# Is the MLE sufficient?

## Is the MLE sufficient?

f(x; θ). If g is a one-to-one function, and ˆθ is the MLE of θ then g(ˆθ) is the MLE of g(θ). If ˆθ is the unique MLE and T = t(X) is sufficient, then ˆθ is a function of t. If the MLE is itself sufficient, it is minimal sufficient.

How do you know if a statistic is sufficient?

In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”.

How do you prove minimal sufficient?

Definition 1 (Minimal Sufficiency). A sufficient statistic T is minimal if for every sufficient statistic T and for every x, y ∈ X, T(x) = T(y) whenever T (x) = T (y). In other words, T is a function of T (there exists f such that T(x) = f(T (x)) for any x ∈ X).

### What is a sufficient population?

A sufficient statistic summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.

What is the connection between a sufficient statistic and an MLE?

A theorem relating the two concepts indicates that if a maximum likelihood estimate (MLE) for a parameter is unique, then it is a function of every sufficient statistic.

Is MLE always consistent?

This is just one of the technical details that we will consider. Ultimately, we will show that the maximum likelihood estimator is, in many cases, asymptotically normal. However, this is not always the case; in fact, it is not even necessarily true that the MLE is consistent, as shown in Problem 27.1.

## What is factorization theorem?

Theorem (Factorization theorem): The statistic is sufficient for if and only if the model p ( x | θ ) can be factorized as follows: p ( x | θ ) = g ( t ( x ) , θ ) h ( x ) . Proof: CB 6.2.6. ◼ Corollary: The likelihood based on a sufficient statistic is equivalent to the likelihood based on the entire data.

Why is sufficiency important in statistics?

Short Answer: A sufficient statistics carries with it all the information needed to make inference about the population, excluding information that gives sample specific . So given the sufficient statistics, you can do the same inferential analysis without your entire data.

How do you prove that something is not sufficient in statistics?

If you want to show a statistic is not a sufficient statistic , you can compare it with minimal sufficient statistic. Use the fact that a minimal sufficient statistic is a function of any sufficient statistic. Define T=max(X1,⋯,Xn) and U=2n∑ni=1Xi. We want to prove U is not a sufficient statistics.

### What does it mean for an estimator to be sufficient?

Sufficient estimators exist when one can reduce the dimensionality of the observed data without loss of information. Thus sufficiency refers to how well an estimator utilizes the information in the sample relative to the postulated statistical model.

What do you mean by sufficient estimator?

Quick Reference. An estimator of a parameter θ that gives as much information about θ as is possible from the sample. The sample mean is a sufficient estimator of the population mean of a normal distribution.

What is the invariance property of MLE?

Invariance property of MLE: if ˆθ is the MLE of θ, then for any function f(θ), the MLE of f(θ) is f(ˆθ). Also, f must be a one-to-one function. The book says, “For example, to estimate θ2, the square of a normal mean, the mapping is not one-to-one.” So, we can’t use invariance property.