A quick question; is it possible to say in a way analogous to the single variable case that a multiv

Nataly Best

Nataly Best

Answered question

2022-01-22

A quick question; is it possible to say in a way analogous to the single variable case that a multivariable function is "asymptotically equivalent" to a second multivariable function? For example, consider the function of n1,n2R given by
Var(μ^)=σ2(n1+2n2)(n1+n2)2.
where σ2 is a constant.
Can we say that Var(μ^)1n1+n2 and then conclude that Var(μ^)0 as n1 and n2?
lim(x,y)(,)x+2y(x+y)2
does not exist. Am I wrong to think of Var(μ^) as a function of two variables?

Answer & Explanation

utgyrnr0

utgyrnr0

Beginner2022-01-23Added 11 answers

The two previous answers are perfectly right, thus I just would like it emphasize the statistical perspective. Note that when you are talking about a variance of an estimator, n1 and n2 are sample sizes, hence they are (strictly) positive integers, i.e., n1,n2N. As such, when you take the limit, you cannot consider any possible route (as Wolfram does) in R2. The extreme cases in this variance are n1>>n2 or n2>>n1, that dont
Karli Kaiser

Karli Kaiser

Beginner2022-01-24Added 9 answers

You can conclude Var(μ)0 as n1 and n2, but you can't say Var(μ)1n1+n2 because when n1>>n2 it is approximatly 1n1+n2 but when n1<<n2 it is near 2n1+n2

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