The likelihood ratio is the ratio of the likelihoods of two models (or a null and alternative parameter value within a single model), which may be used to compare or test the models. If either model is not fully specified then its maximum likelihood over all free parameters is used - this is ...

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Are we frequentists really just implicit/unwitting Bayesians?

For a given inference problem, we know that a Bayesian approach usually differ in both form and results from a fequentist approach. Frequentists (usually includes me) often point out that their ...
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I Just Ran Two Million Regressions - Integrated Likelihood

I am currently working on trying to implement a method used in a popular paper titled "I Just Ran Two Million Regressions". The basic idea behind it is that there are certain cases where it is not ...
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likelihood ratio test (mixed model, nlme in R) not significant, but big R²?

I am carrying out linear mixed models with only one factor as predictor. I start with a likelihood ratio test (LRT) to see if adding the factor (to the null model consisting of the random effect ...
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Probability that the p-value is less than 0.05 if H0 is true?

Suppose that under $H_0$, a measurement $X$ is $N(0,\sigma^2 )$, and that under $H_1$, $X$ is $N(1,\sigma^2 )$ and that the prior probability $P(H_0) = P(H_1)$. With $\sigma$ = 1, what is the ...
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Application of Wilks' theorem to poisson data

My parametrized astrophysical model predicts a mean bin count $e_i(\theta_1,\ldots,\theta_p)$ for each of $N$ non-overlapping energy regions. For a set of observed counts $k_i$ the joint likelihood is ...
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31 views

How do I report the results of likelihood ratio test from glmmADMB?

I want to test if an interaction is significant. My data are strongly overdispersed and contain repeated measures so I have a negative binomial GLMM model in ...
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17 views

Maximum Likelihood Estimates using Negative Log-Likelihood

My query is for an assignment where the first question is to determine if the rate of failure on mechanical equipment is different after a change in manufacturer. In order to answer this I have done a ...
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6 views

“Likelihood Ratio” for multiclass

If i have 2 classes, the likelihood ratio gives me a boundary like the image below. What if i have 3 or more classes ? What test can I use to classify an input data ? What if it is a mixture of ...
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39 views

Bayes optimal classifier vs Likelihood Ratio

I am getting slightly confused by all the probabilistic classifiers. The bayes optimal classifier is given as $ max (p(x|C)p(C)) $ and if all classes have equal prior then it reduces to $ max (p(x|C)...
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15 views

“Multiclass Likelihood Ratio” - Help please?

I am reading up on this paper which is basically about classifying pixels as a target or background using the likelihood ratio test. But LRT is only for 2 classes. What if i have more than 2 classes ? ...
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2 views

Generalized Likelihood Ratio of 2 signals with various unknowns

I am attempting to derive the likelihood ratio of the 2 expressions where S and C are known, a and $\sigma$ are unknown. This is the paper (1st link), equation no. 54. I am able to derive (51). If ...
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9 views

Likelihood Ratio Test for many classes / hypothesis

I am reading this paper on image processing where the authors use the likelihood ratio test to determine if a pixel belongs to the target or the background. I was wondering if it is possible to ...
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21 views

Composite-likelihood ratio test

I am interested in a number of articles (such as Kim and Stephan 2002 and follow-up articles) that use composite likelihood ratio tests to infer selection pressure on linked (phased) genetic data. I ...
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3answers
266 views

Does likelihood ratio test control for overfitting?

I have two nested logistic regression models, A and B. A is nested under B. Let's say B has $K$ more features than A. B has a higher log likelihood than A. However the improved likelihood of B is due ...
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2answers
64 views

Support of likelihood ratio test statistic

Say I'm testing $H_0: Y \sim \text{Exp}(1)$ against $H_1: Y \sim \text{U}(0, 1)$. I believe this gives me the following likelihood ratio test: $$ t^*(y) = \frac{p_1(y)}{p_0(y)} = \frac{1}{e ^ {-y}} ...
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Critical region of likelihood ratio test

This is problem # 5 from RSS's 2014 Graduate Diploma Module 2: $$P(X_j=k) = \begin{cases} (1-p)^3 & k=0\\ 3p(1-p) & k=1\\ p^3 & k=2\\ 0 & \text{otherwise} \end{cases}$$ $Y_k = \sum_{...
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15 views

Testing Proportional Odds Assumptions

I am working in R with a response variable that is the letter grade the student received in a specific course. The response is ordinal, and, in my opinion, seems logically proportional. My ...
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93 views

How to deal with zero-valued probability?

