Actuarial Outpost Mahler Likelihood Ratio Test question
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#1
02-21-2007, 06:54 PM
 dmw Member CAS Join Date: Mar 2005 Posts: 699
Mahler Likelihood Ratio Test question

This question is from Mahler's guide to Fitting Loss Distributions, 2007, section 13, Likelihood Ratio Test.

In question 13.14:

You are given the following 5 claims: 40, 150, 230, 400, 770.
You assume the size of loss distribution is Gamma.

Determine the result of using the likelihood ratio test in order to test the hypothesis that alpha = 3 and theta = 200 versus the alternative, alpha = 3.

A. Reject at the .005 sig. level
B. Reject at the .010 sig. level, but not at the .005 sig. level
C. Reject at the .025 sig. level, but not at the .010 sig. level
D. Reject at the .050 sig. level, but not at the .025 sig. level
E. Do not reject at the .050 sig. level

The answer is D. The number of degrees of freedom used to solve this question is 1 = 1 - 0

My questions are: Which hypothesis has 1 parameter. , which hypothesis has 0 parameters and why for each one? I thought that H0 would have 2 params. and H1 would have 1 param. But, that goes against the rules of the Likelihood Ratio Test, which, according to Mahler is: "H0 is the hypothesis that the distribution with fewer parameters is appropriate. The alternative hypothesis H1 is that the distribution with more parameters is appropriate."

Thanks.
#2
02-21-2007, 07:01 PM
 YoungestEverFSA Member Join Date: Oct 2006 Posts: 178

I think the likelyhood ration test tests whether finding a parameter from the data (which should be closer and therefore fit the data better than an arbitary figure), makes a SIGNIFICANT difference.

In this case, in the example where you are given both alpha and theta, you are not estimating any parametres FROM THE DATA, and therefore the number of estimated parameters is 0. In the second case, you are estimating only theta FROM THE DATA, and therefore the number of estimated parametres is 1.
#3
02-22-2007, 01:38 PM
 Howard Mahler Member CAS Join Date: May 2002 Location: Boston, MA Posts: 1,892

Youngest is exactly correct.

Come back in a week and try either 13.21 (4,11/03, Q.28) , 13.22 (4, 11/05, Q.25), or 13.8 to 13.11.
#4
03-03-2019, 04:34 PM
 gauchodelpaso Member SOA Join Date: Feb 2012 College: Eastern Michigan U Posts: 135

Well, here I am 12 years later... trying to understand the significance. And I've been 100 times with hypothesis testing and still don't get it right, must be my brain convolutions are convoluted...
Let's take this example. I understand that the more parameters fitted should give a better likelihood, or loglikelihood. And that the question is about the significance. And following the method (accept to the right, and reject to the left, pretty ingenious, as long as the table is ordered accordingly). What adds to the confusion is that we use 1-probability as we are in the right end of the distribution.

Looking at the chisquared table we get for 1 degree of freedom (rounding chisq)
P .9 .95 | .975 .99 .995
1-P .1 .05 | .025 .01 .005
chisq 2.7 3.8 | 5.0 6.6 7.9

Double the difference is 4.94 and lies in between (1-P) = .05 and (1-P) = .025, where I placed the |.
That makes it reject at .05 and accept at .025.

What is the exact meaning of this significance? Assuming that Ho is the one with less parameters fitted, the one we are told the model with both parameters given (alpha=3, theta=200), and not taken from fitting the data, versus H1 the one that they give us alpha=3 and we determine theta = 106 from the sample. And we see that H1 certainly has a better loglikelihood (-34.87) than Ho (-37.34).

My question is:

What's the meaning of accepting Ho at .025, or 2.5%, and rejecting Ho at .05, 5%?

That there is a 2.5% chance that the unfitted distribution may be correct, if I run it 1000 times there may be 25 runs that will be correctly representing the data, but not 5% or 50 runs? Or something like of that nature?

It seems like 2.5% is really a remote possibility.

I understand that when we reject at 5%, we also reject any higher possibilities, 10%, 20% etc.

And when we accept at 2.5%, we also accept the possibilities of lower probabilities, like 1%, 0.5%.

So accepting at 2.5% means that is the highest probability that Ho is true, with this data sample and parameters given. Well, actually something in between 2.5 and 5% is the highest value, probably closer to 2.5%( as per Excel formula 2.624%).

I am sorry is this created more noise that is tolerable.

Please let me know of any other interpretations, thank you
(in here I used "accept" instead of probably the more accepted "not reject")
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#5
03-03-2019, 04:44 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,237

"5% significance" means that there is a 5% probability of observing your sample orsomething more extreme if H_0 is true. It means "An event that occurs 5% of the time occurs a significant proportion of the time; it's not an anomaly - so we're not convinced H_0 is false. But something that happens only 4% of the time is an anomaly, extremely unusual, if H_0 is true, so we don't think H_0 is true."
#6
03-04-2019, 07:14 PM
 gauchodelpaso Member SOA Join Date: Feb 2012 College: Eastern Michigan U Posts: 135

Thank you Dr Weishaus again!
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German
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Prelims: 1/P - 2/FM - 3F/MFE - LTAM - STAM
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#7
03-04-2019, 07:31 PM
 gauchodelpaso Member SOA Join Date: Feb 2012 College: Eastern Michigan U Posts: 135

so we accept(or can't reject) Ho at 2.5% probability that may be true, but we reject at 5% because we can't have that level of significance with this data. Is that correct?
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#8
03-04-2019, 07:54 PM
 Colymbosathon ecplecticos Member Join Date: Dec 2003 Posts: 6,121

Quote:
 Originally Posted by gauchodelpaso so we accept(or can't reject) Ho at 2.5% probability that may be true, but we reject at 5% because we can't have that level of significance with this data. Is that correct?
Never accept!

At 2.5% we reject one random sample (assuming that H0 is true) out of 40. (We make 1 mistake out of 40 tries.)

At 5.0% we reject that random sample and one more. (We make two mistakes out of 40 tries.)

If H0 is NOT true, ?? (That is what power tries to address.) Confidence levels address only H0 (and whether it is a 1- or 2-sided test.)
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#9
03-04-2019, 09:15 PM
 gauchodelpaso Member SOA Join Date: Feb 2012 College: Eastern Michigan U Posts: 135

I guess the jury may still be out among non-statisticians about reject or can't accept being different. To me it's the same thing "I can't accept somebody stealing" or "I reject somebody stealing". Unless there are more categories, like being indifferent to somebody stealing. I may have to brush on my logic course. And perhaps there is an indifferent area in Hypothesis testing. Like one for accepting, one indifferent and one rejecting. Yes, I am very confused.
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#10
03-04-2019, 09:23 PM
 gauchodelpaso Member SOA Join Date: Feb 2012 College: Eastern Michigan U Posts: 135

Why can't I get a straight answer to my questions???? I give very specific examples, and I am answered with a different, or seemingly different, example... I know I may be dense... but how much is 1 and 1?, well if you apply the convolution theorem and get the limit at infinity it may be the square root of 4, but just the positive one, on given Fridays.
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Prelims: 1/P - 2/FM - 3F/MFE - LTAM - STAM
VEE: Economics - Corporate Finance - Applied Statistics