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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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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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19 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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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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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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Proof of Consistent estimators for mean and variance [on hold]

Show that the statistic $\bar {x}$ a consistent estimator for mean $\mu$ $\frac{n}{n-1}s^2$ is a consistent estimator for variance $\sigma ^2$
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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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10 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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16 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
31 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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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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14 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 ...
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26 views

Finding BLUE of $\theta$ where $X_1,\ldots,X_n$ have common pdf $f(x)=\frac{1}{2\theta}e^{-|x|/\theta}$

Let $X_1,...,X_n$ have the common pdf $$f(x)=\frac{1}{2\theta}\exp\left(-\frac{|x|}{\theta}\right)$$, where $x$ can be any real number and $\theta$ is positive. I’m trying to construct the best ...
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Finding standard errors of maximum likelihood estimates [migrated]

Suppose we use Maximum Likelihood estimation to estimate certain parameters in a model. Furthermore, suppose that the log likelihood function can not be solved analytically and thus must be optimised ...
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Estimate parameters through transformation and linear regression (statistics)

A model for a chemical process is yi=V_m/(k + xi) ,where xi and yi, the predictor (independent) and response(dependent)variable, respectively, are obtained from data. V_m and k are the two ...
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Parameter estimation for Stochastic differential equation

I have a process $X(t)$ definied on some finite time horizont $[0,T]$ and I know that my process satisfies the following SDE: $dX(t)=\mu dt + \sigma dB_t$. where $B$ is a standard brownian motion. ...
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Definition of BLUE

I am tasked with finding the best linear unbiased estimator (BLUE) for the population mean based on $X_1,...,X_n$ iid $Poisson(\lambda)$. My question is, am I supposed to find the best linear unbiased ...
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Minimal sufficiency and completeness

Let $X_1,X_2... X_n$ be i.i.d $N(\theta,\theta^2)$.Then why is $(X_{bar}, S^2)$ not complete despite the fact that $f(x,\theta)$ belongs to $k$-parameter exponential family and jointly minimal ...
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How do I choose functions to get a better M-estimator?

Suppose $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}$ are random vector and random variable. Suppose I have $f_1(\cdot)$ and $f_2(\cdot)$ such that $E[Y|X]=f_1(\theta_0^T X)$ and $E[Y|X]=f_2(\theta_0^T X)$...
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MLE of $\sigma_x^2$ with $W=X-\delta$

Consider $X_1,X_2,...,X_n$ a random sample of an statistic population that is modeled by the density function:$$f(x)= \frac{1}{\lambda}e^{-(X-\delta)/\lambda} I_{[\delta,\infty)}$$ Calculate MLE ...
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28 views

Maximum likelihood for $\lambda$ and $\delta$

Consider $X_1,X_2,...,X_n$ a random sample of an statistic population that is modeled by the density function:$$f(x)= \frac{1}{\lambda}e^{-(X-\delta)/\lambda} I_{[\delta,\infty)}$$ Obtain maximum ...
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Understanding of sufficient statistics

I have came across a question regarding sufficient statistics, but I cannot understand it under this context. Let $P$ be a finite family with densities $p_i, i = 0, \dots , k$ and for any x, let $S(x)...
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1answer
25 views

CAN and BAN estimators of $\sigma^2$

Let $X_1,X_2,..X_n \sim^{\text{i.i.d}} N(0,\sigma^2)$ Show that $T_n=\frac{1}{n}\sum_{i=1}^{n} X_i^2$ is a Consistent and Asymptotically Normal estimator(CAN) as well as the Best Asymptotic Normal ...
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When is a sufficient statistic also complete?

A sufficient statistic $T$ for the parameter $\theta$ is said to be complete if and only if: $$ E(g(T)) = 0 \ \ \forall \theta \Leftrightarrow g(T) = 0$$ Then it's said that a vector$\mathbf{T}$ ...
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36 views

Method of moments estimator for distribution with density $p_{\theta}(x)=\theta(1+x)^{-(1+\theta)}$

Let $X_{1},...,X_{n}$ be a sample from probability distribution with density $p_{\theta}(x)=\theta(1+x)^{-(1+\theta)}$ with $x>0$ and $0$ elsewhere, with $\theta>1$ unknown. Determine the method ...
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Was von Neumann right that 'with four parameters you can fit an elephant'?

I was looking into free parameters and found this previous nice answer to What is a “free parameter” in a computational model? Is it possible to actually demonstrate that with x free parameters, any ...
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42 views

Optimizing a normalized estimator for a linear channel

Assume $y=Hx+n$, how to design an optimal linear filter $W$ such that $$\mathbb{E}\left(\frac{\|x^{\dagger}Wy\|^2}{\|Wy\|^2}\right)$$ is maximized, where $\frac{Wy}{\|Wy\|}$ can be reviewed as a ...
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1answer
44 views

Can this be solved with Least Squares?

I have images (data is represented by grayscale values from 0 to 255) and the total sum of the grayscale values represent the mass of the object. I know that every grayscale is related to mass, given ...
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32 views

Cramer-Rao Lower Bound for a Conditional Likelihood Function

I'm here looking for assurance that my interpretation is correct. Let the likelihood function under consideration be a conditional likelihood given by $$p(r|x;\theta)$$ where $r$ is some random ...
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How to calculte the Fourier Transform of a sovable chaos waveform?

