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Point Estimation

Using sampling if a single value is estimated for the unknown parameter of the population, then this process of estimation is called point estimation. We shall discuss two methods of point estimation below:

 

I. Method of Maximum Likelihood

 

Let x1,x2,…, xn be a random sample from a population whose p.mf (discrete case) or p.df

(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 (xn, θ)

 

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

Here θ is called Maximum Likelihood Estimator (MLE).

 

Properties of MLE

 

  1. MLE is not necessarily unbiased.
  2. MLE is consistent, most efficient and also sufficient, provided a sufficient estimator exists.
  3. MLE tends to be distributed normally for large samples.
  4. If g(θ) is a function of θ and e is an MLE of θ, then g(θ) is the MLE of g(θ).

 

Example 4. 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,…, xn  from the population of X

 

Solution: Consider the following likelihood function:

Taking log on both sides we obtain

Hence the MLE of p is  1/ 1+x

Example 5. 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,…, xn, from the population of X.

 

Solution. Consider the following likelihood function :

Taking log on both sides we obtain

 

ln L = n in A. + (A.- 1) In (x1,x2,…, xn)

Taking log on both sides we obtain

 

ln L = n in A. + (A.- 1) In (x1,x2,…, xn)

Also,    Hence the MLE of A. is  

Example 6. 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.

=> P= ½,which is the MLE

II. Method of Moments

 

In this method, the first few moments of the population is equated with the corresponding moments of the sample.

Then                                        µ’r = m’r

 

Where

=> P= ½,which is the MLE

II. Method of Moments

 

In this method, the first few moments of the population is equated with the corresponding moments of the sample.

Then                                        µ’r = m’r

 

Where            

The solution for the parameters gives the estimates. But this method is applicable only when population moments exist.

 

 

Example 7. Estimate the parameter p of the binomial distribution by the method of moments when n is known).

 

 

Solution. Here,

which is the estimated value.