Refers to an estimator of a population parameter that "hits the true value" on average. That is, a function of the observed data $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$ if $E(\hat{\theta}) = \theta$. The simplest example of an unbiased estimator is the sample mean as an ...

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9 views

Help in deriving unbiased estimator for a measure

There is a metric $H$ defined as = $\sum_{i=1}^{N} \min |u_i - u_j| * ..*|u_{i+d-1} - u_{j+d-1}|$ where $u$ is a multi dimensional vector of dimension $d$ and $u_i,u_j$ $\in \mathcal{R}^d$ are the ...
1
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0answers
5 views

Can someone provide an intuitive explanation of how Stein's Unbiased Risk Estimator can be used for signal denosing?

How does SURE just knows what the noise content of a signal is and how can such an estimator eliminate the noise? Note: I'm not very familiar with estimation theory in statistics
2
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1answer
33 views

Calculating the expected value and variance of an estimator of a normal quantile

I don't quite understand how to use the estimator function and the variance function and plug in the sample mean. I expected that we would plug in the value $\bar X - 1.645s$ into $E(s)$ and $V(s)$. ...
1
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1answer
49 views

Stein unbiased estimate of risk as surrogate for MSE

I am having trouble understanding Stein's unbiased risk estimate (Stein, C. M. (1981). Estimation of the Mean of a Multivariate Normal Distribution. The Annals of Statistics, 9(6), 1135–1151. ...
2
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0answers
34 views

Finding $Var(S^2), E(S^4),$ and unbiased estimator for $\sigma^4$ from random, normal samp

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution $N(\mu,\sigma^2)$ and let $S^2$ be the sample variance. (a) Find $V(S^2)$ and derive $E(S^4)$. (b) find an unbiased ...
1
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2answers
48 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
1
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0answers
27 views

Is this consistent and/or unbiased?

Does OLS produce inconsistent and/or bias estimators? The first example has the beta raised to a power. $$ y = β_0 + (β_1^2 x_1) + (β_2 x_2) + u $$ The second example has (beta + 1)estimator. $$ y ...
1
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1answer
39 views

Finding the unbiased variance estimator in high dimensional spaces

The problem comes from linear regression. Assume the regression function is linear, i.e. $$ f(X) = \beta_0+\sum_{j=1}^pX_j\beta_j $$ .Given a set of training data $(x_1, y_1),\ldots,(x_N,y_N)$,we try ...
0
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2answers
30 views

efficiency - bias trade-off

Under which conditions would a researcher choose optimally when there is a trade-off between the variance and bias of an estimator? I hope this question is not too broad... Any help would be ...
3
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62 views

Unbiased estimator with minimum variance for $1/\theta$

Let$ X_1, ...,X_n$ be a random sample feom a distribution $Geometric(\theta)$ for $0<\theta<1$. I.e, $$p_{\theta}(x)=\theta(1-\theta)^{x-1} I_{\{1,2,...\}}(x)$$ Find the unbiased estimator ...
2
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2answers
68 views

Obtaining an estimator via Rao-Blackwell theorem

Let $X_1,...,X_n$ be iid with pdf $$f(x|\theta) = exp(\theta -x) I(x)_{(\theta, \infty)}$$ It is asked to find an unbiased estimator for$ \theta $ that is function of a sufficient statistical for ...
2
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1answer
111 views

Asymptotically unbiased estimator using MLE

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across. Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; ...
5
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2answers
75 views

How would one formally prove that the OOB error in random forest is unbiased?

I have read this statement many times but have never come across a proof. I would like to try to produce one myself but I'm not even sure on what notation to use. Can anyone help me with this?
1
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0answers
59 views

Is it necessary to transform an econometric model in order to have only diagonal elements in the error covariance matrix?

Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $$y_{jis} = x_{jis} \beta + \upsilon_{jis}$$ $j$ refers to school, $i$ refers to ...
3
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0answers
51 views

Combining unbiased estimators with unknown variance

Say we are given a sequence of independently (but not identically) distributed random variables $X_1,...,X_n$ which are known to be bounded, $X_t \in (a,b)$, and to have the same mean, $\mathbb{E}X_t ...
4
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1answer
75 views

Should sampling happen with or without replacement

When doing a simple random sample to estimate population mean for some statistic, how do I know whether sampling happens with or without replacement? It feels wrong to use replacement, because 1) my ...
2
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2answers
71 views

Why we shouldn't be obsessed with unbiasedness

In my Bayesian statistics class, my professor makes the remark that we should not be obsessed with unbiased estimator. First: I understand this statement in the sense of trading biasedness for ...
4
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0answers
36 views

R packages that work with biased samples

I'm working with a biased sample of web users. I'm only able to track responses of users who have navigated my site in a certain way, and I'd like to run an analysis to determine how certain factors ...
5
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3answers
135 views

Unbiased estimate of population standard deviation: is sqrt(2) a superior correction?

