To differentiate the likelihood function we need to use the product rule along with the power rule:
L ( p ) Σ xip-1 Σ xi (1 – p)n – Σ xi – (n – Σ xi )pΣ xi (1 – p)n-1 – Σ xi
We rewrite some of the negative exponents and have:
L ( p ) (1/p) Σ xipΣ xi (1 – p)n – Σ xi – 1/(1 – p) (n – Σ xi )pΣ xi (1 – p)n – Σ xi
[(1/p) Σ xi – 1/(1 – p) (n – Σ xi)]ipΣ xi (1 – p)n – Σ xi
Now, in order to continue the process of maximization, we set this derivative equal to zero and solve for p:
0 [(1/p) Σ xi – 1/(1 – p) (n – Σ xi)]ipΣ xi (1 – p)n – Σ xi
Since p and (1- p) are nonzero we have that
0 (1/p) Σ xi – 1/(1 – p) (n – Σ xi). Get a real answer from a real personGet help from our friendly experts. Aptech helps people achieve their goals by offering products and applications that define the leading edge of statistical analysis capabilities. We need to put on our calculus hats now, since in order to maximize the function, we are going to need to differentiate the likelihood function with respect to \(p\). 25*0.
How to Be Continuity Assignment Help
25 = 0. Therefore, it is computationally faster than Newton-Raphson method.
where
x
{\displaystyle {\bar {x}}}
is the sample mean.
If one wants to demonstrate that the ML estimator
{\displaystyle {\widehat {\theta \,}}}
converges to θ0 almost surely, then a stronger condition of uniform convergence almost surely has to be imposed:
Additionally, redirected here (as assumed above) the data were generated by
f
(
;
0
)
{\displaystyle f(\cdot \,;\theta _{0})}
, then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. .