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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2
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2answers
63 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
votes
0answers
27 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
votes
3answers
86 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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0answers
5 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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0answers
5 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!
0
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0answers
14 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
11 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 ...
0
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0answers
31 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 ...
0
votes
1answer
62 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
16 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
61 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, ...
2
votes
0answers
83 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
73 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
45 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, ...
1
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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
votes
2answers
172 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
votes
0answers
16 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
vote
1answer
105 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
24 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
71 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 ...
1
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0answers
127 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
votes
0answers
210 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 ...
1
vote
0answers
33 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
vote
1answer
169 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
80 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
votes
0answers
41 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
126 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
votes
1answer
77 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
vote
0answers
111 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
vote
1answer
121 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
votes
0answers
50 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 ...
9
votes
0answers
235 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, ...
5
votes
1answer
281 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
votes
2answers
512 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 ...
2
votes
0answers
134 views

Improving an unbiased estimator using the Rao-Blackwell theorem [closed]

Given data following a Bernoulli distribution $Y_1,...Y_n \sim B(1,p)$, we want to measure $\theta = \text{Var}(Y_1)$ I found this unbiased estimator $S^2 = \frac{\sum{(y_i-\bar{y}})^2}{n-1}$ I need ...
5
votes
2answers
73 views

is there any difference between taking more samples and a sample with more observations?

I'm quite confused with the difference between taking the average of more samples and taking the average of a sample with more observations. Do you get unbiased estimates by taking more sample of ...
3
votes
2answers
800 views

Unbiased estimator of variance of binomial variable

$Y_{1...n}\sim \operatorname{Bin}(1,p)$, iid, and I need to find an unbiased estimator for $\theta=\operatorname{var}(y_i)$. I did some calculations and I think that the answer is ...
4
votes
0answers
93 views

Is it possible to have an estimator that is unbiased and bounded?

I have a parameter $\theta$ which lies between $[0,1]$. Let us say that I can run an experiment and obtain $\hat{\theta} = \theta + w$, where $w$ is a standard Gaussian. What I need is an estimate of ...
2
votes
1answer
674 views

Correct equation for weighted unbiased sample covariance

I'm looking for the correct equation to compute the weighted unbiased sample covariance. Internet sources are quite rare on this theme and they all use different equations. The most likely equation ...
3
votes
1answer
86 views

Bias of maximum likelihood estimators for logistic regression

I would like to understand a couple of fact on maximum likelihood estimators (MLEs) for logistic regressions. Is it true that, in general, the MLE for logistic regression is biased? I would say ...
1
vote
1answer
218 views

Showing that $s^2$ is an unbiased estimator of $\sigma^2$ [duplicate]

$s^2 = \sum \frac{(x_i - \bar{x})^2}{n-1} $ which apparently equals $ \frac{\sum(x_i^2) +n\bar{x}^2 - 2n\bar{x}^2}{n-1}$. Does this just come from expanding the numerator and using the fact that ...
0
votes
1answer
88 views

Parameters, Estimates

I lack some knowledge in the concepts of parameters, estimates and moment (math and stats). I can't find an online easy-to-understand source of information about these concepts. Would you help me with ...
1
vote
0answers
30 views

Trying to understand an example where unbiased estimators don't exist

I am new to statistics especially in the topic of estimators and sufficient statistic. I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". ...
0
votes
0answers
65 views

Bias in EM estimation for a mixture of normal distributions

Are the parameter estimates for a mixture of two normal distribution using EM algorithm biased or unbiased? More specifically, if I use the EM-algorithm to obtain ML estimates of $μ_1$, $μ_2$, ...
2
votes
0answers
59 views

Unbiased estimate of the semi-partial correlation

Is the sample semi-partial correlation a biased estimate of the population semi-partial correlation? If it is biased, what is an unbiased estimator of the population semi-partial correlation? Are ...
3
votes
1answer
53 views

Will two estimators converge to the same answer?

Say I have two estimators for the same quantity and using the same model, $E[f(X)]$. I also know that these two estimators are consistent, meaning, if we have a lot of data, they will be close to the ...
3
votes
0answers
185 views

Unbiased hypothesis tests

Is there some textbook or expository account showing that the definition of "unbiased test" bears the same sort of relation to "unbiased estimator" that interval estimation generally bears to ...
2
votes
0answers
171 views

Determining variance of an U.M.V.U.E

Given $X_1...X_n$ i.i.d. Bernoulli R.V.s with parameter $\theta$, I found an UMVUE for $\tau(\theta)=\theta^2+\theta$ with Rao-Blackwellization. That process seemed fairly straight-forward to me, and ...
3
votes
3answers
191 views

Counterexample for the sufficient condition required for consistency

We know that if an estimator is an unbiased estimator of theta and if its variance tends to 0 as n tends to infinity then it is a consistent estimator for theta. But this is a sufficient and not a ...
2
votes
4answers
526 views

Why must one trade off between bias and variance?

Apparently, a learning algorithm must make a trade off between bias and variance when producing a hypothesis. Bias means systematic deviation from data. Variance refers to the error due to ...