1. STATISTICAL INFERENCE PART I POINT ESTIMATION

2. STATISTICAL INFERENCE
Determining certain unknown properties of a probability distribution on the basis of a sample (usually, a r.s.) obtained from that distribution 
Point Estimation:
Interval Estimation: ( ) 
Hypothesis Testing:

3. STATISTICAL INFERENCE
Parameter Space (): The set of all possible values of an unknown parameter, ; .
A pdf with unknown parameter: f(x; ), .
Estimation: Where in ,  is likely to be?
{ f(x; ),  }
The family of pdfs

4. STATISTICAL INFERENCE
Statistic: A function of rvs (usually a sample rvs in an estimation) which does not contain any unknown parameters.
Estimator of an unknown parameter : A statistic used for estimating .
An observed value

5. Method of Moments Estimation, Maximum Likelihood Estimation
METHODS OF ESTIMATION
Method of Moments Estimation, Maximum Likelihood Estimation

6. METHOD OF MOMENTS ESTIMATION (MME)
Let X1, X2,…,Xn be a r.s. from a population with pmf or pdf f(x;1, 2,…, k). The MMEs are found by equating the first k population moments to corresponding sample moments and solving the resulting system of equations.
Population Moments
Sample Moments

7. METHOD OF MOMENTS ESTIMATION (MME)
so on…
Continue this until there are enough equations to solve for the unknown parameters.

8. EXAMPLES Let X~Exp(). For a r.s of size n, find the MME of .
For the following sample (assuming it is from Exp()), find the estimate of :
11.37, 3, 0.15, 4.27, 2.56, 0.59.

9. EXAMPLES
Let X~N(μ,σ²). For a r.s of size n, find the MMEs of μ and σ².
For the following sample (assuming it is from N(μ,σ²)), find the estimates of μ and σ²:
4.93, 6.82, 3.12, 7.57, 3.04, 4.98, 4.62, 4.84, 2.95, 4.22

10. DRAWBACKS OF MMES
Although sometimes parameters are positive valued, MMEs can be negative.
If moments does not exist, we cannot find MMEs.

11. MAXIMUM LIKELIHOOD ESTIMATION (MLE)
Let X1, X2,…,Xn be a r.s. from a population with pmf or pdf f(x;1, 2,…, k), the likelihood function is defined by

12. MAXIMUM LIKELIHOOD ESTIMATION (MLE)
For each sample point (x1,…,xn), let
be a parameter value at which
L(1,…, k| x1,…,xn) attains its maximum as a function of (1,…, k), with (x1,…,xn) held fixed. A maximum likelihood estimator (MLE) of parameters (1,…, k) based on a sample (X1,…,Xn) is
The MLE is the parameter point for which the observed sample is most likely.

13. EXAMPLES
Let X~Bin(n,p). One observation on X is available, and it is known that n is either 2 or 3 and p=1/2 or 1/3. Our objective is to estimate the pair (n,p). x
(2,1/2)
(2,1/3)
(3,1/2)
(3,1/3)
Max. Prob.
1/4
4/9
1/8
8/27 1
1/2
3/8
12/27 2
1/9
6/27 3
1/27
P(X=0|n=2,p=1/2)=1/4 …

14. MAXIMUM LIKELIHOOD ESTIMATION (MLE)
It is usually convenient to work with the logarithm of the likelihood function.
Suppose that f(x;1, 2,…, k) is a positive, differentiable function of 1, 2,…, k. If a supremum exists, it must satisfy the likelihood equations
MLE occurring at boundary of  cannot be obtained by differentiation. So, use inspection.

15. MLE
Moreover, you need to check that you are in fact maximizing the log-likelihood (or likelihood) by checking that the second derivative is negative.

16. EXAMPLES
1. X~Exp(), >0. For a r.s of size n, find the MLE of .

17. EXAMPLES
2. X~N(,2). For a r.s. of size n, find the MLEs of  and 2.

18. EXAMPLES
3. X~Uniform(0,), >0. For a r.s of size n, find the MLE of .

19. INVARIANCE PROPERTY OF THE MLE
If is the MLE of , then for any function (), the MLE of () is
Example: X~N(,2). For a r.s. of size n, the MLE of  is By the invariance property of MLE, the MLE of 2 is

20. ADVANTAGES OF MLE
Often yields good estimates, especially for large sample size.
Usually they are consistent estimators.
Invariance property of MLEs
Asymptotic distribution of MLE is Normal.
Most widely used estimation technique.

21. DISADVANTAGES OF MLE
Requires that the pdf or pmf is known except the value of parameters.
MLE may not exist or may not be unique.
MLE may not be obtained explicitly (numerical or search methods may be required.). It is sensitive to the choice of starting values when using numerical estimation. 
MLEs can be heavily biased for small samples.
The optimality properties may not apply for small samples.

22. SOME PROPERTIES OF ESTIMATORS
UNBIASED ESTIMATOR (UE): An estimator is an UE of the unknown parameter , if
Otherwise, it is a Biased Estimator of .
Bias of for estimating 
If is UE of ,

23. SOME PROPERTIES OF ESTIMATORS
ASYMPTOTICALLY UNBIASED ESTIMATOR (AUE): An estimator is an AUE of the unknown parameter , if

24. SOME PROPERTIES OF ESTIMATORS
CONSISTENT ESTIMATOR (CE): An estimator which converges in probability to an unknown parameter  for all  is called a CE of .
For large n, a CE tends to be closer to
the unknown population parameter.
MLEs are generally CEs.

25. EXAMPLE
For a r.s. of size n,
By WLLN,

26. MEAN SQUARED ERROR (MSE)
The Mean Square Error (MSE) of an estimator for estimating  is
If is smaller, is the better estimator of .

27. MEAN SQUARED ERROR CONSISTENCY
Tn is called mean squared error consistent (or consistent in quadratic mean) if E{Tn}20 as n.
Theorem: Tn is consistent in MSE iff
i) Var(Tn)0 as n.
If E{Tn}20 as n, Tn is also a CE of .

28. EXAMPLES
X~Exp(), >0. For a r.s of size n, consider the following estimators of , and discuss their bias and consistency.
Which estimator is better?