# Questions tagged [maximum-likelihood]

For questions that use the method of maximum likelihood for estimating the parameters of a statistical model with given data.

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### Restricted covariance matrix [closed]

I want to define linear restrictions on a symmetric $3\times3$ covariance matrix such that matrix $A$ is equal to matrix $B$. \begin{align} A &= \begin{bmatrix} \sigma^2_{\nu_0} & \cdot & \...
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### Predicting if an Optimization Algorithm is "Doomed to Fail?

Suppose we have a linear combination weighted (with weights $\pi_i$ ) sum of Normal Distributions (Mixture Distributions https://en.wikipedia.org/wiki/Mixture_distribution): \begin{align*} p(x|\theta) ...
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### Deriving Maximum Likelihood Estimators for a Linear Model with Exponential Error Term

I am currently working on a problem where I need to derive the maximum likelihood estimators for a linear model with an exponential error term. Here's the problem: A machine sequentially performs two ...
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### How to find the likelihood of a uniform random variable with support of length 1, but where we don't know where it starts.

It is said for the above question the answer is Option A) A horizontal line between $x_1$ and $x_1-1$. I am unable to understand how the likelihood is being expressed in terms of $x_1$ rather than it ...
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### MLE of $\theta$ in $N(\theta, \theta^2)$ and Asymptotic Distribution of $\hat{{\theta}}_{\text{MLE}}$

Question Let $( X_1, X_2, \ldots, X_n )$ be an independent random sample from $N(\theta, \theta^2)$ where $\theta \neq 0$. Find the MLE for $\theta$ and find the asymptotic distribution of the MLE ...
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### Peak location of the maximum likelihood estimator's sampling distribution

Let's say we obtained a point maximum likelihood estimation $\hat{\theta}_\mathrm{MLE}\left(\mathbf{x}\right)$ from a set of measurements $\mathbf{x} = \left[x_1, x_2, \cdots, x_n \right]$ that ...
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### Integration issue with the Gamma statistical model

I need to verify if an MLE is biased for this Gamma statistical model. \begin{align*} \mathbb{E}\left[\frac{1}{\bar{X}}\right]&=\int^\infty_0\frac{1}{s}\frac{(n\beta)^{2n}}{\Gamma(2n)}s^{(2n-1)}e^{...
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### Asymptotic confidence interval using MLE and Fisher Information

We have observed x1, x2, ..., xn, independent samples from a Poisson distribution with an unknown mean λ > 0. Let $z_{1-α/2}$ denote the $1-\frac{α}{2}$ quantile of the standard normal distribution....
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Suppose I have a log-likelihood of the form $$\mathcal{L} = \sum_{i = 1}^{n} a_i + \sum_{j = 1}^{m} b_j,$$ where $a_i$ and $b_j$ are some independent log-probabilities. The problem is, the second sum $... 1 vote 1 answer 50 views ### Find the maximum likelihood estimator for θ We have a simple random sample of size n from a distribution with pmf 𝑝(𝑥) =$\theta{(1-\theta)}^{x-1}$for 𝑥 = 1,2, …. Find the MLE[𝜃] My try:$ L\ =\ \theta{(1-\theta)}^{1-1}\times\theta{(1-\...
I have a question regarding the correction of my exercise: Exercise 6. Let $Y_1,\dots,Y_n$ be i.i.d. such that $Y_i$ equals $1$ with probability $p$ and $-1$ with probability $1-p$, for all $i\in[n]$....