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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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How can I find the parameters for a Gaussian exponential waveform?

I have a double Gaussian shaped waveform `eta = H1 exp(-C1(x-X1)^2)-H2 exp(-C2(x-X2)^2` where x is given and ...
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Finding umvue of normal pdf [on hold]

Let $X_1,X_2,\ldots,X_n$ be iid $\mathcal N(\mu,1)$ RVs.Let $T(\mathbf X) =\sum_{i=1}^nX_i$.Show that $\phi(x;t/n,n-1/n)$ is UMVUE of $\phi(x;\mu,1)$ where $\phi(x;\mu,\sigma^2)$ is the PDF of a $\...
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24 views

Estimator for binomial distribution

I have a question from my introduction to mathematical statistics book. I'm working on the following problem. We have an urn with a ratio of white balls to black balls of $\frac{p}{1-p}$. We count ...
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1answer
18 views

Expected value of two sums

I am trying to derive the expected value of the following form: $E[{\frac{\sum_{i=1}^{n} {x_i}{y_i}}{\sum_{i=1}^{n} {x_i}^2}}] = $ I understand that x is a constant, and that y is a random variable. ...
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24 views

Linear Least square estimate of $x^3$ given $x$ and the moments.

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments E$(x^n)=\mu_n$. I need to find the ...
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1answer
38 views

Introduction Mathematical Statistics, finding the Mean Square Error of estimators

I'm working on a problem from my introduction to mathematical statistics course. So far, I've done the following work: Let $X_{1},...,X_{m}$ and $Y_{1},...,Y_{n}$ be independent samples form the ...
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1answer
49 views

Let $X_1,X_2…X_n$ be a random sample from $N(\mu,\sigma^2)$. Find the umvue of $\mu^3$. [closed]

Note: I know lehman scheffe theorem and that sample mean is umvue of $\mu$. But how can we find the UMVUE of hiher powers of $\mu $?
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19 views

How to compare asymptotic distribution of MLE and OLS estimators?

I have the following problem on my Statistics Problem Set: Consider the model \begin{equation*} y_{i}=x_{i}^{\prime }\beta +u_{i}, \end{equation*} where $\left( x_{i}^{\prime },u_{i}\right) $ ...
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29 views

Low-rank approximation of covariance matrix

I am reading a paper in which the author expresses the log-likelihood function for a gaussian as $L(\Theta^{(k)}) = -N$ log det $R^{(k)}$ - tr $\left[{R^{(k)}}^{-1} \hat{R}\right]$ where $N$ is ...
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Estimate parameters of Wishart matrix.

Given a sequence of real Wishart matrices $W_1 , \cdots , W_k \sim \mathcal{W}_m(n,\Sigma)$ where $\Sigma$ is a singular matrix. Are there good estimates for the degrees of freedom? The MLE for $\...
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1answer
40 views

Estimate degrees of freedom in sample variance.

Given a sequence of independent identically distributed random variables $X_1,\ldots,X_m \sim \chi^2_n / n$ is there literature on estimates for the degrees of freedom $n$? In an attempt to find the ...
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Estimation of parameters from independent but non-identical random variables

I am looking for some reference (books/papers/slides, etc) for estimating parameters from independent but non-identical random variables by using order statistics. The model is the following. There ...
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1answer
125 views

UMVUE of $\frac{\theta}{1+\theta}$ and $\frac{e^{\theta}}{\theta}$ from $U(-\theta,\theta)$ distribution

Let $X_1,X_2,\dots, X_n$ be rvs with pdf: $$f(x\mid \theta)=\frac{1}{2\theta}I(-\theta<x<\theta)$$ Find UMVUE of $(i)\dfrac{\theta}{1+\theta}$ and $(ii)\dfrac{e^{\theta}}{\theta}$. Note ...
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1answer
23 views

Show that the sample variance is an unbiased estimator of $\lambda$ for the Poisson distribution

I am trying to show that the sample variance is an unbiased estimator of $\lambda$ for a Poisson distribution. Let $(X_1, \dots, X_n)$ be a random sample from a Poisson distribution with mean $\...
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1answer
58 views

Beta distribution as a member of the exponential family

I come across the beta distribution quite frequently when solving exercises for my statistics class. However, I have not been able to fully grasp how to work with it. Exponential family form is: $$...
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1answer
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Gamma distribution family and sufficient statistic

Let X $=X_1,...X_n$ be a random sample iid from the probability density function: $$ f(x;\theta)=\frac{\Gamma(\theta)\sin(\pi\theta)\theta^{1-\theta}}{\pi}e^{-\theta x}x^{-\theta}$$ $x>0, 0<\...
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2answers
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ML estimation with given samples

Let $X_i,...,X_n$ be a random independent sample from a distribution with pdf $$ f(x;\theta)= (\theta + 1)x^{-(\theta+2)},$$ where $x>0$, and $\theta > 0$. What is the ML estimate for the ...
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1answer
37 views

derive sufficient statistic from a random independent sample from a weibull distribution

