Actuarial Outpost Likelihood ratio test
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

#1
07-14-2007, 01:00 PM
 Ruud Van Nistelrooij Member Join Date: Sep 2005 Posts: 148
Likelihood ratio test

how to determine the Chi-square degree of freedom???
#2
07-14-2007, 01:19 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

You're smart, read the text or study manual.

Seriously though--help comes to those who help themselves. What I mean by this is that the quality of the feedback you get on a question depends on the amount of detail you provide. If I asked you, "How do you calculate integrals," what kind of response would that generate? The scope is too broad. On the other hand, if I asked you, "How do you calculate

$\int_0^{\infty} \frac{\log(1+x)}{1+x^2} \, dx$

then at least the response will be specific. Your question is too vague because an adequate response would basically involve quoting the text or study manual--essentially, to teach you the entire topic.
__________________
Spoiler:
"No, Moslems don't believe Jesus was the messiah.

Think of it like a movie. The Torah is the first one, and the New Testament is the sequel. Then the Qu'ran comes out, and it retcons the last one like it never happened. There's still Jesus, but he's not the main character anymore, and the messiah hasn't shown up yet.

Jews like the first movie but ignored the sequels, Christians think you need to watch the first two, but the third movie doesn't count, Moslems think the third one was the best, and Mormons liked the second one so much they started writing fanfiction that doesn't fit with ANY of the series canon."

-RandomFerret
#3
07-14-2007, 01:23 PM
 JohnLocke Member SOA Join Date: Mar 2007 Posts: 13,464

You're asking a lotta questions about 4/C. If you get a good manual (like ASM), it should have all the answers you seek.
#4
07-14-2007, 01:24 PM
 Ruud Van Nistelrooij Member Join Date: Sep 2005 Posts: 148

Quote:
 Originally Posted by atomic You're smart, read the text or study manual. Seriously though--help comes to those who help themselves. What I mean by this is that the quality of the feedback you get on a question depends on the amount of detail you provide. If I asked you, "How do you calculate integrals," what kind of response would that generate? The scope is too broad. On the other hand, if I asked you, "How do you calculate $\int_0^{\infty} \frac{\log(1+x)}{1+x^2} \, dx$ then at least the response will be specific. Your question is too vague because an adequate response would basically involve quoting the text or study manual--essentially, to teach you the entire topic.

just like question 14 of exam C speing 2007, this question about likelihood ratio test. the test statistic = -2log^

i do not undestand the following statement:
"the test statistic −2 log Λ will be asymptotically χ2 distributed with degrees of freedom equal to the difference in dimensionality of Θ and Θ0." reference: http://en.wikipedia.org/wiki/Likelihood-ratio_test

Last edited by Ruud Van Nistelrooij; 07-14-2007 at 01:29 PM..
#5
07-14-2007, 01:43 PM
 Ruud Van Nistelrooij Member Join Date: Sep 2005 Posts: 148

Quote:
 Originally Posted by Ruud Van Nistelrooij thank you for your advise just like question 14 of exam C speing 2007, this question about likelihood ratio test. the test statistic = -2log^ i do not undestand the following statement: "the test statistic −2 log Λ will be asymptotically χ2 distributed with degrees of freedom equal to the difference in dimensionality of Θ and Θ0." reference: http://en.wikipedia.org/wiki/Likelihood-ratio_test
btw, a question irrelevant to SOA exam

is Lakewood city in New Jersey close to New York city?
#6
07-14-2007, 01:44 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
 Originally Posted by Ruud Van Nistelrooij thank you for your advise just like question 14 of exam C speing 2007, this question about likelihood ratio test. the test statistic = -2log^ i do not undestand the following statement: "the test statistic −2 log Λ will be asymptotically χ2 distributed with degrees of freedom equal to the difference in dimensionality of Θ and Θ0." reference: http://en.wikipedia.org/wiki/Likelihood-ratio_test
The Wikipedia article examines the topic in greater detail than is needed or tested on the exam. However, I will address the article and apply the concepts as they are used in Exam 4/C. Hereafter, we will discuss the likelihood ratio test as it is tested for the exam.

We use the LR test when we want to select a parametric model to fit empirical data. However, the number of parameters in the model is not predetermined; for example, we can use a 1-parameter model (e.g., exponential or single-parameter Pareto), a 2-parameter model (gamma, Pareto), or an N-parameter model. We might expect that as the number of parameters is increased, we are more likely to obtain a model whose MLE fit of the data is better--at the cost of the complexity of the model.