I have a few models which I would like to combine later on; for each model I have the sensitivity and specificity. I then calculate for each model the likelihood ratio to get a feel of how well the ...
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74 views

Is it correct to compare likelihood ratio indices between logistic regression and multinomial logistic regression models?

In the paper "Including Transfer-Out Behavior in Retention Models: Using the NSLC Enrollment Search Data" (http://www.studentclearinghouse.org/colleges/files/ST_UofMD_casestudy.pdf) the author ...
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9 views

Should I use an Odds Ratio or a Likelihood Ratio to report prevalence of x in a Case-Control study?

I am trying to report the prevalence data for a case control study. I first ran a $\chi^2$ test on a 2x2 table of the prevalence data. I got a $\chi^2$ and Likelihood ratio result in SPSS. It ...
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34 views

Queries regarding 'logistftest' function of logistf package for model comparison

I am a user of r package ‘logistf’, I have 2 queries regarding the ‘logistftest’ function when doing model comparisons, I would appreciate if anyone can give helps. I am comparing two models (one is,...
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24 views

How to deal with big data sets (big $n$) in R? (beginner question) [duplicate]

Say that data is assumed to be exponentially distributed with mean $\zeta$. The likelihood function then incldues the factor $\exp{- \sum x_i/\zeta}$. For a big data set, this is potentially a really ...
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19 views

Likelihood ratio test seems to show little difference between models with AICc difference of 3

I'm running a multinomial logistic regression analysis of the behavioural responses of deer to camera traps using no reaction, reaction and strong reaction as dependent variables and season, camera ...
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17 views

Statistical significance of highly correlated explanatory variables

Let's assume I am predicting a Y with 2 x-es: x1 and x2. Let's also assume that x1 and x2 are highly correlated. Let's also say fit 1 uses both x1 and x2 to predict Y while fit 2 only uses x2. ...
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91 views

Log-likelihood ratio test vs. information criteria for model selection

I am trying to select the model with the best fit among GARCH(1,1), ARMA(2,2) and GJR-GARCH(1,1) models for a time series of log returns. The results from IC (Akaike, Bayesian) and likelihood-ratio ...
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22 views

Implementing the Wald Test in PEST Procedure

I'm trying to code a PEST sequence in MATLAB but am unsure about how and when the Wald test is actually implemented. Having read the MATLAB help page for the test, I think my confusion lies with what ...
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35 views

How to infer the likelihood ratio between two examples shown to a Neural Network?

How can I construct a neural net enabling me to efficiently estimate the likelihood of some input relative to some other input? (i.e. a likelihood ratio) An example: Let's say we train this network ...
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25 views

Log likelihood is sufficient statistic

We have two hypotheses: \begin{align} \mathcal H_0&: X \sim P\\\mathcal H_1&: X \sim Q\end{align} Define $F = \log \frac{dP}{dQ}$ to be used for the likelihood ratio test with threshold $\...
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11 views

Profile Likelihood Algorithm Assumptions

Given a statistical model $$\{p_\theta(y) : \theta \in \Theta \}$$ and same data $y$. The log likelihood function is then $l(\theta)=\log(p_\theta(y))$. For a one parameter hypothesis $\theta_j=\beta$ ...
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16 views

References for how to bound the approximation error of a likelihood ratio test?

In likelihood inference, if your log-likelihood is quadratic, then the key values of likelihood theory (Fisher Information, Wilks LR Statistic) have exact probabilistic interpretations. However, if ...
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97 views

Mean-variance spanning test

Is anyone familiar with a mean-variance spanning test? We would like to do a mean-variance spanning test of the diversification benefits of commodities in a portfolio. What you do is that you run a ...
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1answer
29 views

Generalized likelihood test question

This example appears in Rice's stats book. In the first rectangle, why do we have to maximize the denominator? and why do we use MLE as the denominator to maximize it? I know that MLE best reduces ...
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1answer
41 views

