Chapter-3 page-40

Hi,

2 Lines from Chapter-3 under heading "7.2 Assessing the fit":

If the Markov property holds we expect nijk to be an observation from
a Binomial distribution with parameters nij and pjk .

It would be very helpful if you explain same.

Regards,
Ashok

 

I am explaining the line of top of Page 40

and using example Q 3.17 as an aid.

Let's say we want to find the distribution of \(N_{12}\) given \(N_1\).
i.e. no. of transitions from state 1 to 2 given Number of times the process was in state 1.

Now if the process is in state 1, after a transition takes place it can either remain in state 1, or go to state 2 or 3.

Transition to state 2 can be taken as a success, and transition to state 1 and 3 a failure.

So distribution of \(N_{12}\) given \(N_1\) is \(Bin(N_1, ~ p_{12})\).
Now we want to estimate \(p_{12}\) from the data given.

11
12
12
13
13
13

These are the observations from the data set. So, 6 times the process was in state 1 and after a transition took place, 2 times it went to state 2.

So \(\widehat{p}_{12}= \frac{2}{6}\)
Similarly \(\widehat{p}_{11} = \frac{1}{6} and ~\widehat{p}_{13}= \frac{3}{6}\)

Therefore distribution of \(N_{12}|N_1\) is \(Bin(N_1,~ \frac{2}{6})\).

or in general distribution of \(N_{ij}|N_i\) is \(Bin(N_i,~ p_{ij})\).

You can apply the same logic to triplets as well, which was your doubt.

 

confused

I have a querry on Q. 3.17

The question states:

You have been given the following series of data from a 3 state process:

1,3,2,2,1,3,3,2,3,1,2,3,2,1,1,2,2,1,3,3.

i)calculate the values of n(i), n(i,j) and n(i,j,k)

ii) Estimate the one step transition probabilities.



a) The answers give n(3)=6, however, i count 7 entries.

b)The transition probability P(33) is given as 2/6,which is n(33)/n(3),implying n(3,3)=2 however, the answers state n(33) as 1, since final entry 3,3 is ignored.

So, is n(3,3) 2 or 1?

Why is n(3)=6?

Anyone kindly assist.

 

When estimating p(3,3), we count the number of times 3 is followed by 3 and then divide this by the number of times that 3 is followed by anything.

The number of times that 3 is followed by anything is 6. We don't count the 3 at the end of the list as it's not followed by anything.

 

Get it now.

Thanks