Linked Questions

1
vote
0answers
131 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. ...
100
votes
4answers
103k views

Difference between standard error and standard deviation

I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error?
21
votes
5answers
5k views

Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution?

It came as a bit of a shock to me the first time I did a normal distribution Monte Carlo simulation and discovered that the mean of $100$ standard deviations from $100$ samples, all having a sample ...
13
votes
4answers
9k views

An example of a consistent and biased estimator?

Really stumped on this one. I would really like an example or situation where an estimator B would be both consistent and biased.
10
votes
4answers
3k views

How does one explain what an unbiased estimator is to a layperson?

Suppose $\hat{\theta}$ is an unbiased estimator for $\theta$. Then of course, $\mathbb{E}[\hat{\theta} \mid \theta] = \theta$. How does one explain this to a layperson? In the past, what I have said ...
15
votes
4answers
4k views

Intuitive understanding of the difference between consistent and asymptotically unbiased

I am trying to to get an intuitive understanding and feel for the difference and practical difference between the term consistent and asymptotically unbiased. I know their mathematical/statistical ...
17
votes
1answer
52k views

How to show that an estimator is consistent?

Is it enough to show that MSE = 0 as $n\rightarrow\infty$? I also read in my notes something about plim. How do I find plim and use it to show that the estimator is consistent?
18
votes
3answers
2k views

Asymptotic consistency with non-zero asymptotic variance - what does it represent?

The issue has come up before, but I want to ask a specific question that will attempt to elicit an answer that will clarify (and classify) it: In "Poor Man's Asymptotics", one keeps a clear ...
7
votes
1answer
19k views

Bias of the maximum likelihood estimator of an exponential distribution

The maximum likelihood estimator of an exponential distribution $f(x, \lambda) = \lambda e^{-\lambda x}$ is $\lambda_{MLE} = \frac {n} {\sum x_i}$; I know how to derive that by find the derivative of ...
9
votes
3answers
2k 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 ...
9
votes
1answer
2k views

Fisher consistency versus “standard” consistency

My question relates two types of consistency. In particular, how does the Fisher consistency differ from standard notions of consistency, such as convergence in probability to the generative parameter....
12
votes
1answer
4k views

What's the difference between asymptotic unbiasedness and consistency?

Does each imply the other? If not, does one imply the other? Why/why not? This issue came up in response to a comment on an answer I posted here. Although google searching the relevant terms didn't ...
0
votes
2answers
3k views

Proof that $\mathrm{E}(s^2) = \sigma^2 \cdot N/(N-1)$

What's the derivation for expected value for sample variance for a sample taken from simple random sampling without replacement, i.e., how do we show that $$\mathrm{E}(s^2) = \sigma^2 \frac{N}{N-1}$$...
5
votes
1answer
537 views

Does inconsistency imply biasedness?

If an estimator $\theta$ is inconsistent, can I always conclude that $\theta$ is also biased?
8
votes
0answers
1k views

Why don't asymptotically consistent estimators have zero variance at infinity?

I know that the statement in question is wrong because estimators cannot have asymptotic variances that are lower than the Cramer-Rao bound. However, if asymptotic consistence means that an estimator ...

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