Estimation

Score, Fisher Information and Estimator Sensitivity

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As we have seen in the previous articles, that the estimation of a parameter from a set of data samples depends strongly on the underlying PDF. The accuracy of the estimation is inversely proportional to the variance of the underlying PDF. That is, less the variance of PDF more is the accuracy of estimation and vice … Read more

Theoretical derivation of MLE for Gaussian Distribution:

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[ratings] 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 … Read more

Theoretical derivation of MLE for Exponential Distribution:

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[ratings] 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 } &s=2 $ Calculating the Likelihood The log likelihood is given by, Differentiating and equating … Read more

Theoretical derivation of Maximum Likelihood Estimator for Poisson PDF:

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[ratings] 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 … Read more

Maximum Likelihood Estimation (MLE) : Understand with example

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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

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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

Estimation Theory : an introduction

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Key focus: Understand the basics of estimation theory with a simple example in communication systems. Know how to assess the performance of an estimator. A simple estimation problem : DSB-AM receiver In Double Side Band – Amplitude Modulation (DSB-AM), the desired message is amplitude modulated over a carrier of frequency f0. The following discussion is … Read more