# Thread: Deriving the sampling (or asymptotic) distribution

1. ## Deriving the sampling (or asymptotic) distribution

Assume we have the following function:

where

is a constant

is a constant

for are random variables

for are random variables

is defined such that it is value that satisfies

I wish to derive a sampling (or asymptotic) distribution for the statistic .

By sampling distribution I mean the following:

The solution to doesn't have a closed-form solution, but it is obvious that the resulting value of depends on and , so can be treated as a random variable that depends on the random variables and . Then for every observations of and , we have a corresponding value that satisfies , what is the sampling distribution of ?

By asymptotic distribution I mean the following:

Assume we have instances of and , that is, groups of and . Then we solve and have observations of , that is, . What is the distribution of as ?

Also assume you are allowed the following assumptions to achieve the above:

1) You can make any distributional assumptions regarding and , e.g., and are independent from each other, also , for are independently and identically distributed.

2) Rather than making distributional assumptions about and , assume you can make some assumptions about the processes and , e.g., both processes are stationary (or weakly stationary) etc.

2. ## Re: Deriving the sampling (or asymptotic) distribution

Usually when you are talking about the asymptotics of a certain statistic, it should be a sequence indexed by the sample size . However, now it seems that the group you defined are independent to each other so that are just a sequence of i.i.d. random variables which does not related to any asymptotic result. Am I missing anything?

Besides, if we define

then the function becomes

So what why do we need to write it out like that?

Last but not least,

So the actually relation can be simplified to ?

3. ## The Following User Says Thank You to BGM For This Useful Post:

TrueTears (02-19-2014)

4. ## Re: Deriving the sampling (or asymptotic) distribution

You are quite right. Yes so the relation to which you have simplified to is the one I am after. Assume we fix , how can we derive the distribution of ?

5. ## Re: Deriving the sampling (or asymptotic) distribution

Any further ideas?

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