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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How to choose between exponential and gamma distributions

I have same data and I would like to choose a model for it. To start with I fit an exponential distribution and a gamma distribution. Now I wanted to do a simple likelihood ratio test . However, I ...
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38 views

Normalized likelihoods

AIC (BIC) model selection methods are widely used. These methods can select non-nested models unlike likelihood ratio type selection that requires model to be nested. The AIC has definition ...
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Likelihood Ratio as statistical test for Transfer Entropy with MatLab

I estimated a Transfer Entropy value TE(XY). Now I want to establish the statistical significance of the estimated value. Therefore I used the method of shuffled surrogates to estimate a shuffled ...
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40 views

Likelihood ratio test to determine if average number of accidents has dropped?

I can't find any worked (non-trivial) practical example for a likelihood ratio test, believe me I have spent hours looking. Here is a question I've been trying to complete but I can't get any further. ...
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Motivating likelihood ratio test vs Wald test for paper reviewer

I've got back reviews for a paper I've submitted, with the following problem. I have two logistic regression models, say y ~ A, and y ~ A + B, where B is a factor with several levels. I have ...
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39 views

Constrained Maximization and Likelihood Ratio Tests for Nested Linear Models

Suppose $\boldsymbol \beta \in \mathbb{R}^k$ is a vector of coefficients for a generalized linear model with $g \left[ E(Y|X) \right] = X\beta$ for a link function $g$ and I wish to test the composite ...
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42 views

Generalized log likelihood ratio test for non-nested models

I understand that if I have two models A and B and A is nested in B then, given some data, I can fit the parameters of A and B using MLE and apply the generalized log likelihood ratio test. In ...
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222 views

Question about Dynkin Lehmann Scheffe Theorem

I'm self-studying for an examination, and I would like to understand how to use the Dynkin Lehmann Scheffe theorem for an applied question. I am using Bickel and Doksum's "Mathematical Statistics" ...
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29 views

Clarification on interpreting Wald's test and Likelihood ratio tests

I am running multinomial logistic regression analysis on my data. The response variable is the number of calves produced each year (0,1, or 2). I am trying to evaluate the influence of the X ...
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68 views

Goodness of fit likelihood ratio test with zero values

I have a vector of observed frequencies that have zero values in some cells and a vector of expected frequencies generated by a model. I would like to do a likelihood ratio test rather than a chi ...
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42 views

Hypothesis Testing of endpoint of Uniform distribution

Let $c$ be a positive constant. Suppose I have $X$~$Unif(0,\theta)$ and I wish to test the null hypothesis $H_0: \theta \leq c$ against the alternative hypothesis $H_1:\theta > c$. Suppose I did a ...
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233 views

Why doesn't Wilks' 1938 proof work for misspecified models?

In the famous 1938 paper ("The large-sample distribution of the likelihood ratio for testing composite hypotheses", Annals of Mathematical Statistics, 9:60-62), Samuel Wilks derived the asymptotic ...
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39 views

fitting LMEMs for repeated measures with no correlation between intercept and slope

For a simulation study, I contrast the power of different LMEMs for repeated measures. To get p-values, I use likelihood ratio tests where I compare a model including a fixed treatment effect with one ...
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1answer
106 views

What are the regularity conditions for Likelihood Ratio test

Could anyone please tell me what the regularity conditions are for the asymptotic distribution of Likelihood Ratio test? Everywhere I look, it is written 'Under the regularity conditions' or 'under ...
3
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1answer
79 views

Wald test and Likelihood ratio test, where do the confidence intervals on the regression coefficients come from?

So I'm trying to build my own Wald test and likelihood ratio test code within a machine learning pipeline. I can get the final fitted logistic regression coefficients from liblinear. I'm coding in ...
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36 views

Likelihood ratio tests versus signed likelihood ratio tests

What is the difference between likelihood and signed likelihood ratio tests (SLRT)? What is the use of having the "sign" function in it? In short, what is the advantage of having SLRT over LRT? ...
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57 views

How to formulate a likelihood ratio test for an exponential distribution? [duplicate]

Find the form of the likelihood ratio test of $H_0 : \lambda = \lambda_0$ against $\lambda\neq\lambda_0$ when $X_1,X_2,...,X_n$ is a random sample from $Ex(\lambda)$. Simplify it as much as ...
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203 views

