Point estimation is the process of estimation where sampling is used when single value is estimated for the unknown parameter of a population. There are two methods of point estimation which are used. They are:

**Method of maximum likelihood**

**Properties of maximum likelihood**

- It is consistent, most efficient and sufficient with the provision that there exists a sufficient estimator.
- It is not unbiased.
- It is generally distributed normally for large samples.
- If g(is a function of and e is the maximum likelihood of it, then g(is an maximum likelihood of g(

Let x₁, x₂, …,be a random sample from a population whose pmf (discrete case) or pdf (continuous case ) is f (x, θ) where θ is the parameter. Then construct the likelihood function as follows:

L = f (x₁, θ) .f (x₂, θ) … f (, θ)

Since log L is maximum when L is maximum, therefore to obtain the estimate of θ, we maximize L as follows,

And is known as MLE or maximum likelihood estimator.

**Example 1:**

*A discrete random variable X can take up all non-negative integers and P (X= r) *= *p (1- p) ^{r} (r *=

*0, 1, 2, .. .)Here, p (0*<

*p*<

*I) is the parameter of the distribution. Find the MLE of p for a sample of size:*x₁, x₂, …,

*from the population of X.*

**Solution:**

Consider the following: L= P(X= x₁). P(X= x₂)… P(X=* )*

= p

=

=

Taking log on both sides,

= 0

=

=

at

=

Hence the MLE of *p *is 1/ 1+x

**Example 2:**

A random variable X has a distribution with density function: *f* (x) = λ x ^{λ -1}(0 < x < 1) where A is the parameter. Find the MLE of A for a sample of size n. x₁, x₂, …,, from the population of X.

**Solution:**

Consider the following

L= (x₁).*f (*x₂)…*.f (*)

= λ x₁ ^{λ -1 }.λ x₂ ^{λ -1 }…*.*λ

^{ = }

Taking log on both sides,

ln L = n ln A. + (A.- 1) ln (*x _{1},x_{2},…, *

*)*

*Also, *

Hence, MLE of A is

**Example 3:**

X tossed a biased coin 40 times and got head 15 times, while Y tossed it 50 times and got head 30 times. Find the MLE of the probability of getting head when the coin is tossed.

**Solution:**

Let P be the unknown probability of getting a head.

Using binomial distribution, Probability of getting 15 heads in 40 tosses

Probability of getting 30 heads in 50 tosses

The likelihood function is taken by multiplying these probabilities.

L=

Log L = log [ + 45log (1-p)

which is the MLE.

**Method of moments**

In this particular method, the first few moments of a population is equated with the equivalent moments of the same sample.

Then, µ’_{r }= m’_{r}

Where ) and

The method is only applicable when the population moment exists and the solution for the parameters gives the estimates.

**Example:**

Estimate the parameter p of the binomial distribution by the method of moments when n is known.

**Solution:**

np=

p= and that is the value estimated.

**Links of Previous Main Topic:-**

- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation

**Links of Next Statistics Topics:-**

- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory