Questions tagged [estimator]

For questions about estimators; an estimator is an approximations of a parameter that is using the avaiable data. There could be multiple estimators for a particular parameter.

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80 views

MVUE for a function of variance of Normal Distribution

Let $X_1, X_2, ..., X_n$ be a random sample from a $N(\theta_1,\theta_2)$ distribution. Find the uniformly minimum variance unbiased estimator of $3{\theta_2}^2$. Using factorization theorem, I found ...
2
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1answer
39 views

Asymptotic Properties of OLS estimators

This is an econometrics exercise in which we were asked to show some properties of the estimators for the model $$Y=\beta_0+\beta_1X+U$$ where we were told to assume that $X$ and $U$ are independent. ...
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1answer
148 views

If best unbiased estimator exists then it's maximum likelihood estimator?

Our teacher proved in class that if the best unbiased estimator exists, then it is an MLE using a theorem that if $\hat{\theta}-\theta$ is proportional to the score of $\theta$ with probability $1$, ...
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22 views

What does symmetrised tuple mean?

I was reading the paper "Remarks on some nonparametric estimates of a density function" by Murray Rosenblatt (1956) and in one part he writes that an estimate $S(y;X_1,...,X_n)$ of the density ...
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0answers
18 views

On the pooled variance in the estimating difference of means

Suppose that we need to compare means of two sample spaces, say P, Q. Let $\mu$ and $\nu$ be the expectation value of $P$ and $Q$, respectively. Consider $m$ independent random samples from $P$ and $n$...
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25 views

Finding approximate variance of the logistic transform of an estimator for the survival function

So, let $\hat{S}(t)$ be the Kaplan–Meier estimator of an unknown survival function $S(t)$. Consider the logistic transform of this estimator: $$\hat{S}_{\text{logistic}}(t) = \log \left(\frac{\hat{S}(...
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0answers
30 views

Unbiased estimator of linear regression

I need to check if an estimator $\hat\beta = \frac{1}{n}\sum\limits_{i=1}^{n} \frac{Y_i-\bar{Y}}{X_i-\bar{X}}$ of regression $ Y_i = \alpha +\beta X_i + \epsilon_i, i = 1,...n $ is unbiased. My idea ...
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0answers
33 views

Prove an estimator is biased

I want to prove that an estimator in biased. For a big n, like $10^{13}$ we define a distribution of values $x_i \in (0,1]$ for $i \in [n]$. We want to estimate with some error $\epsilon$ the value ...
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33 views

Given an MLE $\hat{\theta}$, how to show $f(\hat{\theta})$ is also an MLE?

Given an injective function $f: \Theta \rightarrow \mathbb{R}$ and an MLE $\hat{\theta}$ of $\theta ^*$, how do I prove that $f(\hat{\theta})$ is an MLE of $f(\theta^*)$. I know that because $f$ is ...
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1answer
490 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let $X=(X_1,X_2,...,...
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0answers
88 views

Optimal unbiased estimator

I have the sample $X_1,...,X_n$ of i.i.d. from $U(\theta - 1/2; \theta +1/2)$. It is well known that $T = (X_{(1)}; X_{(n)})$ is a sufficient but not complete statistic, because $X_{(n)}-X_{(1)} - (n-...
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60 views

Point Estimator.

When the point estimator under consideration has a pdf , the $P[T=\tau(\theta)]=0 $ , where $\tau(.)$ is some function of parameter $\theta$ and $T$ is an estimator of $\tau(\theta)$. But I did ...
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33 views

Determine all $\overrightarrow{a}$ for which $T_\overrightarrow{a}$ is an unbiased estimator for the variance $Var(X)$ of $X$.

consider a random variable $X$, whose mean value is known 𝜇 and stochastically independent repetitions $X_1,...,X_n$ of $X$. For each vector $\overrightarrow{a}=(a_1,...,a_n) \in \mathbb{R}_{n} \text{...
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9 views

conservative two-sided equal-tailed confidence interval

Consider a random sample $X_1,...,X_n$ from a Bernoulli distribution with unknown parameter $p$ that describes the probability that $X_i$ is equal to $1$. The maximum likelihood ($ML$) estimator for $...
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1answer
52 views

Mean square error of MLE

$\textbf{X}=(X_1,...,X_n)$ is a random sample from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \...
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17 views

Deriving OLS estimator dependent on X being orthogonal

Let $X \in \mathbb{R}^{n \times d}$ be a predictor matrix with orthonormal columns and $y \in \mathbb{R}^n$ as output vector and vector (OLS estimator) $\beta_{LS} \in \mathbb{R}^d$ for estimating a ...
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1answer
26 views

how do i find the variance of an estimator?

