Questions tagged [minimum-variance]

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What are the uniformly minimum variance unbiased estimators (UMVUE) for the minimum and maximum parameters of a PERT distribution?

I believe the answers to this question are the sample minimum and the sample maximum, but I have not been able to find a reference or proof of this.
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UMVUE for a Uniform distribution [duplicate]

How did we derive the PDF and CDF highlighted in green? Thanks
learn_to_code1's user avatar
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Minimizing variance of sequence of independent but not identically distributed random variable

I tried to work on the problem Let $(X_n)$ be a sequence of independent random variables with $E[X_n]=\mu$ and $Var[X_n]=n$ for every $n \in \mathbb{N}$. Find the statistic of the form $\sum_{i=1}^...
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For a given $Y$ , what is the minimum variance of a $X$ such that $E[Y|X]=X$?

Suppose $Y$ is a given (real-valued continuous) random variable. We define any variable $X$ as exogenous to $Y$ if $\forall X: E[Y|X]=X$. The question is this: For a given $Y$ , What is the minimum ...
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Foreacast Combinations: derivation of minimum MSE / variance approach

I am just despairing of the derivation of the minimum variance procedure. The method of the combination of forecasts was first established in 1969 by Bates and Granger. They also invented the minimum ...
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Finding UMVUE for exponential sample [duplicate]

Let $X_1,...,X_n$ be a random sample of i.i.d. exponential distribution with probability density function $$f(x|\theta)=\frac{1}{\theta}exp(-\frac{x}{\theta}), \ x\geq0$$ Let $S_n=\sum_{i=1}^nX_i$ and ...
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Experimental Design: Choose Data Points to Minimize Quadratic Term Variance in Multiple Regression

$\newcommand{\eps}{\varepsilon}\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Suppose that you wish to fit a model $$Y=\beta_0+\beta_1x+\beta_2x^...
Adrian Keister's user avatar
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Prove that the variance of the Generalized Least squares estimator is less than the variance of the OLS estimator

Suppose, we consider the following regression model, $$Y = X\beta + \varepsilon$$ where $\varepsilon$ ~ $N(0, \sigma^2V)$ and V is a known $n\times n$ non-singular, positive definite square matrix. ...
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Proof Sample Variance is Minimum Variance Unbiased Estimator for Unknown Mean

I am trying to prove that the unbiased sample variance is a minimum variance estimator. In this problem I have a Normal distribution with unknown mean (and the variance is the parameter to estimate so ...
Susy A.'s user avatar
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1 answer
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Error in Derivation for Control Variate Variance?

I'm trying to derive the variance for a control variate estimator, but I seem to be missing a term that allows me to end up with the covariance in the final answer. Let $f(x)$ be my function and let $...
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How to find an minimum variance unbiased estimator for an integer parameter?

Consider multiple observations $x[n]$ for an integer parameter $A$ under White Gaussian Noise $w[n]$: $x[n]=A+w[n]; \quad$ $n=0,1,...,N−1$ with $w[n] \sim N(0,σ^2)$. Is it possible to have an minimum ...
Thiruppathirajan's user avatar
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1 answer
505 views

Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
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The UMVUE of ratio of parameters for two uniform distributions,

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are ...
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Understanding Rao-Blackwell [duplicate]

From Casella and Berger: Let $W$ be an unbiased estimator of $\tau(\theta)$ and let $T$ be a sufficient statistics for $\theta$. Define $\phi(T) = E[W|T]$. Then $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
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1 answer
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Better to Minimize Absolute Error or Sum of Squared Error?

I have an Excel model which predicts the number of customers for a given month. The prediction depends on a churn rate. I have the absolute error (actual vs predicted), along with squared error and ...
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1 answer
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Showing that the minimum-variance estimator is the OLS estimator

Recap of required theory Consider the following regression: $$y_i = \alpha + \beta x_i + u_i \tag{1}$$ where $y_i$ are iid and $x_i$ are deterministic (i.e. fixed). We know that the OLS estimator $\...
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How to measure how "good" or accurate a probability distribution is? Entropy, variance or what?

How can one measure the accuracy of the probability distribution of, say, a physical magnitude? I know one good candidate is the entropy, which measures the amount of information one has about the ...
Godoy's user avatar
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1 answer
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How does one minimize the standard deviation to find optimal parameters? [closed]

When doing a generalized least squares fit for a line, one computes the residuals as (y - (m*x + b))**2, where (x,y) are the ...
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2 answers
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How is this minimum variance worked out for this importance sampling estimator?

I was stuck with the function 17.13 in the open source book deep learning on page 590. For short, the question is that, For the importance sampling estimator: $$\hat s_q = \frac{1}{n}\sum_{i=1, x^{i}...
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0 answers
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Maximum likelihood estimator and minimum variance challenge

A random sample of size $n_1$ is to be drawn from a normal population with mean $\mu_1$ and variance $\sigma^2_1$. A second random sample of size $n_2$ is to be drawn from a normal population with ...
Clong123's user avatar
3 votes
1 answer
80 views

Good parameter estimates vs good computed moment estimates

Suppose I have a distribution from a known parametric family f(x; θ). I have a sample from that distribution. From the sample, I estimate values for the parameters. Suppose I have estimators that I ...
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3 votes
1 answer
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Are MVUEs and MLEs always functions of a minimal sufficient statistic?

Is it the case that both minimum variance unbiased estimators (MVUEs) and maximum likelihood estimators (MLEs) are always functions of a minimal sufficient statistic? If so, how do we know? If not, ...
Bratt Swan's user avatar
2 votes
1 answer
271 views

How does one can guarantee that any unbiased estimator is MVUE due to it containing a minimal sufficient statistic?

Sufficiency is okay. But I don't really get it why the fact guarantees it has minimal variance? Can anyone explain to me somewhat intuitively?
Jason Park's user avatar
4 votes
1 answer
1k views

Minimum-variance unbiased linear estimator

Suppose that it is known that the mean of RV $X_i$ is $\mu_i\theta$, (i = 1, 2,..., n), where $\mu_i$ are known constants, whereas $\theta$ is unknown. Let $\Sigma$ be the variance matrix of the ...
Dony's user avatar
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Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic

Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$ a)Show that $U_2=\mathbb{E}(...
James Snyder's user avatar
9 votes
0 answers
2k views

Methods of Proving that a UMVUE does not exist?

Are there efficient methods of showing when a UMVUE does not exist? I can think of the trivial case when no unbiased estimators exist at all. But that's not really interesting. I feel like this ...
TheChosenOne's user avatar
5 votes
1 answer
297 views

A Proof of Tukey's Inequality

Suppose that $W_1,W_2,...,W_n$ are uncorrelated unbiased estimators of a parameter $\theta$. Consider $W=\sum_{i=1}^na_iW_i$ such that $E(W)=\theta$ and $Var(W_i)=\sigma^2_i$, where the $a_i$'s are ...
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2k views

Uniform minimum variance unbiased estimator [duplicate]

Let $X_1, X_2, ..., X_n$ be an iid random sample from a Poisson$(\lambda)$ distribution: a) find the UMVUE of $\theta$ = $\lambda^k$ for $k > 0$ a known integer b) find the UMVUE of $\tau = e^{-\...
David's user avatar
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10 votes
1 answer
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Optimal importance sampling with ratio estimator

We want to approximate the following expectation: $$\mathbb{E}[h(x)] = \int h(x)\pi(x) dx$$ Where $h(x)$ is an arbitrary function and $\pi(x)$ is a distribution, also for simplicity, let's assume ...
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