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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 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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36 views

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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31 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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20 views

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
25 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
32 views

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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32 views

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
54 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
40 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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26 views

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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23 views

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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19 views

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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41 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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27 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
27 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....
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1answer
53 views

How to quantify the confidence on a Bayesian posterior probability?

Consider a physical system that depends on a parameter $0\leq \phi <\infty$. I want to (i) find the probability that this parameter is smaller than a critical value: $\phi\leq \phi_c$, and (ii) ...
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20 views

Parameter estimation (Uniform Distribution)- Urgent Help Needed

I want to estimate a parameter $k$, from my data $(y_1,y_2, ....,y_n)$ which are independent and identically distributed. $k=a$, $\;$$\;$$\;$$\;$$\;$$\;$ if $y_i$ is uniformly distributed on $[a,b]$ ...
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1answer
19 views

Unbiased estimate for a parameter

They ask me to estimate any parameter and I do not if the solution is correct: The life time X of a battery is considered to be a random variable with density function $f (x; Θ) =\frac{ 2 }{ Θ²} (Θ -...
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8 views

Bayesian parameter estimation with a pre-computed grid of function calls

I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, \ldots x_k$ (describing things like the ...
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1answer
34 views

Unbiased estimate for a parameter.

The time T that it takes to execute an optimization algorithm is assumed to be a random variable with the parameter distribution Θ > 0 $f (t; Θ) = \frac{t} {Θ^2} e ^\frac{-t}{Θ} $ if t > 0 Let T1, ...
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1answer
42 views

Estimate relationship between two Bernoulli random variables

$X$ and $Y$ are Bernoulli random variables $X$ and $Y$ are not independent $x_{t} = P(X_t = 1)$ and $y_{t} = P(X_t = 1)$ for time $t$. Is it possible to estimate $P(Y = 1 | X = 1)$ from many pairs of $...
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1answer
16 views

Confidence interval…

State whether or not the following is true. Explain why: Someone says that a computed result (0.06,0.07) of a 95% confidence interval of a parameter, β, implies that P[0.06 < β < 0.07] = 0.95. ...
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37 views

Finding Best Unbiased Estimator of Uniform Distribution

Let $X_i$, $i=1,...,n$ be iid with $f(x,\theta) = \frac{1}{2\theta}$ for $-\theta<x<\theta$. Find the best unbiased estimator of $\theta$ if one exists. So I first tried $T(X)=X_{(n)}$, which ...
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15 views

Sample moments Intuitive explanation

Can anyone intuitively explain to me the definition I have posted above? Also, what is the purpose of this definition i.e. how do you use it?
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21 views

Confusion regarding OPEF and CRLB

I am a little confused on this , We know that under suitable regularity conditions,the Cramer-Rao lower bound is attained by the variance of an unbiased estimator $T(X)$ of $g(\theta)$ iff the family ...
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1answer
18 views

Statistics: Point estimation

If we have a sample of $x=2$ from a $Po(6 \cdot \lambda)$ distribution. How do we calculate $\lambda*$ and $d(\lambda*)$? I think that $\lambda* = \frac{2}{6} = \frac{1}{3} $ but I am not sure ...
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1answer
29 views

Find UMVU estimator for $e^{-3 \theta}$ given a complete sufficient statistic $X \sim Pois(\theta)$ with $\theta>0$.

My attempt: We know, since $X\sim Pois(\theta)$ that $\mathbb{P}_{\theta}(X=x)=e^{-\theta}\theta^{x}/x!$. A given tip is that we must recall that $e^{x}=\sum^{\infty}_{k=0}\frac{x^{k}}{k!}$. I know ...
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1answer
22 views

Showing that an estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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16 views

Showing that a estimator is consistent

Let $X_1,X_2,\ldots,X_n$ be a random sample from $\mathcal{N}(\theta,1)$. Consider the following (randomized) estimator of $\theta$ given a sample of size $n$: $$ \hat{\theta}_n = \bar{X} + \begin{...
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20 views

2D Maximum Likelihood Fit

I have read a couple of places that it is possible to do a 2D (or 3D) maximum likelihood fit, but I can't seem to understand how this would work. Suppose I'm considering a probability distribution ...
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2answers
44 views

Check if the estimator is unbiased

For $X_i\sim U[0,a]$ where $i=1,2,\dots,n$ so, $E(X_i)=\dfrac a2$. Is $a'=\max\{X_1,X_2,\dots,X_n\}$ an unbiased estimator of $a$? This is what I thought. Since $a'=\max\{X_1,X_2,\dots,X_n\}=X_k$ ...
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27 views

Smoothly merging two parametric curves

Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s \geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I ...
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34 views

Deriving UMVUE for $\mu\sigma^k$ when both $\mu$ and $\sigma$ are unknown and are the parameters of a normal distribution

