Let X=θ+N be the output of a noisy channel where the input is the parameter θand N is a zero-mea

Let X=θ+N be the output of a noisy channel where the input...

Key Concepts

Consistent Estimator

A consistent estimator is one that converges in probability to the true parameter as the sample size increases indefinitely. In other words, as more data become available, the probability that the estimator is arbitrarily close to the true parameter approaches one.

Transformation of Random Variables

Transforming random variables involves finding the probability density function (pdf) of a new variable which is a function of one or more other random variables. When the estimator is expressed in terms of another random variable, such as the sample mean, properties of the transformation (like linearity and known distribution of the original variable) are used to determine the distribution of the estimator.

Unbiased Estimator

An unbiased estimator is one for which the expected value of the estimate is equal to the true value of the parameter being estimated. This means that on average, across many samples, the estimator will correctly recover the parameter without systematic error.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method for estimating the parameters of a statistical model by finding the values that maximize the likelihood function, which represents the probability of the observed data given the parameters. In the context of normally distributed data, this typically leads to solutions that utilize the sample mean as the estimator for the location parameter.

Gaussian Distribution

The Gaussian or normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and variance. It has key properties such as symmetry about the mean, a specific spread determined by the variance, and the fact that linear transformations of Gaussian random variables remain Gaussian.

Likelihood Function

The likelihood function is a function of the parameters of a statistical model given observed data, representing the joint probability of the observed data as a function of the parameters. Maximizing this function provides the maximum likelihood estimates for the parameters, and in many problems, including those with additive Gaussian noise, the overall likelihood simplifies due to the product structure of independent observations.

Transcript

00:01 Hi, here it is given that random samples x1, x2, and so on xn follows normal distribution with mean theta and variance sigma square where sigma square is known.

00:20 That means it is the known quantity.

00:23 First we have to find the mle have mle of theta, theta -hat mle.

00:30 And again, second question, we have to find the mla of theta when theta is restricted.

00:37 When theta lies between 0 to infinity, 0 less than equals to theta less than infinity, for that case we have to find theta hat mle restricted.

00:53 Okay, so we start now as each xi follows normal theta sigma square.

01:03 So the pdf of this distribution will be f x theta equals to 1 by root 2 pi sigma e to the power minus half x minus theta whole divided by sigma square where minus infinity less than x less than infinity and also minus infinity less than theta less than infinity.

01:34 And also sigma square greater than zero.

01:36 Sigma square is known and zero otherwise.

01:43 Okay.

01:44 Now, to find the mla of theta, we first construct the likelihood function.

01:49 Now, the likelihood function is denoted by l theta x -curd.

01:55 That is equals to product of i runs from 1 to n f x -i -theta.

02:05 Now, equals to 1 by root 2 pi sigma, volt to the power n, e to the power minus half summation i runs from 1 to n, xi minus theta, whole divided by sigma square.

02:27 Now, this is the likelihood function of theta given x curve.

02:31 Now, maximizing this likelihood function is equivalent to maximizing the log likelihood function.

02:38 Function...

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