ $<n$, $...
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28 views

Estimating time varying parameters from data

We have measurements for a dependent variable $y$ at different times $t$ along with the values of three independent variables ($p$, $q$, and $r$) at those times. Say, $y(t)$ is the value of the ...
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Estimating a parameter from a sequence of signals with non i.i.d. noise

I have the following statistics/probability question: Let $A$ and $B$ be two finite sets of real numbers. Let $n \geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily ...
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Find the UMVUE of $b^{\mu}$

Let $X_1,X_2,..X_n$ be a random sample from Cauchy$(\mu,1)$ population. Find the UMVUE of $b^{\mu}$ where $b$ is any positive real number. Now actually calculating sample mean won't work here because ...
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How to find the MLE of a series of unknown parameters?

I encountered a problem when I was doing my statistics homework. The question is stated as follows: Consider the model in which $D_{1},...,D_{n}$ are i.i.d random variables and that for $i = 1,...,n$,...
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Testing certain estimators for consistency

Let $X_1, X_2, \ldots , X_n$ be a random sample from $U(0, θ)$, where $θ > 0$ is the unknown parameter. Let $X_{(n)} = max\{X_1,X_2, \ldots , X_n \}$. Then which of the following is (are) ...
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EM algorithm for a specific example

We consider $X \in R^{n\times d}_+$ as well as $n\times d$ mutually independent latent variables $z_{ij}$ where $z_{ij}$ has a Poisson distribution of parameter $X_{ij}w_j$ for $i\in \{1,..,n\}$ and $...
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Method of moments estimator for $\theta^2$ when $X_i\sim p_{\theta}(x)=\frac{2x}{\theta^2}1_{0{\leq}x{\leq}\theta}$

Let $X_1, ..., X_n$ be from a sample from a probability distribution: $$p_{\theta}(x) = \frac{2x}{\theta^2}1_{0{\leq}x{\leq}\theta}$$ , where $\theta > 0$ is an unknown parameter. I have found ...
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1answer
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confidence interval with MLE estimator

$f_{\theta}(x) = 2 \theta x e^{- \theta x^2} $ on the interval $(0, \infty)$. T is the MLE estimator of $\theta$. We construct the confidence interval of $\theta$ $(aT, bT)$, where $a$ and $b$ are ...
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1answer
42 views

Diverging Integral with Bessel Function

I am looking for the solution to the integral: $$\int_{a}^{\infty} x J_n(\alpha x)\;dx$$ where $a<< \alpha$ and $n$. I get something out of Mathematica for $a=1,2,0.1...$ in terms of the ...
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Show that $e^{-\mu+\frac{\sigma^2}{2}} $ is estimable if $\frac{\mu}{\sigma}$ is estimable.

Suppose $X_1,X_2,...,X_n \sim^{i.i.d} N(\mu,\sigma^2)$.Show that $e^{-\mu+\frac{\sigma^2}{2}} $ is estimable if $\frac{\mu}{\sigma}$ is estimable. I am utterly confused, in fact I can think of this ...
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UMVUE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{(\ln\theta)\theta^x }{\theta -1}$

I'm having some trouble finding the UMVU estimator of $\theta$ for the following distribution $$f(x)=\frac{(\ln\theta)\theta^x }{\theta -1} \text{ for } x \in (0,1)$$ Specifically, I know $T_n=\sum ...
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33 views

Mean estimator of a Gaussian variable with positive mean for quadratic loss

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian, and $X\sim\mathcal{N}(\theta,1)$, in which $\theta>0$ is positive but othrewise unknown. We want to estimate $\theta$ ...
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What is the distribution of the sample mean of a log-normal distribution

As titled. It is straightforward to estimate the mu and sigma parameters of the log-normal distribution using samples. However, it is not apparent what the theoretical distribution is for the mean of (...
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1answer
56 views

Minimum variance unbiased estimator of exponential distribution

The given model is $\text{Exp}(\mu,\sigma),\;\mu\in\Bbb{R},\sigma\gt0$ whose pdf is $f(x\text{;}\theta)={1\over \sigma}e^{-{{(x-\mu)}\over \sigma}}I_{(\mu,\infty)}(x)$ I easily found $(X_{(1)},\bar{...
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1answer
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Estimating posterior distribution when realization is unknown

I'm working on a problem that can be reduced down the following scenario: Consider two people, Alice and Bob. Alice has a prior probability distribution on a variable (which is known by Bob as well) ...
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2answers
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Is estimator $\dfrac{\bar X}{1+\bar X}$ of $\theta$ is consistent?

Let $X_1,X_2,X_3.....X_n$ be a random sample from a population X having the probability density function $$ f(x;\theta) = \begin{cases} \theta x^{\theta -1} & \text{if $0 \le x\le$ 1} \\0 &...
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1answer
82 views

Calculating UMVUE for Poisson distribution

I've started to learn methods of finding UMVUE distribution. I found some nice examples on this site. I got stuck while evaluating UMVUE for Poisson distribution for a parametric function $g(\theta)...
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1answer
42 views

Estimation of parameter by the method of moments

Let $X_1,X_2,\dots X_n$ be a random sample from the density $$f(x;\theta)=e^{-(x-\theta)} e^{-e^{-(x-\theta)}}, \quad -\infty<x<\infty ,\quad -\infty<\theta<\infty$$ Find the method of ...
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Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
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Std Deviation of a point estimate which is the sum of two normally and independently distributed random variables

The problem States: Given $\bar x= 41$ and $\bar y= 40.7$. $σ_x= 0.1$ and $σ_y= 0.19$ $X \sim N[\mu_X; \sigma^2]; Y \sim N[\mu_Y ; \sigma^2]$; with $\mu_X > 0,\; \mu_Y > 0, \sigma > 0$ ...
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Estimating parameters of “unknown” non-linear function

I've got data of dice rolls. Every "experiment" consists of $n$ (independent) rolls with a three-sided dice (I'll call the results $A, B, C$ from now on). The chance to roll an $A$ in a single roll is ...
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44 views

Maximum estimator in upper Chernoff bound

I have the following exercise about Chernoff bounds: Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,\sigma^{2})$. Show that ...
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32 views

Evaluating variance of scale parameter estimators

Let $\{Y_{i}\}_{i=1}^{n}$ be a random sample of the random variable $Y_{i}\sim \mathcal{N}(0,\sigma^{2})$, we define the following estimators for $\sigma^{2}$ $U=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-...
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1answer
28 views

Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation.

Let $V_1$ be the variance of the estimated mean from a stratified random sample of size $n$ with proportional allocation. Assume that the strata sizes are such that the allocations are all integers....