Background on bias correction constants The standard deviation is calculated like this: $$ SD = \left(\frac{1}{N-constant} \sum_{i=1}^N (x_i - \overline{x})^2\right)^{1/2} $$ Following Wikipedia's ...
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8 views

What I have to undergo testing before proceeding to time series analysis? and after?

I have to conduct a time series regression analysis. Before proceeding I tested for stationarity and I verified that the variables are distributed according to the normal probability distribution. ...
0
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6 views

More examples of inconsistent, but unbiased data and vice versa? [duplicate]

I'm starting to understand the distinction, but I'm having trouble envisioning an example where one's estimates have one, but not the other of these properties. Thanks for the help!
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1answer
33 views

Identifiability and unbiasedness

How do you show that if my model parameter $\theta$ (scalar) is U-estimable (i.e. if there exists an unbiased estimator of $\theta)$, then $\theta$ is identifiable? This makes sense intuitively, but ...
0
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0answers
17 views

How to analyze data from two-tiered experiment

I have an experiment where each subject undergoes M sessions of treatment drug & control drug. I have N subjects. How do I 'match/pair' my data together? In particular, lets say that d(n,m) is ...
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43 views

Symmetric distribution and unbiased estimator

If we have 10 random variables, each distributed with a $Cauchy(t,1)$ distribution and we have an estimator for $t$. When using the R program to simulate mean ...
1
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1answer
93 views

Is the bootstrap estimate of the mean biased when a single extreme value is in the sample?

My sample includes $n$ random observations, while $n-1$ of these observations are in the range (0-1) there is also one observation that gets very high value. For example, a sample of prices where ...
1
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0answers
22 views

Admissibility and domination for estimators

Watching a video by the "mathematicalmonk" on the web, I was wondering how to answer this kind of questions: Given $X_1,\ldots,X_n\sim \mathcal{N}\left(\mu,\sigma^2\right)$. Assume that $\mu$ is ...
0
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0answers
64 views

Bootstrap and var of median

I've 2 different questions: A. Sample with n variables, while $n-1$ variables are in the range (0-1), and one of the variable is very high: $10^{1000}$. If I'll use bootstrap to estimate the mean, ...
3
votes
0answers
104 views

Best method to estimate the mean of a normal distribution?

Let $X = ( x_1, ..., x_n ) $ be $n$ samples from a normal distribution with unknown mean. What is the best estimator for this mean? I can think of at least 2 unbiased estimators: The empirical mean ...
2
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0answers
86 views

For the binomial distribution, why does no unbiased estimator exist for $1/p$?

Suppose that $X$ ~ $Binomial(n,p)$ for $0 < p < 1$ Why does no unbiased estimator exist for $1/p$? My approach: We try to find the structure of $E_p(U(x))$, where $U(x)$ is any estimator of ...
4
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0answers
53 views

Half of the mean as biased estimate - Stein's paradox

i read this paper with interest yesterday http://statweb.stanford.edu/~ckirby/brad/other/Article1977.pdf and stumbled upon the statement that, if and only if, the population mean is close to zero, ...
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0answers
17 views

Consistency vs. unbiasdness [duplicate]

What is the difference between $ \lim_{n \to \infty} \ \mathrm{E}_{\theta}(T_n(X)) = \theta$ and $ T_n(X) \xrightarrow{p} \theta \ $ for $\ n \xrightarrow{} \infty$ ? (unbiasdness vs. ...
4
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2answers
196 views

Trying to understand unbiased estimator

After reading this post, I still don't thoroughly understand what an estimator is. Suppose the samples $D_i={(x_1,y_1),...(x_n,y_n)}$ are drawn randomly from function $$f(x)=sin(2\pi x),$$ so my ...
0
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0answers
17 views

Find the least squares estimator of the parameter B (beta) in the following regression model: y= B + u What is the variance of the estimator? [duplicate]

Find the least squares estimator of the parameter B (beta) in the following regression model: y= B + u What is the variance of the estimator?
1
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1answer
120 views

Distribution of $\bar{X^2} $ when $X\sim N \left( \theta, \sigma^2 \right) $

How can I derive the distribution of $$\bar{X^2}\quad \text{when}\quad X\sim N \left( \theta, \sigma^2 \right) $$ The context of this question is an exercise requiring me to show that $\bar{X^2}- ...
0
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0answers
32 views

Calculating estimator of the variance of sample mean

Assuming equal variances determine the value of the best estimator of the variance of sample mean 1 From my exhibit/minitab I have Sample 1: N = 1, Mean = 123.7, StDev = 19.8, SE Mean 5.5 T-Test of ...
0
votes
1answer
87 views

Have I applied the delta method correctly?

