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I have a question regarding the definiton of estimators. In the german wikipedia it says that the distribution is determined by g($X_1,...,X_n$) where by g is the estimator function and $X_1,..X_n$ are the observed random variables, and g is evaluated for all possible samples.

Now my question is if the distribution of the estimator is evaluated for a fixed n, i.e. does each sample size has its own estimator distribution? Or is the distribution constructed for variable n?

Furtheron I am interested in the following: If I have a dataset of around 200 observations given, and I want to construct the distribution of an estimator for different sample sizes, is it recommended to use sampling without replacement or with replacement (i.e. bootstrapping)? The data is normally distributed, and the estimator to be calculated is the estimator for the t-test.

Thanks

Pugl
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1 Answers1

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The sample size affects the distribution. For example, the sample mean of an i.i.d. sample from a standard Logistic distribution does not follow a normal distribution - but as the sample size increases the distribution tends to the normal (by the Central Limit Theorem).

In this post, I show that the distribution of the "t-value", which is the Student's distribution for finite sample size, tends to the normal as the sample size increases.

In this post I show that the asymptotic distribution of the estimator of $\theta$ of a sample of i.i.d. Uniforms $U(0,\theta)$ tends to the exponential distribution -but it is not Exponential for finite sample size.

These changes do not happen abruptly but gradually. So the distribution of an estimator depends on $n$.

Of course, in some cases, the distribution family of the estimator is invariant to the sample size -e.g. the sample mean of a sample of normals: it is also normal for any sample finite sample size, and also asymptotically.

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Alecos Papadopoulos
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  • Hi. Thanks,yes I understand that, but it is not exactly my question. So, one constructs a distribution for a fixed n when talking about the distribution of an estimator (and the the same for a sample size of m >> n, whereby the distribution then might be different), right? – Pugl May 13 '14 at 12:31
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    One constructs a distribution for an _abstract_ $n$ and then see how it changes as $n$ changes. – Alecos Papadopoulos May 13 '14 at 12:32
  • I see. Given I have a dataset of ca. 200 and I want to know how my estimator changes with sample size, which option would be better: 1) Bootstrapping (which as I understand is sampling with replacment) or 2) Sampling for all possible samples without replacement, and then calculating a distribution for all these possible samples for each possible sample size n? – Pugl May 13 '14 at 12:40
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    I wouldn't want to advise on this, I do not consider myself as being especially proficient in the theory behind simulation methods. Since it appears that your interest lies more on applied derivations of distributions, I suggest that you update your post to include these questions, and also tag it appropriately so that it attracts users that can help you with them. – Alecos Papadopoulos May 13 '14 at 12:57