Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)

by

It was mentioned in one of the earlier articles that CRLB may provide a way to find a MVUE (Minimum Variance Unbiased Estimators). Theorem: There exists an unbiased estimator that attains CRLB if and only if, Here \( ln \; L(\mathbf{x};\theta) \) is the log likelihood function of x parameterized by \(\theta\) – the parameter … Read more

Cramér-Rao Lower Bound: Introduction

by

Key concept: Cramér-Rao bound is the lower bound on variance of unbiased estimators that estimate deterministic parameters. Introduction The criteria for existence of having an Minimum Variance Unbiased Estimator (MVUE) was discussed in a previous article. To have an MVUE, it is necessary to have estimates that are unbiased and that give minimum variance (compared … Read more

Theoretical derivation of MLE for Gaussian Distribution:

by

As a pre-requisite, check out the previous article on the logic behind deriving the maximum likelihood estimator for a given PDF. Let X=(x1,x2,…, xN) are the samples taken from Gaussian distribution given by Calculating the Likelihood The log likelihood is given by, Differentiating and equating to zero to find the maxim (otherwise equating the score … Read more

Theoretical derivation of MLE for Exponential Distribution:

by

As a pre-requisite, check out the previous article on the logic behind deriving the maximum likelihood estimator for a given PDF. Let X=(x1,x2,…, xN) are the samples taken from Exponential distribution given by $latex f(x;\theta) = \theta e^{-\theta x } $ Calculating the Likelihood The log likelihood is given by, Differentiating and equating to zero … Read more

Theoretical derivation of Maximum Likelihood Estimator for Poisson PDF:

by

Suppose X=(x1,x2,…, xN) are the samples taken from a random distribution whose PDF is parameterized by the parameter $latex \theta $. If the PDF of the underlying parameter satisfies some regularity condition (if the log of the PDF is differentiable) then the likelihood function is given by Here $latex f_N(x_N;\theta) $ is the PDF of … Read more

Maximum Likelihood Estimation (MLE) : Understand with example

by

Key focus: Understand maximum likelihood estimation (MLE) using hands-on example. Know the importance of log likelihood function and its use in estimation problems. Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The core idea behind MLE is to find the parameter values that maximize the likelihood … Read more

Estimator Bias

by

Estimator bias: Systematic deviation from the true value, either consistently overestimating or underestimating the parameter of interest. Estimator Bias: Biased or Unbiased Consider a simple communication system model where a transmitter transmits continuous stream of data samples representing a constant value – ‘A’. The data samples sent via a communication channel gets added with White … Read more

QAM modulation simulation — rectangular constellations in Matlab and Python

by

Digital Modulations 101 — 5 lessons (same on every page in this path) 1. BPSK → 2. QPSK → 3. M-PSK sim → 4. QAM sim (you are here) → 5. EVM Tools: BER vs Eb/N0 · EVM → SNR · Eb/N0 ↔ SNR Also on the Matlab → Python path (lesson 3 of 4): … Read more