Suppose $X_i$ is a random independent sample from a Weibull distribution $$ f(x) = \frac{\beta}{\theta^\beta}x^{\beta-1}\exp\big(-(\frac{x}{\theta})^\beta\big)$$ Find a sufficient statistic for ...
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1answer
25 views

derive asymptotic distribution of the ML estimator

Let $x$ be a random variable with probability density (pdf) $$f(x)= (\theta +1)x^\theta $$ where $\theta >-1$. The expressions for its mean and variance are $$E(X)= \frac{\theta + 1}{\theta +...
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Problem understanding the concept of random sample

I'm reading a book on statistics and the definition given of a random sample doesn't make much sense to me. Here's the idea that I have of a random sample: in a given population, any sample of size $...
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1answer
63 views

Efficiency of estimators and UMVUE

(1) An estimator is efficient when it reaches the Cramer Rao Lower Bound. (2) If an estimator reaches the CRLB, then it is the UMVUE. (3) The UMVUE is always unique. If these points are correct, can ...
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23 views

need help identifying a formula for “pseudo-entropy”

Maintaining some old code, I've come across: $$\text{pseudo-entropy} = -x \log(x) + x ^{0.45} \cdot (1 - x) ^ {16}$$ I simply need a name for this formula so I can read up on what it's supposed to ...
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1answer
41 views

Find a sufficient statistic. [closed]

Suppose that $X_1,\ldots,X_n$ is a random sample from a distribution with pdf $$ f(x;\theta)=\frac{\theta^3}{2}x^2e^{-\theta x}, \quad 0<x<\infty$$ where $0<\theta<\infty$ Find a ...
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24 views

calculate variance of unbiased estimator in Rayleigh distribution

Given : $\hat{\theta} = \frac{\sum X_i^2}{2n} $ and $ E(X^2) = 2\theta $. $\hat{\theta}$ is unbiased estimator for $\theta$ based on i.i.d sample from $f(x;\theta)= \frac{x}{\theta}e^{\frac{-x^2}{2\...
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2answers
42 views

constructing a 95% confidence interval - manipulating inequalities

given the asymptotic distribution of $\hat{\theta_1}$ construct a 95% confidence interval for $\theta$ for large samples: $\hat{\theta_1} = \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}}$ I ...
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1answer
75 views

Minimum mean squared error of an estimator of the variance of the normal distribution

I am trying to find the estimator of the variance $\sigma^2$ of a normal distribution with the minimum mean square error. From reading up, I know the unbiased estimator of the variance of a Guassian ...
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1answer
48 views

Getting different parameters of distribution when using different methods

I have $n$ observations of variable $X$ and now I want to estimate parameters $a$ and $\lambda$ of gamma distribution ($a$ is more known as $\alpha$ and $\lambda$ as $\beta$ but that is how I was ...
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1answer
42 views

Mean squared error calculation [closed]

If $ X_1,...,X_n$ ~ $N(\mu, \sigma^2)$ where $\mu$ is known and $\sigma^2$ is unknown, calculate the MSE of $V^2$ $V^2 = \frac1n \sum_{X_i}^n Var(X_i) =\sigma^2$ Therefore: $MSE(V^2) = Var(V^2) = \...
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1answer
20 views

Effect of scaling a probability distribution on median of distribution.

Say a probability distribution has density function $f(x),x\in[a,b]$ and cumulative distribution function as $F(x)$. Consider the scaled distribution $g(x)=\frac1\theta f(\theta x),x\in[a\theta,b\...
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Bayes estimator dominates unbiased estimator

Let $\theta>0$ and $Y\sim N(\theta,1)$, and we want to estimate $\theta$ from $Y$ . Here, we already know $\theta>0$. Let \begin{equation*} \pi(\theta)=\begin{cases} 1\ \ (\theta \geq 0)\\ 0\ \ ...
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Estimation with iid uniformly RVs

Problem Let $Y_i, X, V_i$ be scalar random variables $\forall i=1,2,\dots, p$, defined as follow: $X\sim \mathcal{N}(\bar{x}, \sigma_x^2)$, i.e. $X$ is a Gaussian variables with expected value $\bar{...
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How do I choose the polynomials for a stochastic filter? - Transfer functions + Extended Least Square

I'm buildning a Mimimum Variance Controller(MVC) but I having som trouble to select the stochastic filter. First of all! To build a MVC, you need a ARMAX model, in other words polynomial who look ...
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two forms of spectral density, how to explain their equivalence?

The mathematical definition of the spectrum of a stationary process $\mathbf{x}(t)$is to take the Fourier transform of a finite segment $$\mathbf{X}_T (u) = \frac{1}{T} \int_{-T/2}^{T/2} \mathbf{x} ...
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2answers
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Three values for quadratic equation?

What is the number of value of $a$ for which the equation $$(a^2-1)x^2-(a^2-3a+2)x+a^2-8a+7=0$$, in $'x'$ possess, three distinct roots? I got confused over the fact that how does a quadratic ...
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Can a weakly consistent estimator beat a strongly consistent one?