I have UMVUE $$\tilde\theta = \frac{(n-1)(U-n)}{(U-1)(U-2)}$$ for $P(Y=2)=\theta(1-\theta)$ where $U=\sum_{i=1}^n Y_i$ $Y_i \sim \text{geometric} (\theta)$ I am using delta method to find the ...
2
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0answers
166 views

How does one show that there is no unbiased estimator of $\lambda^{-1}$ for a Poisson distribution with mean $\lambda$?

Suppose that $ X_{0},X_{1},\ldots,X_{n} $ are i.i.d. random variables that follow the Poisson distribution with mean $ \lambda $. How can I prove that there is no unbiased estimator of the quantity $ ...
2
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0answers
261 views

Show that a given estimator is biased and consistent

Let the model be $\log(W) = a + bX + U$ where $E(U) = 0$. We are allowed to assume that $\operatorname{Cov}(X,U) = 0$, and want to show $e^{xb^\text{ols}}$-1 is a consistent estimator for $e^{xb}$-1 ...
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37 views

Decision Fusion using Multiple Data Sources with different update rates and accuracy

If you have two data sources which report their decision at different data rates, and have different "trustworthiness" how can I combine them so that I can take advantage of both of them? I'm still ...
1
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1answer
260 views

How do I check for bias of an estimator?

I need to check if an estimator $\hat\theta$ for the parameter $\theta$ is biased. Theory says I should compare the expected value of $\hat\theta$ versus the expected value of $\theta$. I assume the ...
0
votes
1answer
83 views

Simple OLS with two samples

I want to obtain an unbiased estimator for $b_1$ in a simple regression like that: $Y_i = B_0 + B_1X_i + u_i$ when I have two samples, always same size for Y and X, but once the sample size is l and ...
0
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0answers
43 views

Choice of variance estimator

Consider the problem of the choice of estimator of $\sigma^2$ based on a random sample of size $n$ from a $N(\mu,\sigma^2)$ distribution. In undergraduate, we were always taught to use the sample ...
0
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0answers
164 views

b1 estimator properties in intercept-only model

For the model $Y_i=B_1+u_i,i = 1, 2, …, n$ (This model does NOT have an independent variable), how do you: Derive the OLS estimator of $B_1$ Prove that $B_1$ is a linear estimator. Prove that $B_1$ ...
0
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1answer
110 views

Question on expected value of an estimator

I have been given the pdf: $$f_{\theta}(x)=\frac{x}{\theta^2}e^{-\frac{x^2}{2\theta^2}}$$ and I want to know if the MLE estimator $\theta$ is unbiased. Attempt: I want to maximize the function so I ...
1
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0answers
179 views

Unbiased estimator of variance for samples *without* replacement

This is a follow-up question on that one: Could Bessel's correction make sample variance estimation even more biased? I understand that you need Bessel's correction to get an unbiased estimate of ...
1
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1answer
138 views

Could Bessel's correction make sample variance estimation even more biased?

It is well known that Bessel's correction creates an unbiased estimator of variance. What it basically does is divide by $n-1$ instead of $n$. Now what I did is that I chose a few number, like ...
2
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0answers
79 views

Unbiasedness condition in ordinary kriging and simple kriging

I have this confusion. In ordinary kriging we have used the unbiasedness condition which gave the sum of weights equal to one. However, in the case of simple kriging we have no such conditions why? I ...
11
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1answer
361 views

Does a median-unbiased estimator minimize mean absolute deviance?

This is a follow-up but also a different question of my previous one. I read on Wikipedia that "A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as ...
5
votes
1answer
483 views

When to use sample median as an estimator for the median of a lognormal distribution?

I myself would always use geometric mean to estimate a lognormal median. However, in the industry world, sometimes using the sample median gives better results. The question thus is, is there a cutoff ...
3
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2answers
719 views

How do I use the standard regression assumptions to prove that $\hat{\sigma}^2$ is an unbiased estimator of $\sigma^2$?

I'm working through an econometrics textbook and it's proving that $$ \sigma^2 = E(\hat{\sigma}^2) = \frac{SSR}{n-2} $$ I followed the proof (an example of which is shown on talkstats) until it ...