Efficient Estimators by applying CRLB

It has been reiterated that not all estimators are efficient. Even not all the Minimum Variance Unbiased Estimators (MVUE) are efficient. Then how do we quantify whether the estimator designed by us is efficient or not?

An efficient estimator is defined as the one that is
* Unbiased (mean of the estimate = true value of the parameter)
* Attains Cramer-Rao Lower Bound (CRLB).

How to Identify Efficient Estimators?

As mentioned in the previous article, the second partial derivative of log likelihood function of the observed signal model may be (not true always) written in a form like the one below.

If we can write the CRLB equation in the above form, then the estimator is an efficient estimator.

Example:

In an another previous article, CRLB for an estimator that estimates the DC component from a set of observed samples (affected with AWGN noise) was derived. The intermediate step that derived the above requirement for the scenario is given below

From the above equation, it can be ascertained that the efficient estimator exists for the case and it is given by . The efficient estimator is simply given by sample mean of the observed samples.

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For further reading

[1]	An Introduction to Estimation Theory
[2]	Bias of an Estimator
[3]	Minimum Variance Unbiased Estimators (MVUE)
[4]	Maximum Likelihood Estimation
[5]	Maximum Likelihood Decoding
[6]	Probability and Random Process
[7]	Likelihood Function and Maximum Likelihood Estimation (MLE)
[8]	Score, Fisher Information and Estimator Sensitivity
[9]	Introduction to Cramer Rao Lower Bound (CRLB)
[10]	Cramer Rao Lower Bound for Scalar Parameter Estimation
[11]	Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)
[12]	Efficient Estimators and CRLB
[13]	Cramer Rao Lower Bound for Phase Estimation
[14]	Normalized CRLB - an alternate form of CRLB and its relation to estimator sensitivity
[15]	Cramer Rao Lower Bound (CRLB) for Vector Parameter Estimation
[16]	The Mean Square Error – Why do we use it for estimation problems
[17]	How to estimate unknown parameters using Ordinary Least Squares (OLS)
[18]	Essential Preliminary Matrix Algebra for Signal Processing
[19]	Why Cholesky Decomposition ? A sample case:
[20]	Tests for Positive Definiteness of a Matrix
[21]	Solving a Triangular Matrix using Forward & Backward Substitution
[22]	Cholesky Factorization - Matlab and Python
[23]	LTI system models for random signals – AR, MA and ARMA models
[24]	Comparing AR and ARMA model - minimization of squared error
[25]	Yule Walker Estimation
[26]	AutoCorrelation (Correlogram) and persistence – Time series analysis
[27]	Linear Models - Least Squares Estimator (LSE)
[28]	Best Linear Unbiased Estimator (BLUE)

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