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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Likelihood Ratio Tesr [on hold]

Let X1;X2; :::;Xn be a random sample from N(0; 2 = ) distribution where 0 < < 1 and 0 is known. Show that the likelihood ratio test of H0 : = 0 versus H1 : 6= 0 can be based upon the ...
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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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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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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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80 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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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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57 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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68 views

Likelihood Ratio for two-sample Normal distribution, equality of means hypothesis

I have been trying to solve the following exercise Let $X_1,X_2,\ldots X_n$ and $Y_1,Y_2,\ldots, Y_n$ be two independent random samples from two normal distributions $N(\mu_1,\sigma^2)$ and ...
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66 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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54 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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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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39 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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92 views

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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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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1answer
109 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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36 views

Hypothesis test on variance of normal sample

This question is a follow-up to this discussion: ...
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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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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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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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177 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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151 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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290 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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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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1answer
72 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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26 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 ...
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85 views

Application of likelihood ratio test to test the Markov property

Do you know a reference (freely available on the web) where the likelihood ratio test is applied in order to test for the Markov property? The setting is a directly observable discrete Markov-chain ...
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1answer
84 views

Likelihood ratio for normal distribution

Let $X_1, \dots , X_n \sim \mathrm{N}(\mu,\sigma^2)$. $\sigma^2$ is known. We want to test $\mathrm{H}_0: \mu = 0$ versus $\mathrm{H}_1: \mu > 0$. For the likelihood ratio I got: $\Lambda_1 = ...
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Likelihood-ratio test in the big picture

I am trying to understand the big picture of statistical tests. That is how to choose the correct statistical test under different situations. I found this table where choosing a statistical test is ...
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Finding weighted ratio between two cases

There is some process running (for example, network traffic generation). It has two different input conditions which when applied create 2 process cases (#1 and #2). There is a need to compare these ...
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Challenging Likelihood Ratio Test

Derive Likelihood Ratio Test of size $\alpha$. H$_0$: $\theta=\theta_0$ H$_1$:$\theta \neq \theta_0$ \begin{equation} {f}(x,\theta , c) = \theta(x-c)^{\theta-1} \ {c<x<c+1} \end{equation} I ...
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441 views

How to interpret and compare models in Cox regression?

I am trying to interpret the results of a Cox regression; I am doing a PhD in medicine. I love statistics but my question is still pretty basic, I think, and I did not find an answer in previous ...
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1answer
126 views

The test statistic in the likelihood ratio test for nested linear models

Imagine that we have a family of probability disributions with p.d.f $f_{\theta}(z)$ where $\theta \in \Theta$. We also know that there is a linear dependence between parameters. As a consequence we ...
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212 views

Why can't likelihood ratio tests be used for non-nested models?

More specifically, why do the likelihood ratio tests have asymptotically a $\chi^2$ distribution if the models are nested, but this is no longer the case for the not-nested models? I understand that ...
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132 views

Likelihood ratio test - sample size issue

I am a statistician running into an odd problem, and it feels like I am missing a key point here. I have a true model (generated by me so I know truth), and a PREDICTED model that estimates certain ...
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173 views

Negative binomial - likelihood ratio and likelihood ratio chi-squared

I ran a hierarchical negative binomial regression analysis, and got information relative to the log likelihood and likelihood ratio chi-squared in the output. I have the following questions regarding ...
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Likelihood-ratio test for different distributions

I aim to discriminate between three populations by using a maximum likelihood classifier. The idea is that every population has a unique distribution $\hat{f_i} (x)$. The pdf $\hat{f_i}(x)$ has been ...
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107 views

Is likelihood ratio test the only way to build hypothesis tests?

Usually we can construct likelihood ratio for testing the Null hypothesis and alternative hypothesis: The likelihood ratio test $ P( l(\beta_{1}) / l(\beta_{2}) ) < \alpha $ is the rejecting ...
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107 views

Maximum Likelihood question

I need to obtain the functional (i.e.: statistic obtained from the empirical distribution function) associated to the most powerful test to contrast $H_0: f_o$ vs. $H_1: f_1$, using a sample of size ...
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1answer
200 views

Likelihood ratio test to compare two predictions

I have two predictions from two different types of methods. "predictedHousePrices1" is a continuous variable and the output of a prediction from a RandomForest model, "predictedHousePrices2" is the ...
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274 views

Likelihood ratio test versus AIC for model comparison

When performing model comparison, why does the likelihood ratio test require two nested models while this is not required when using the AIC?
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67 views

Partial likelihood ratio test under complex sampling in R

I am seeking help in calculating the likelihood ratio test (LRT) under complex sampling in R. The test is described by Lumley & Scott(2013). I tried to reproduce the results from their example on ...
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183 views

Likelihood ratio test in R for categorical variables

I am working with behavioral data of male sea lions, with a binomial model to understand the effect of different variables in determining the location where the encounters between males occur (Land ...
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50 views

Is this a nested model?

I have a dataset with explanatory variable $X$ and response variable $Y$. Besides, there is a two-level factor $S_{j}, j=1,2$. The random error variance might be different between these two strata. I ...
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1answer
125 views

Is this a nested model and can I use the likelihood ratio test?

I have a data set, total_data and I applied a model to it. For instance, the model has one parameter $\beta$, and I calculated the log-likelihood of the fitted model (using maximum likelihood method). ...
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24 views

Orthogonal contrasts given MLEs and log-likelihoods

I have a parametric distribution for which the MLE must be obtained numerically, with optimization routines. My functions return the $\hat\beta$s and the log-likelihood. To test the hypothesis that ...
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727 views

How to calculate the confidence intervals for likelihood ratios from a 2x2 table in the presence of cells with zeroes

I am analysing a diagnostic test (against a gold standard, using a 2x2 table). I want to calculate likelihood ratios (sensitivity / (1-specificity) etc) however I have several sets of data with 0 ...
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268 views

Likelihood ratio test for a random variable following the Gamma Distribution

Assuming we have a random variable $X\sim \operatorname{Gamma}\left(\alpha,\beta \right )$:$\frac{1}{\Gamma (\alpha )\beta ^{\alpha}}x^{\alpha-1}e^{\frac{-x}{\beta }}$ I'd like to test the ...