Recently I am stucking in frequency estimation of a solvable chaos waveform. Its analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi) $$ where $u_m \sim U(-...
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Distinguish between gamma and log-normal distributions based on 95th percentile of a random variable

I know mean and variance of a skewed positive random variable $X$ analytically. in literature both gamma and log-normal distributions can be fitted to such a random variable. I know that to find the ...
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1answer
29 views

Unbiased estimator having a deviation less than $0.05$

In a mass production of items produced indepedently of eachother the probability of an item being defect is $p$. An unbiased estimator for $p$ is $\hat p = \frac{X}{n}$ where $X$ = amount of items ...
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1answer
23 views

Finding $a$ and $b$ such that $\hat u$ is an unbiased estimator

A chemist wants to decide the amount of a certain substance $\mu$ in a specific type of food. In the lab, the chemist has two measuring intruments $A$ and $B$. The results from the instruments can be ...
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Probability of average times called being inside an interval

Assume that when a man calls a restaurant to order some food, his call has a probability $0.2$ of reaching through. Every attempt is independent and we can assume that the restaurant is open every day....
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variance of estimator $\hat{\mu}_1=\bar{X_1}$

From a population with mean $\mu$ and variance $\sigma^2$, an independent random sample of size $n_1$ is extracted. The mean sample is $\bar{X_1}$. The next estimator is proposed: $$\hat{\mu}_1=\...
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Nonparametric Bayesian estimation of several black-box functions of different variables from their noisy sums

In order to introduce my problem, let’s start with the nonparametric estimation of a single unknown/black-box function $f:{\Omega _f} \to \mathbb{R}$ of a discrete variable $x$ in a finite domain ${\...
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Find the ML and LSE estimation of b in a Rayleigh distribution

Im having a hard time with a homework for a Statistical Course im taking. The question is as follows. " A Rayleigh distributed stochastic variable X have the density function $$ f_X(x)=\frac{x}{b^2} e^...
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Proof Verification: $\tilde{\beta_1}$ is an unbiased estimator of $\beta_1$ obtained by assuming intercept is zero

Consider the standard simple regression model $y= \beta_o + \beta_1 x +u$ under the Gauss-Markov Assumptions SLR.1 through SLR.5. Let $\tilde{\beta_1}$ be the estimator for $\beta_1$ obtained by ...
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What is an estimator for the “number of trials” given observed successes and the success probability?

The binomial distribution with $n$ trials, $k$ successes and success probability $p$ is given by $$P(k;n,p) = \binom{n}{k} p^k (1-p)^{(n-k)}, \quad k \in \{0,...,n\}$$ Suppose that we observe $k$ ...
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Deriving a normal log-likelihood ratio with a restricted parameter.

Suppose $X|\theta\sim N(\theta,1)$ and that $\theta$ is restricted, $\theta\geq 0$. Consider $H_0: \theta=\theta_0$ vs. $H_1: \theta\neq\theta_0$. Derive the log-likelihood ratio and its distribution ...
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1answer
20 views

How to use MLE for estimators

Suppose that $X_1, X_2, \ldots, X_n \sim N_p(\mu_x, \Sigma_x)$. If we assume that $\mu_x = k_1\mu_0$ where $\mu_0$ is known and $\Sigma_x$ is known. Derive the maximum likelihood estimator of $k_1$. ...
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34 views

Method of Moments estimator for Uniform $(-\theta,\theta)$

Let $x_1=2,x_2=1,x_3=\sqrt{5},x_4=\sqrt{2}$ be the observed values of a random sample of size $4$ from a distributions with probability density: $f(x|\theta)=\frac{1}{2 \theta} , -\theta \le x \le \...
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1answer
35 views

Bayes estimator for probability distribution with given prior density.

I'm working on a mathematical statistics problem. Let $X_{i},...,X_{n}$ be a sample from a probability distribution with density $p_{\theta}(x)=\theta x^{\theta-1}$ for $0 \leq x \leq 1$ and $0$ ...
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43 views

Asymptotic Normality of MLE Poisson Parameter (Lambda)

I'm need help with asymptotic normality of MLE. Example: $X_1,..., X_n$ iid $X_i$ ~ $Poisson(\lambda) $ Likelihood: $L(\lambda)=\prod_{i=1}^{n} \frac{e^{\lambda}\lambda^{x_i}}{x_i!}$ Log-...
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2answers
37 views

Method of moments estimator for $\theta^{2}$.

I'm working on a problem from my mathematical statistics book and the following is asked from me. Let $X_{1},...,X_{n}$ be a sample from a probability density function with density $p_{\theta}(x)= \...
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1answer
36 views

MLE estimation for two parameter pareto (With slightly different PDF)

The PDF is given as $f(X=x) = \frac{\alpha \lambda^\alpha}{(\lambda + x)^{\alpha + 1}} \forall_x > 0, \alpha > 0, \lambda >0$ I found the log-likelihood to be: $ln(L(\alpha,\lambda)) = n \...
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1answer
125 views

Matrix Regression for linear ODE system

Background I have the following homogeneous ODE system as an Initial Value Problem: $$ y'=A\cdot y\quad\wedge\quad y(0)=y_0 $$ where $y\in\mathbb{R}^{N\times 1}$ is the unknown vector and $A\in\...
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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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1answer
50 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 draw ...
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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 ...