**Definition 1**: Suppose a random variable *x* has a frequency function *f(x; θ)* that depends on parameters *θ =* {*θ*_{1}*, θ*_{2}*, …, θ _{k}*}. For a sample {

*x*

_{1}

*, x*

_{2}

*, …, x*} the

_{n}**likelihood function**is defined by

Here we treat *x*_{1}*, x*_{2}*, …, x _{n}* as fixed. The

**maximum likelihood estimator**of

*θ*is the value of

*θ*that maximizes

*L(θ)*. We can then view the maximum likelihood estimator of

*θ*as a function of the sample

*x*_{1}

*, x*_{2}

*We will commonly represent the maximum likelihood estimator of*

*, …, x*._{n}*θ*as

*θ-*hat, written

**Definition 2**: Let *θ-hat* be the maximum likelihood estimator of *θ* for the sample {*x*_{1}*, x*_{2}*, …, x _{n}*}. Suppose we want to test a null hypothesis regarding the parameters in

*θ*and let

*θ’-*hat be the maximum likelihood estimator of

*θ*for the sample {

*x*

_{1}

*, x*

_{2}

*, …, x*} when the null hypothesis is true. Now define

_{n}**Observation**: Here *λ *can be viewed as a function of *x*_{1}*, x*_{2}*, …, x _{n.}*. Clearly

and so 0 ≤ *λ* ≤ 1. For any given *x*_{1}*, x*_{2}*, …, x _{n.}*, values of

*λ*is near 1 correspond to the null hypothesis being true. The closer that is to 1 the more reasonable it is to accept the null hypothesis, while the farther away from 1 the more reasonable it is to reject the null hypothesis. Thus

*λ*can be used as a statistic for testing the validity of the null hypothesis.

Suppose *g*(*λ*) is the frequency function for *λ *where *g*(*λ*) doesn’t depend on any unknown parameters, then for any *α*, there is a value *λ*_{0}* *such that the cumulative distribution function *F* at *λ*_{0} takes the value *α*. For those familiar with calculus, this means that

*λ* is the **likelihood ratio test statistic** for the hypothesis H_{0}. We reject the null hypothesis if and only if the value of *λ ≤ λ*_{0}* *where λ is the value of *λ* for the sample {*x*_{1}*, x*_{2}*, …, x _{n}*} and

*λ*

_{0 }is the value such that

*F*(

*λ*

_{0})

*= α.*

Question:

What is the reason for having a #value answer for the Max Likelihood part of the Chi-test?

Victor,

It is an error of some sort, but I would have to see the data to give you a more specific answer. If you send me an Excel file with your data, I could answer more specifically (my email address is available via Contact Us).

Charles

All my appreciation.

_Excel has more to offer then i tough.

_College Stat-Math concept were far gone and now i need those for employment.

Thank you for presenting quite a body of knowledge and so reachable.

Denis