Let $X_1,...,X_n$ be iid $N(\mu,\sigma^2)$, and define $f$ as $f( \theta)=f(\mu,\sigma)=\mu\sigma^k$. I'm attempting to find the UMVUE for $f(\theta)$ via the Lehmann-Scheffe approach, i.e. I'm ...
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38 views

Finding MVUE (unbiased estimator with minimum variance)

Let $X_1,...,X_m$ be i.i.d. sample with $N(\mu_1,\sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(\mu_2,2\sigma^2)$. Let $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− ...
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26 views

find 95% confidence interval

Consider the estimation of α in $\frac{2}{\alpha^2}(\alpha -x)$ , $0<x<\alpha$ Suppose you only have one observation. Find a 95% confidence interval using the statistical method. To solve it,...
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27 views

Find a lower bound of 95% confidence for θ, of $U(0,θ)$ distribution

Consider a random sample of observations of a $U(0,θ)$ distribution. Find a lower bound of 95% confidence for θ My idea was: Let $X_ {1}, X_ {2}, ..., X_ {N}$ be a random sample of a uniform ...
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11 views

how affect the variance and the expected value add a variable in a linear model

if the correct model is $Y=X_{1}\beta_{1} + X_{2}\beta_{2} + \varepsilon$, with variance $Var(Y)=\sigma^{2}I$.how change the variance and the expected value of $\beta^{*}_{1}$, in $Y=X_{1}\beta^{*}_{1}...
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1answer
48 views

Pivots for exponential distribution

In my mathematical statistics course I got the following problem: Let $X_{1}, ... , X_{n}$ be an i.i.d. sample from the Exp($\lambda$) distribution. Construct two different pivots and two con fidence ...
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Maximum likelihood estimation in a Poisson convolution

Suppose that $X$ and $Y$ are i.i.d. Poisson random variables, with mean $\nu$. The parameter $\nu$ is unknown and we would like to estimate it. We only are given the single data point $$ X-Y. $$ What ...
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Distance functions for stochastic simulations in R

I have been using approximate Bayesian computation to parameterize deterministic models. However now I am using stochastic models (Tau-leap method) I don't know how to use these data in distance ...
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1answer
48 views

Deriving the UMVUE for Rayleigh scale parameter

Let $X_1,...,X_n$ be iid with the pdf given by $f(x|\theta)=2\theta^{-1}xe^{-x^2/\theta}$ for $x>0$. My task is to find the UMVUE for $\theta$, and I’m given the following hint: “$U(X)=\sum_{i=1}^...
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1answer
27 views

Deriving Rao-Blackwellized version of unbiased estimator

Let $X_1,...,X_n$ be iid Poisson($\lambda$) with $n\geq 4$. We are given the unbiased estimator $T(X)=I(X_1=0 \cap X_2=0 \cap X_3=0)$ for $f(\lambda)=e^{-3\lambda}$, and my task is to derive the Rao-...
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35 views

Most powerful test for discrete uniform Neyman Pearson Lemma

This is with regard to the question whose link is given below- Most powerful test for discrete uniform I obtained the most powerful test function as- $\phi(x)$ = 1 if X < 3 ; ...
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6 views

find a Confidence interval in terms of sample correlation

i'm reading about linear models, and i tried to solve the next problem, but i'm really lost about how to start. i was reading in other books but i really have no idea how to tackle it. can you give me ...
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Which loss function does the maximum likelihood estimator minimize?

I'm trying to understand Maximum Likelihood estimators in the context of general estimation theory. I know Bayesian estimator minimizes mean squared loss, MAP estimator minimizes all-or-nothing loss (...
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11 views

Division of two population variances

Why do we divide variances of two samples / population while estimating while for mean and proportion we take difference of two population .What is the reason behind division of variance?
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17 views

Parametric Estimation with Different Distribution Populations

I am looking for some reference for a parametric estimation problem with different populations. Suppose the parameter we are interested in is $\theta$, and we have data, say, from two probability ...
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1answer
66 views

Cramér-Rao Lower Bound for estimator of mean in Exponential distribution

Let $X_{1},...,X_{n}$ be a random sample of size $n\geq3$ from the exponential family with mean $1/\theta$. (1) Find a sufficient statistic $T(X)$ for $\theta$ and write down its density. (2) Obtain ...
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18 views

Intuition about conjugate priors and parameter estimation

I have a problem that I am starting to work in a field where I need lots of non-rigorous probability theory for modelling.One large stumbling block for me is concept of conjugate Priors of random ...
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15 views

Estimation of 2 parameters with Maximum likelihood and a function depending on 2 random variables

I have the following PSF (Point Spread Function) (Moffat PSF) : I want to estimate the parameters $\alpha$ and $\beta$ ($\theta=[\alpha,\beta]$ represents the vector of parameters to estimate) with ...