Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [maximum-likelihood]

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

In statistics, maximum-likelihood estimation (MLE) is a method of estimating the parameters of a statistical model given data. The method of maximum likelihood corresponds to many well-known estimation methods in statistics. Maximum likelihood, also called the maximum likelihood method, is the procedure of finding the value of one or more parameters for a given statistic which makes the known likelihood distribution a maximum. The maximum likelihood estimate for a parameter $\mu$ is denoted $\widehat{\mu}$.

1482 questions
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MAP Solution for Linear Regression - What is a Gaussian prior?

I am looking at some slides that compute the MLE and MAP solution for a Linear Regression problem. It states that the problem can be defined as such: We can compute the MLE of w as such: Now they talk about computing the MAP of w I simply can't…
user1436508
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model fitting for partial-nonparametric GARCH model using R

I was trying to reproduce the partial non-parametric garch model from paper on page 1764 https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1540-6261.1993.tb05127.x The problem is how to derive the likelihood function? For example, I divide the…
Teng Li
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Fisher information of binomial distribution - question about expectation

I know that this has been solved before, but I am specifically asking about how to solve the expectation: The second derivative of the log-likelihood function is: $$\frac{d^2\ell\theta}{d\theta^2} = -\frac{x}{\theta^2} - \frac{n - x}{(1 -…
SEL
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Maximum likelihood estimators for gamma distribution

I'm having trouble with an exercise about maximum likelihood estimators. Specifically, the exercise gives me values of a protein which was found in 50 adults. We assumed that the data follow a gamma distribution: $X \sim \Gamma(r,\lambda)= \frac…
gensdimi
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Why does setting the derivative of a likelihood function equal to 0 maximize the likelihood function?

I'm learning from a statistics tutorial which defines a likelihood function as \begin{align} L(1,3,2,2; \theta)=27 \cdot \theta^{8} (1-\theta)^{4} \tag{1} \end{align} and then the tutorial sets the derivative of (1) to zero to find the value of…
soplus2018
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Joint MLE of n datasets

Assume we have count data of the number of plants for $n$ sites. Also, assume that the mean among sites is different, whereby $\mu_1 \neq \mu_2 \neq \mu_3, \cdots,\mu_{n-1} \neq \mu_n$, and each of our $n$ datasets are Poisson distributed. Let's say…
Constantin
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Sum of a Sum of a Squared Difference

How did the author jump from the second equation to the third equation? I suspect there’s a rule I’m forgetting that allows for this, any help is appreciated.
Seraphim
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Which probability distributions are the MLEs known to be consistent and asymptotically normal?

MLEs are pretty useful for estimating parameters of probability distributions when they are consistent and asymptotically normal. But I suck too much at math and proofs to prove that the relevant regularity conditions hold. For example, one…
BatWannaBe
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Maximum likelihood estimator of $b$, given $n$ draws from the minimum distribution of two values from $U[0,b]$ are observed

I see how the MLE of the upper bound for a uniform distribution is obtained, given $n$ draws are observed. Now I wonder the following: What is the maximum likelihood estimator of $b$, given $n$ draws from the minimum distribution of two values from…
bonifaz
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Finding MLE for a discrete distribution

Given discrete random variables $X_1, \ldots X_n$ with probability mass function: $$ f(x;\theta) = \begin{cases} \theta & x= -1 \\ (1-\theta)^2 \theta^x & x = 0, 1, \ldots \\ \end{cases} $$ Prove that the MLE of $\theta$…
drgxfs
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Log-Maximum-Likelihood depends on log base?

I am given the following PDF: $$f_X(z) = \begin{cases} \frac{ak^a}{z^{a+1}}, & \text{for $z \ge k$} \\ 0, & \text{for $z \lt k$} \end{cases}$$ For some parameters $k$ and $a$. I am then told to find the Maximum-Likelihood estimator for $a$ if we…
yagod
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Find the maximum likelihood of $f(x|β) = \frac{1}{β}\,e^{-x/β}$

Let X1, · · · , Xn be an i.i.d. sample from an exponential distribution with the density function. Find the maximum likelihood of $$f(x|β) = \frac{1}{β}\,e^{-x/β}$$ First, I need to find: $$p(x|β) = \prod_{i=1}^n \frac{1}{β} \,e^{-x_i/β}$$ but…
llamaro25
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Determining the generalised likelihood ratio for a test determining whether the $x_{i}$ share a common parameter

I need to determine the generalised likelihood ratio in order to test whether a set of values $X$ are such that each $x_{i} \in X$ are given by a common poisson distribution $Po(\lambda)$ or alternatively they each are given by their own…
Flose
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Parameter estimation using method of maximum likelihood. What am I doing wrong?

Question from Michael Baron's Probability and Statistics for Computer Scientists, 2nd edition: Data: 3 7 5 3 2. Assume data is produced from Geometric distribution. Estimate $p$, the geometric distribution parameter. (Geometric distribution: $P(x)…
O. Altun
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About MLE of $\sigma$ with Normal Distribution

I tried to find answers the questions below, but I could not get clear answers for them. For a random sample of size n, $x_1, x_2, ..., x_n$ from a Normal distribution where $\sigma^2$ is unknown. It is quite easy to derive answers with the MLE of…
coalt
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