Expansion of Taylor series for statistical functionals

In summary, the conversation discusses the expansion of a functional, \theta, around a distribution function, f_1. The desired expansion is \theta(f_1 + f_2) = \theta(f_1) + \frac{d \theta}{d (something)} *f_2 + o(1). It is unclear what the "derivative-equivalent" is and how it can be calculated in a specific situation. The concept of a Lie group of transformations is mentioned, but it is not clear how it relates to the problem at hand. The conversation ends with a request for help and direction.
  • #1
Testguy
6
0
Hi

By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is.

My situation is that I have a functional, say \theta which depend on the distribution function given to it. I want to expand \theta(f_1 + f_2) around the distribution function f_1. I think it should look like

\theta(f_1(y) + f_2(y)) = \theta(f_1(y)) + \frac{d \theta}{d (something)} *f_2(y) + o(1),

but I do not know what the derivative-equivalent actually is.

What is the definition of this and how can it be calculated in a given situation? Or am I totally out of bounds with such a formula?

Can someone help me or at least point me in the right direction?

Any help is appreciated.
 
Physics news on Phys.org
  • #2
Testguy said:
I want to expand [itex] \theta(f_1 + f_2) [/itex] around the distribution function [itex] f_1 [/itex]. I think it should look like

[itex] \theta(f_1(y) + f_2(y)) = \theta(f_1(y)) + \frac{d \theta}{d (something)} *f_2(y) + o(1) [/itex].

Does "distribution function" mean a probability distribution function? If so, wouldn't the argument of [itex] \theta [/itex] have to be [itex] \frac {f_1 + f_2}{2} [/itex] or some other combination that produced a probability density or cumulative probability function?


but I do not know what the derivative-equivalent actually is.

What is the definition of this and how can it be calculated in a given situation? Or am I totally out of bounds with such a formula?

I'm not an expert on functionals, so I can't say whether you are out of bounds. When a Lie group of transformations [itex] T(x,h) [/itex] acts on a set X, it also acts (for a given value of the parameters h) to take a function [itex] f(x) [/itex] to another function [itex] g(x) = f(T(x,h)) [/itex]. There are many things known about that scenario. I don't know whether you could arrange for a group to take [itex] f_1 [/itex] to [itex] f_1 + f_2 [/itex].

It isn't clear what variable's powers appear in the Taylor series that you want and it isn't clear what variable the "o(1)" applies to.

If you define a set of parameterized transformations ( such as [itex] T(f_1,f_2,h) = (1-h) f_1(x) + h (f_1(x) + f_2(x) ) [/itex], you might make sense of a derivative with respect to [itex] h [/itex].
 

What is the Taylor series expansion?

The Taylor series expansion is a mathematical tool used to approximate a complex function as a sum of simpler functions, often polynomials. It is named after the mathematician Brook Taylor and is widely used in various fields of science.

How is the Taylor series expansion used in statistical functionals?

The Taylor series expansion is used in statistical functionals to approximate complex functions that involve statistical operations, such as means, variances, and correlations. This allows for a more efficient and accurate calculation of these functions.

What are the benefits of using the Taylor series expansion for statistical functionals?

Using the Taylor series expansion for statistical functionals allows for a more precise and efficient calculation of these functions, as well as providing a better understanding of their behavior. It also allows for the derivation of new statistical functionals by expanding existing ones.

Are there any limitations to using the Taylor series expansion for statistical functionals?

Yes, there are limitations to using the Taylor series expansion for statistical functionals. It may not provide accurate results for functions with complex behavior or for large datasets. Additionally, it may not be suitable for functions that are not smooth or have discontinuities.

How can we improve the accuracy of the Taylor series expansion for statistical functionals?

To improve the accuracy of the Taylor series expansion for statistical functionals, higher order expansions can be used. This involves including higher order terms in the expansion, resulting in a more precise approximation. Additionally, combining the Taylor series expansion with other numerical methods can also improve accuracy.

Similar threads

How to choose the correct function to use for a Taylor expansion?
    • Introductory Physics Homework Help
Replies
7
Views
714
I How to find the moments using the characteristic function?
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
929
I Relation with Hessian and Log-likelihood
    • Set Theory, Logic, Probability, Statistics
Replies
12
Views
1K
MHB Finding MLE for $\theta$ of Statistical Product Model
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
A Covariance of Posterior Predictive Distribution
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
950
Asymptotic expansions of the sine function
    • Calculus and Beyond Homework Help
Replies
2
Views
921
MHB Calculate the density - Unbiased estimator for θ
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
734
I Time series and why would we remove seasonality and trend?
    • Set Theory, Logic, Probability, Statistics
Replies
10
Views
906
I My textbook is deriving wave speed on a string under tension
    • Mechanics
Replies
19
Views
1K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
    • General Math
Replies
3
Views
672

Hot Threads

Recent Insights

Back
Top