Suppose we have two estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ of $\theta$, both with the same bias. If we have $$ \begin{align} &\hat{\theta}_1 \xrightarrow{a.s.}\ \theta \\ &\hat{\...
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1answer
42 views

How should I write the difference equation of a polynomial equation?

I going to estimate the polyomial $R^*$ and $S^*$ from $$ y(t)= \frac{R^*}{A_o^*(z^{-1}) A_m^*(z^{-1})}u(t) + \frac{S^*}{A_o^*(z^{-1}) A_m^*(z^{-1})}y(t)$$ $A_o^*(z^{-1})$, $A_m^*(z^{-1})$ polynomals ...
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2answers
107 views

Taking derivative of objective function

Given $x_t$ and $c_n$, the objective function is defined as follows $$J (\theta) := \sum_{t=1}^L \Big( x_t - \sum_{n=1}^N c_n \, \underbrace{\exp \left( -\left(\frac{t-n}{\theta}\right)^2 \right)}_{=:...
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1answer
105 views

Maximum Likelihood Estimator (MLE) of $ \theta $ for the PDF $ f( x; \theta) = \frac{1}{2}(1+\theta x)$

I need to find de maximum likelihood estimator of $\theta$ for $f(x)=\frac{1}{2}(1+\theta x)$, $-1 \leq x \leq 1$ I start with: $L(\theta)=f(x_1,\theta)f(x_2,\theta)\cdots f(x_n,\theta)$ $$L(\theta)=...
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1answer
36 views

estimate the midpoint of an interval given by n random variables

$X_1,...,X_n$ be independent, uniformly distributed random variables on the interval $[a,b]$ for unknown $a,b \in \mathbb{R}$ and $a < b$. The midpoint of the interval is supposed to be estimated ...
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1answer
46 views

How can I estimate a discrete transfer function? Recursive Least Square

This is going to be a large fun question about practical estimation for real world problems. Assume that we have a poor damped system described with this transfer function. $$G(s) = \frac{4.5}{1 + 0....
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80 views

Optimize: $\arg\min_{\theta>0} \quad \big\| x - \sum_{n} c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$

Can somebody help me get started with the following problem? I want to solve: $$\theta^* := \arg\min_{\theta>0} \quad \big\| x - \sum_{n=1}^N c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$$ where $x$ ...
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Does the posterior approach the form of a conjugate prior in any meaningful sense?

The Bernstein von-Mises theorem says that, "the posterior distribution for unknown quantities in any problem is effectively asymptotically independent of the prior distribution (assuming it obeys ...
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50 views

Expected value or MAP to summarize a posterior distribution

Suppose we have a posterior distribution over a parameter, what would be the right way to summarize it? On one hand, we have the MAP (maximum a posterior), where we would get the mode (maximum) of the ...
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33 views

Pareto estimation

The data is the observations $x_1,...,x_n$ from Pareto distributions. Find MLE estimator for $f(x;a,k) = { ka^k \over {x^{k+1}}}$, $x \in (a,\infty)$, $k$ known, $a$ parameter. So... $f(x;a,k) = { ...
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1answer
40 views

Finding error of least-squares coefficients.

I've studied the least-squares method from the Calculus approach, using polynomials of degree $n$: for a set of data points $(x_i,y_i)$, $i=1,...,n$, define a function $E=(y_i-f(x_i))^2$ with $f(x;\...
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2answers
33 views

Show if the estimator is unbiased

Suppose that the random variables $Y_1... Y_n$ satisfy $Y_i = \beta x_i + ϵ_i$ , i = 1...n where $x_i$ are fixed constants and the $ϵ_i$ are iid Normally distributed random variables with mean ...
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1answer
33 views

How to estimate formula by a table

Given some table, where $y=f(x)$, and I believe $f(x)=a*x^3+b*x^2+c*x+d$ But the trick is that values in table are rounded (I dont know is it floor, ceiling or just round to nearest) not every value ...
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1answer
41 views

Is it possible to answer this problem with standard Central Limit Theorem or should we use Lindeberg-Feller CLT?

I have the following problem on my Statistics I problem set: Suppose that $X_t = \mu + U_t$, where $U_t = V_t + \rho V_{t-1}$ and $V_t$ are iid standard normal variables. Apply a CLT to ...
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Optimiziation Regarding GMM Estimator Arellano-Bover

I have been struggling a lot with implementing Arellano-Bover estimator for the following panel model: $Y_{it} = \rho Y_{it-1} + \lambda_i + \varepsilon_{it}$, with $Y$ observed data, $\rho$ ...
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1answer
16 views

How to work out the variance of this estimator?

We want the variance of this biased estimator. The answer is: $$σ^2/4n$$ Why is it not this? $$σ^2/2n$$ I.e. wouldn't it be $$1/2n^2 * n * σ^2$$ Any clarification appreciated. Thanks!