Compare two sets of probabilities to outcome data

Suppose there are two predictive models that both output the probability that the home team wins a given match. Then suppose there is data for thousands of matches, in the format: ...
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23 views

Trouble interpreting the likelihood ratio chi-squared test statistic

I have obtained a likelihood ratio chi-squared test statistic and I don't know if it is significant or not. Do I: Compare my likelihood ratio chi-squared test statistic with the critical value in ...
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1answer
313 views

AIC versus Likelihood Ratio Test in Model Variable Selection

The software that I am currently using to build a model compares a "current run" model to a "reference model" and reports (where applicable) both a chi-squared p-value based on likelihood ratio tests ...
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Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space [duplicate]

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, ..., X_n$, and we would like to test hypothesis $H_0 : \theta ...
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1answer
69 views

Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, ..., X_n$, and we would like to test hypothesis $H_0 : \theta ...
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25 views

Does the deviance have a known sampling distribution or not?

In their book Generalized Linear Models, McCullagh & Nelder seem to imply that the sampling distribution of the deviance is generally not known: This is strange, because the deviance of a model $...
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80 views

Expectation of log likelihood ratio

Given that $X_{1},...,X_{n}$ are i.i.d random variables with joint distribution $f(x\mid \theta) $ with 1 dimensional parameter $\theta$, let $\hat\theta$ be the maximum likelihood estimator of $\...
3
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1answer
96 views

Mysterious results from likelihood ratio confidence bounds on a Weibull reliability estimate

I'm trying to calculate a confidence interval on an estimate of the cumulative distribution of a two-parameter Weibull distribution. (Actually 1-cdf, the survival probability.) I want to do this using ...
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16 views

How to test independence of two markov switching processes using likelihood ratio test

I would like to test for independence of two first order markov switching processes with two states each. I have read this can be done using the LR test (I know that this will be simulation based ...
3
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1answer
219 views

Likelihood Ratio Test statistic for the exponential distribution

I need to test null hypothesis $\lambda = \frac12$ against the alternative hypothesis $\lambda \neq \frac12$ based on data $x_1, x_2, ..., x_n$ that follow the exponential distribution with parameter $...
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3answers
98 views

Null-hypothesis testing and likelihood-ratio testing

In this book Wickens, T. D. (2002). Elementary signal detection theory. Oxford: Oxford University Press. You can read: and this confuses me as I thought that likelihood-ratio testing was a ...
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2answers
329 views

Likelihood Ratio vs Wald test

From what I've been reading, amongst others on the site of the UCLA statistics consulting group likelihood ratio tests and wald tests are pretty similar in testing whether two glm models show a ...
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1answer
54 views

G test with small observed counts (~zero)

I have an experiments that counts how many events happened during a certain period of time. The experiments is repeated many times in different "runs" (I have 65 runs). Here for example the list of ...
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23 views

Likelihood Ratio Criterion in EFA

This is in ref to pp. 54-55 in McDonald,R.P[1] in the context of exploratory factor analysis (EFA) ML estimation. The likelihood ratio criteria, to me, seems to be performing dual roles: I. Providing ...
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1answer
87 views

Likelihood Ratio Test for the variance of a normal distribution

I've found that the asymptotic LR test is used in simple vs bilateral hypothesis test in which it is impossible to actually compute the rejection region, or better, in which we would need to find a ...
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1answer
32 views

Why is the noncentral chi-squared approximate of the deviance bad?

The statistical model under consideration is given by the independent realizations of two independent binomial distributions: $$ x_1 \sim \mathrm{Bin}(n_1, p_1), \quad x_2 \sim \mathrm{Bin}(n_2,p_2), $...
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2answers
70 views

Why is there an intrinsic trade off between the probability of detection and probability of a false alarm in the operating characteristic?

I was reading some notes for fun on Binary Hypothesis testing and it claimed that there happens to be a tradeoff between the probability of detection (also known as the "power"): $$P_D = P(f(y) = H_1 ...
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1answer
41 views

Can the Operating Characteristic for the LRT be derived from minimizing Bayes Risk $ \varphi(f) = \alpha P_F - \beta P_D + \gamma$?

For fun I was reading some notes on Operating Characteristics and it said (paraphrased with notation definition): As Fig. 1 suggests (omitted from question), good detection probability $P_D = P(\...