Page 396 - DMTH404_STATISTICS

Statistics

Notes n
L = f(X ; ) . f(X ; ) . ...... . f(X ; ) = Õ ( f X ; ) .
1 2 n i
i= 1
We have to find that value of q for which L is maximum. The conditions for maxima of L are :
2
dL d L
= 0 and < 0. The value of q satisfying these conditions is known as Maximum Likelihood
d d 2
Estimator (MLE).
Generalising the above, if L is a function of k parameters  ,  , ......  , the first order conditions
1 2 k
¶ L ¶ L ¶ L
for maxima of L are: = = ...... = 0 .
¶ ¶ ¶
1 1 k
This gives k simultaneous equations in k unknowns  ,  , ......  , and can be solved to get
1 2 k
k maximum likelihood estimators.
Sometimes it is convenient to work using logarithm of L. Since log L is a monotonic
transformation of L, the maxima of L and maxima of log L occur at the same value.

Properties of Maximum Likelihood Estimators

1. The maximum likelihood estimators are consistent.
2. The maximum likelihood estimators are not necessarily unbiased. If a maximum likelihood
estimator is biased, then by slight modifications it can be converted into an unbiased
estimator.
3. If a maximum likelihood estimator is unbiased, then it will also be most efficient.
4. A maximum likelihood estimator is sufficient provided sufficient estimator exists.

5. The maximum likelihood estimators are invariant under functional transformation, i.e., if
t is a maximum likelihood estimator of , then f(t) would be maximum likelihood estimator
of f().

Example 2: Obtain a maximum likelihood estimator of p (the proportion of successes) in
n
-
)
a population with p.m.f. given by ( ;f X  = C  X (1 - ) n X , where X denotes the number of
X
successes in a sample of n trials.
Solution.
-
n
Since C  X (1 - ) n X is the probability of X successes out of n trials, therefore, this is also the
X
-
n
likelihood function. Thus, we can write L = C  X (1 - ) n X .
X
Taking logarithm of both sides, we get
n
log L = log C + X log + (n X- )log (1 - )
X
Differentiating w.r.t. p, we get
d log L X n X
-
= 0 + - = 0 for maxima of L.
-
d  1 

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