Neyman-Pearson lemma: critical region and hypothesis testing

Let $X_1,X_2,...,X_n$ be i.i.d r.v's with common p.d.f. $$ \mbox f(x)=\frac{x^5e^{-x/\theta}}{5!\theta^6} $$ where $\theta$ > 0. Show that the Neyman-Pearson lemma produces a test of ...
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35 views

Type I error in t tests and LRT

In situations where unequal variance t-test is appropriate, if we use equal variance t-test, type I error would be inflated. I created a likelihood ratio test (LRT) equivalent of unequal variance ...
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123 views

Likelihood Ratio Test and boundary parameters

The chi-square approximation of the distribution of the test statistic in Likelihood Ratio Test (LRT) may not be reliable if a parameter of a model is on the boundary of a parameter space --- is ...
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42 views

Likelihood ratio test statistic

Let $X_{1},..,X_{n}$ be a random sample from a normal distribution with mean ${\theta}$ and variance 1. Let $H_{0}:{\theta}=0$ and $H_{1}:{\theta}{\neq}0$. Using likelihood ratios, show the critical ...
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42 views

hypothesis test likelihood ratio

Let $X_{1},..,X_{36}$ be a random sample from a bernouilli distribution with parameter $p$. Suppose the sample probability is $1/3$. Using a norml approximation and likelihood ratios, we wish to test ...
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147 views

Log-Likelihood Ratio Test: Difference of Equations

I have noticed that the sign at the front of the log-likelihood ratio formula changes depending on what source I have been looking at. Below are the two formula I have seen, exactly as they were ...
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214 views

What is the relationship between the GINI score and the log-likelihood ratio

I am studying classification and regression trees, and one of the measures for the split location is the GINI score. Now I am used to determining best split location when the log of the likelihood ...
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92 views

One-Sample t Test and Likelihood Ratio Test

Concerning testing the hypotheses about the mean $\mu$ of a single population I know of things like the One-Sample $t$ test where there is a null hypothesis: $H_0: \mu = \mu_0$, and a test statistic ...
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Empty asymptotic confidence interval?

I have a sample $x=(4, 3, 1, 2, 2, 2, 2, 5, 7, 3, 1, 2, 3, 4, 3, 2, 3, 3, 3, 4)$ of size $n=20$. from a binomial distribution with 10 trials and probability of success $p$. I am asked to construct the ...
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101 views

What is it meant by the “rejection region” and “power” of a likelihood ratio test?

Suppose that $X_1,...,X_n$ are i.i.d. data from a $N(\mu, 100)$ distribution. I am trying to find the rejection region for the likelihood ratio test for level $\alpha= 0.10$ of the test: $H_0: \mu = ...
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38 views

Autoregressive Markov chain simulation and the likelihood ratio test for Markov property

I am trying to estimate a Markov chain of second order (Markov chain that fulfills $P[X_t|X_{t-1},X_{t-2}]=P[X_t|X_{t-1},X_{t-2},...,X_{t-p}]$) using an AR(2) process. Once I have simulated the ...
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133 views

Hypothesis testing with Neyman–Pearson (finding cutoff quantile of Poisson dist)

I am currently trying to solve a problem that should be very easy, yet I am stumped, and help would be GREATLY appreciated!! The background of the question is this: Q: The number of typing errors in ...
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23 views

Ways the likelihood ratio test may fail

I have a question related to: Why is a likelihood-ratio test distributed chi-squared? On point 2 of @StasK's answer, he states: The theorem assumes that all the relevant derivatives are ...
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1answer
184 views

Reporting from a likelihood ratio test

I don't know if this belongs here or in StackExchange, it is a mixed but probably pretty simple question. How do I normally report a Likelihood Ratio Test? I would love a good reference in your ...
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128 views

Non-significant p-values for factor levels with only 0s in negative binomial glm using glm.nb() in R

I am trying to fit a negative binomial GLM to fish catch data with month of the year (factor) as my explanatory variable. I have selected the month with the greatest number of catches as my reference ...
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92 views

Comparison between Log-likelihood ratios and beta coefficients

I was asked a question recently which I could not find an answer for and was hoping someone could enlighten me. The question was regarding the significance of a single variable in a linear model. ...
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729 views

Likelihood ratio test - lmer R - Non-nested models

I am currently reviewing some work and have come across the following, which seems wrong to me. Two mixed models are fitted (in R) using lmer. The models are non-nested and are compared by ...
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55 views

Monotone likelihood ratio property: check my proof; also, who proved it first?