If the Estimator was simply the sample mean $s=\frac{\sum{x}}{n}$ taken from a binomial distribution (a random example) how would i calculate the variance of this? I am trying to use the difference ...
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18 views

How to derive formula of variance of sample median?

I wonder how to derive the formula of variance of median of sample. I searched it for a day but couldn't find, so I'm asking help here. You may just notice me the link of it. I saw the result form is ...
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26 views

Computing the bias of the sample autocovariance with unknown mean

In "Introduction to statistical time series" by W. A. Fuller (1976), two definitions of the sample autocovariance with lag $h$ for a signal of length $n$ and unknown mean are proposed, namely $$\...
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18 views

Estimation of statistics of an iid sequence using Kalman filtering

I am given the sequence $v_0, v_1, \ldots$ of i.i.d. Gaussian random variables $\sim \mathcal{N}(\mu,\sigma^2)$ for some unknown parameter $\mu$ ($\sigma$ is known) which I need to estimate given the ...
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10 views

Calculating Total Error In $g(x,y,z) = \frac{x-y}{z}$ Given Errors In Variables $x,y,z$

So I have a function given by: $$ g(x,y,z) = \frac{x-y}{z} $$ where I can only estimate the values of $x,y,z$. I assume that these estimates are uncorrelated. From the Cramer-Rao lower bound, I know ...
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27 views

Finding method of moment estimator from exponentially distributed random sample.

Given a random sample $X_1,...,X_n$ that are $IID$ from an exponential population with a unknown parameter $\lambda>0$ The parameter of interest is $\theta=\frac{1}{2}\sqrt{\lambda}$ Im looking ...
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32 views

Show that the histogram estimator is an unbiased estimator

So the question is asking to show that the histogram estimator $\mathbb{E}_p[\hat{p}_n(x)]$ is unbiased. So I worked and found that the histogram estimator is $$\hat{p}_n(x)=\sum_{j=1}^{m}(\frac{\hat{...
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7 views

Can (G)MM estimator be used to estimate ARMA models?

How to construct (G)MM estimator for the following model? $$ y_t = \beta_0 + \beta_1 y_{t - 1} + \varepsilon_t\\ \varepsilon_t = u_t + \alpha_1u_{t-1} + \alpha_2u_{t - 2} $$ Where $u_t$ is $N(0, \...
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0answers
23 views

Distribution of sample mean squared

I have an iid sample of n Poisson($\theta)$ RVs. I have derived that the MLE for $\phi = \theta^2$ is $\bar{x}^2$. I need to show this estimator is consistent. To show consistency I have to show $Var(...
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16 views

What is the connection between MSE in regression and MSE for prob. distributions?

When assigning a goodness of fit to the least squares regression, one often naturally takes the mean squared error (MSE) or average residual: $$MSE = \frac{1}{n}\sum_{i}{(y_{i}-\hat{f}_{D}(x_{i}))^2}$$...
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1answer
53 views

Utilizing the Factorization Theorem on unknown distributions

I have two unknown distributions \begin{align*} f(x;\theta) &= \frac{1}{x\cdot \ln \theta} & 1 < x < \theta \end{align*} and \begin{align*} f(x;\beta) = \frac{\beta}{(1+x)^{\...
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0answers
19 views

Asymptotical normality of method of moment estimator

Recently I came across a theorem in All of Statistics by Larry Wasserman, which is stated as follow: Let $\widehat{\theta}_{n}$ denote the method of moments estimator of $\theta$. Assume appropriate ...
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41 views

How to find the UMVUE of Bernoulli distribution?

I was trying to find a uniformly minimum-variance unbiased estimator (UMVUE) for $2p^2+3(1-p)$ for $X_i$ i.i.d. $ Ber(p) $ i.e. Bernoulli dist having a probability parameter $p$. Based on my ...
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22 views

can the estimator of beta1 be consistent if the estimator of b2 isnt?

I know how to handle equations like $$y_i = \alpha + \beta x_i + u_i$$ but when I get stuff like $$y_i = \alpha + \beta_1x_1 + \beta_2x_2 + u_i$$ I don't understand how to handle them both, for ...
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0answers
16 views

How are the random variables $Y_i$s identically distributed in this example?

I am reading Probability and Statistics (8th ed.) by Devore and cannot understand some aspects of example 6.5 from page 246: For point estimation we require that the random variables have ...
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13 views

Proof that control function estimation leads to the same result as a 2SLS estimator?