To quantify this notion, we use the LR test. The idea is extraordinarily simple. For various models, we compute the loglikelihood of the MLE fit to the empirical data. Then twice the pairwise difference between the loglikelihoods is chi-square distributed, with degree of freedom equal to the difference in the number of parameters in each model. So if we want to compare a 2-parameter fit to a 4-parameter fit, the LR test gives us a chi-square statistic with df = 4 - 2 = 2. It's that easy.

For practical considerations on the exam, the value of the MLE fit is often given, since it is usually tedious to compute.
__________________
Spoiler:
"No, Moslems don't believe Jesus was the messiah.

Think of it like a movie. The Torah is the first one, and the New Testament is the sequel. Then the Qu'ran comes out, and it retcons the last one like it never happened. There's still Jesus, but he's not the main character anymore, and the messiah hasn't shown up yet.

Jews like the first movie but ignored the sequels, Christians think you need to watch the first two, but the third movie doesn't count, Moslems think the third one was the best, and Mormons liked the second one so much they started writing fanfiction that doesn't fit with ANY of the series canon."

-RandomFerret
#7
07-14-2007, 01:54 PM
 Ruud Van Nistelrooij Member Join Date: Sep 2005 Posts: 148

Quote:
 Originally Posted by atomic The Wikipedia article examines the topic in greater detail than is needed or tested on the exam. However, I will address the article and apply the concepts as they are used in Exam 4/C. Hereafter, we will discuss the likelihood ratio test as it is tested for the exam. We use the LR test when we want to select a parametric model to fit empirical data. However, the number of parameters in the model is not predetermined; for example, we can use a 1-parameter model (e.g., exponential or single-parameter Pareto), a 2-parameter model (gamma, Pareto), or an N-parameter model. We might expect that as the number of parameters is increased, we are more likely to obtain a model whose MLE fit of the data is better--at the cost of the complexity of the model. To quantify this notion, we use the LR test. The idea is extraordinarily simple. For various models, we compute the loglikelihood of the MLE fit to the empirical data. Then twice the pairwise difference between the loglikelihoods is chi-square distributed, with degree of freedom equal to the difference in the number of parameters in each model. So if we want to compare a 2-parameter fit to a 4-parameter fit, the LR test gives us a chi-square statistic with df = 4 - 2 = 2. It's that easy. For practical considerations on the exam, the value of the MLE fit is often given, since it is usually tedious to compute.

in question 14 of spring 07, the distribution is pareto with parameters alpha =2. theta unknown
we test if theta = 3.1 by LR test
why the degree of freedom is 1?
#8
07-14-2007, 02:57 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
 Originally Posted by Ruud Van Nistelrooij thank you for your explanation in question 14 of spring 07, the distribution is pareto with parameters alpha =2. theta unknown we test if theta = 3.1 by LR test why the degree of freedom is 1?
Because you are comparing a 1-parameter (unknown theta) model with a 0-parameter (known theta) model.

To be more precise, the models you are comparing are both 2-parameter Paretos. But the hypothesis is testing between one model where both parameters are known and fixed, versus another model whose theta parameter is variable. So there is exactly 1 degree of freedom, since only one quantity is allowed to change.
__________________
Spoiler:
"No, Moslems don't believe Jesus was the messiah.

Think of it like a movie. The Torah is the first one, and the New Testament is the sequel. Then the Qu'ran comes out, and it retcons the last one like it never happened. There's still Jesus, but he's not the main character anymore, and the messiah hasn't shown up yet.

Jews like the first movie but ignored the sequels, Christians think you need to watch the first two, but the third movie doesn't count, Moslems think the third one was the best, and Mormons liked the second one so much they started writing fanfiction that doesn't fit with ANY of the series canon."

-RandomFerret

Last edited by atomic; 07-14-2007 at 03:08 PM..
#9
07-16-2007, 02:54 PM
 NoName Site Supporter Site Supporter Join Date: Nov 2001 Posts: 5,817

Quote:
 Originally Posted by Ruud Van Nistelrooij btw, a question irrelevant to SOA exam is Lakewood city in New Jersey close to New York city?
#10
07-17-2007, 08:59 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College 1962 (!) Posts: 1,732

To atomic:

Good explanation, but ... : The LR test is valid provided that the model with fewer parameters is a special case of the model with more parameters. It is _not_ valid for comparing an Exponential and a 2-parameter Pareto, for instance.

Jim Daniel
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 01:29 AM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top