I think this is correct: Lemma. Suppose $\ell(x,\theta)$ is a (sufficiently regular) likelihood function; then $$\frac{\partial^2\left(\log \ell(x,\theta)\right)}{\partial \theta \partial x}\ge0$$ ...
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24 views

Maximum power test

Question: In a count that was held a year ago in a reserve , there were 15 tigers counted. 7 females and 8 males, which were marked with numbers from 1 to 15. a. By recent reports, there was fear ...
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Definition of likelihood ratio test

Do I understand this correctly: You fit a model with unknown parameters to a dataset. You choose the parameters so the likelihood of the dataset under the model is maximal. Let this be $L_{max, ...
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355 views

Why the F-test in Gaussian linear models is most powerful?

For a Gaussian linear model $Y=\mu+\sigma G$ where $\mu$ is assumed to lie in some vector space $W$ and $G$ has the standard normal distribution on $\mathbb{R}^n$, the statistic of the $F$-test for ...
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159 views

Logistic regression, goodness of fit interpretation

I'm having some trouble interpretting whether the model is a good fit. Below is an extract from some output from R. ...
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40 views

Hypothesis test on variance of normal sample

This question is a follow-up to this discussion: ...
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76 views

Hypothesis testing and convergence of LRT statistic

This may be a very simple question, but I am not sure about my logic. I have a standard point hypothesis testing scenario, where I collect a sequence of $n$ independent observations ...
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60 views

Designing a uniformly most powerful test.

Let $X_1, ... , X_n$ be iid normally distributed random variables $N(\mu, \sigma^2)$, $\mu \in \Bbb{R}$, $ \sigma^2 > 0$. a) Design a uniformly most powerful test with significance level ...
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How do you control FWER when testing multiple nested models with likelihood ratio tests?

Suppose I have three different models $M_1, M_2$ and $M_3$, these models are nested such that $M_1 \subseteq M_2 \subseteq M_3$. I perform two likelihood ratio tests, $M_1$ against $M_3$ and $M_2$ ...
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80 views

Describing a best critical region of size $\alpha$ for testing $H_0 : \theta = 0$ against the alternative hypothesis $H_1 : \theta =1$

Suppose a sample of size 10 is drawn from a distribution with probability density function $f(x, \theta) = 2x^{\theta}(1-x)^{1-\theta}$ if $0<x<1$ and $0$ otherwise, where $\theta \in ...
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553 views

Likelihood Ratio Test and Wald test provide different conclusion for glm in R

I'm reproducing an example from Generalized, Linear, and Mixed Models. My MWE is below: ...
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273 views

Likelihood Ratio of two-sample Uniform Distribution

Consider two uniform distributions: $$f \left( x, \theta_i \right) =\begin{cases} \frac{1}{2\theta_i} \quad -\theta_i<x<\theta_i, -\infty<\theta_i<\infty \\ 0 \quad \text{elsewhere} ...
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683 views

Likelihood Ratio for two-sample Exponential distribution

Let $X$ and $Y$ be two independent random variables with respective pdfs: $$f \left(x;\theta_i \right) =\begin{cases} \frac{1}{\theta_i} e^{-x/ {\theta_i}} \quad 0<x<\infty, 0<\theta_i< ...
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54 views

Neyman–Pearson lemma for non monotonic spaces

Question: Does the Neyman–Pearson lemma give instructions for how to construct the test when the outcome space is not monotonic? I suspect the answer is NO, but I would like to: Get an affirmative ...
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84 views

Confusion regarding Likelihood Ratio Tests (LRT)

I am trying to construct a LRT to test the hypothesis $H_0: p \ge p_0$ and $H_1: p < p_0$ where $\alpha = .1$ and $p_0 = .6$ and give a critical region. Attempt: $\lambda(x) = \frac{"restricted" ...
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39 views

Testing time homogeneous Markov chains

I am working with different transition diagrams and want to calculate the likelihood ratio statistic for testing time-homogeneous. I saw that there are already some comparable questions, but I still ...