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Topic: Sampling From Finite Population with Replacement
Replies: 28   Last Post: Sep 30, 2010 6:30 AM

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 Ray Koopman Posts: 3,383 Registered: 12/7/04
Re: Sampling From Finite Population with Replacement
Posted: Sep 29, 2010 5:40 PM

On Sep 29, 1:12 pm, Cagdas Ozgenc <cagdas.ozg...@gmail.com> wrote:
> On 29 Eylül, 23:57, Ray Koopman <koop...@sfu.ca> wrote:
>> On Sep 29, 12:03 pm, Cagdas Ozgenc <cagdas.ozg...@gmail.com> wrote:
>>>>>> The population mean is an unbiased estimate of the generator mean.
>>>>>> The sample mean is an unbiased estimate of the population mean,
>>>>>> and therefore of the generator mean.

>>>>>
>>>>> I think you have a point here. But as you can see that there is a
>>>>> problem with consistency.
>>>>>
>>>>> Let's say that generator mean is Mu, and population mean is Mu + Eps.
>>>>> And I take as you suggest Eps is a random error not a systematic error
>>>>> (not a bias).
>>>>>
>>>>> Now as you take more and more sample means, you will see that they
>>>>> will start to gather around Mu+Eps not Mu. Now do we have a random
>>>>> error or a systematic error?

>>>>
>>>> It all depends on whether we're talking about the conditional
>>>> distribution of the sample mean, given the population mean; or the
>>>> unconditional (or marginal) distribution of the sample mean. As an
>>>> estimate of the generator mean, the ssmple mean is conditionally
>>>> biased but marginally unbiased.

>>>
>>> I don't think I am following you. How is all that related to
>>> conditioning?

>>
>> In the marginal distribution all the error is random, and the sample
>> mean is is an unbiased estimate of the generator mean. In the
>> conditional distribution there is both random and systematic error;
>> the sample mean is a biased estimate of the generator mean, with the
>> bias being the unknown but fixed difference between the population
>> mean and the generator mean.

>
> Sorry I didn't make myself clear. Basically I am trying to relate your
> conclusion to my initial question. What does this in general tell us
> about sampling from an infinite population vs sampling from a finite
> population with replacement? Can I conclude that they cannot be
> treated equally? Why is this issue never mentioned in stat texts?

from an infinite population and sampling with replacement from a
finite population. You specified that the population was generated by
a random process and that you wanted to estimate the generator mean.
That means you're doing two-stage sampling, which is something that
most texts (other than those on survey sampling) do not get into,
and that you must also specify whether you are interested in the
conditional or the marginal distribution of the sample mean.