Consider a model with single endogenous regressor: $$ Y_{1i} = X_i^T\beta + Y_{2i}\gamma + \varepsilon_i \\Y_{2i} = Z_i^T\pi + v_i $$ Control function estimation solves the endogeneity problem in the ...
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1answer
42 views

Construct an Unbiased Estimator

I am currently trying to prove that $\hat{\beta}$ is not an unbiased estimator of $\beta$. After proving this I need to to construct a unbiased estimator for $\beta$. I know that $$\hat{\beta} = \frac{...
0
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1answer
20 views

If the Cramer Rao's Lower Bound tends to zero, is the estimator efficient?

I apply a nonlinear transformation to a linear estimator $\hat{\alpha}=f(\hat{\theta})$. Then I find Cramer Rao's Lower Bound, and it asymptothically goes to zero. Does it means that my new estimator $...
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1answer
27 views

How to find asymptotic variance for mle with ln

I have $X_1,...,X_n$ that are iid random variables with density function $$f(x) = {\alpha}x^{\alpha-1}$$ where $0<x<1$ and ${\alpha}$ is an unknown parameter. The MLE estimator I got was $\...
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1answer
20 views

Proving it is a biased estimator

How would I show that if a distribution is equally likely to take values of $1$ or $4$, then the statistic $s_{n-1}$ forms a biased estimator of $\sigma$? My thoughts: Do I find the expression$s_{n-1}...
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0answers
16 views

how to find the efficient estimator

I know in general if we have two unbiased estimator $T_1$ and $T_2.$ then $T_1$ is a more efficient estimator of $T_2$ if $Var(T_1)<Var(T_2) $. However in this problem, it asked me to what that an ...
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0answers
18 views

Computing MMSE estimator in recursive state dynamics

Consider the $p\times p$ matrix $\pmb{A}$ and suppose $\theta_n, n\ge 0$ are $p$-dimensional vectors we want to estimate according to MMSE criterion. We have the recursive relation $\theta_n = \pmb{...
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2answers
111 views

Unbiased estimator for $p^2$. Bernoulli distribution.

Let $X_{1},...,X_{n}$ be a random sample from Bernoulli (p), find an unbiased estimator for $p^{2}$. I think It's the same estimator for $\mathrm{Bin}(n,p)$ so: $${V}(X) = np(1-p) = np - np^2 \\ p^2 ...
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17 views

Error term covariance of two-stage estimators

I am looking into properties of two-stage estimators (2SLS). My setting is as follows: 1) $y_1 = y_2\beta+\epsilon$ 2) $y_2 = z \pi_2 + \eta$. Equation 1 represents the second-stage estimation, ...
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0answers
25 views

Bayesian estimator under 0-1 loss

If you were given the posterior distribution $P(\theta |X)$, as well as $P(X)$, and under no further assumptions, how could the Bayesian estimator be found under the 0-1 loss. I understand that I ...
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0answers
28 views

Checking if estimator $\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}$ is unbiased

Let $P_\theta(X=x) = \left(\frac{\theta}{2}\right)^2(1-\theta)^{1-x^2}$ for $x=-1,0,1$ Let $\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}$ be an estimator of $\theta$. Is it unbiased? So the ...
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0answers
12 views

Calculate the estimator and its standard deviation by Parametric Bootstrap

The third quantile of a distribution function F is the point q such that F(q) = 0.75. Note the q (α; λ) (Gamma distribution). Then, the quantile would be estimated by ^q = q (^α; ^λ). We do ...
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0answers
34 views

Limit at $\theta \rightarrow 0$ is showing that $\frac{1}{n}\sum^n_{i=1}X^2_i$. is not the MLE?

Let $\mathbf X = (X_1, . . . , X_n)$ consist of independent and identically Normal $N(0, θ)$ random variables, with mean $0$ and variance $θ > 0.$ The density of $f_{\theta}(\mathbf x)$ is equal ...
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0answers
14 views

How to show the first order condition of LAD estimator fo median regression is of order $o_{p}(\frac{1}{\sqrt{n}})$?

Given the model $y_{i}=x_{i}^{\prime}\beta_{0}+\epsilon_{i}$ with $E[sgn(\epsilon_{i})|x_{i}]=0$ and $E[sgn(\epsilon_{i}-c)|x_{i}] \neq 0$ if $c=c(x_{i}) \neq 0$, where $sgn(\epsilon)=1-2 \cdot 1(\...