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
1
vote
2 answers

are these equivalent?

I am trying to determine if these two expressions are equivalent: $$\prod_{i=1}^n\frac{\log(\theta)}{\theta - 1} \theta^{x_i} = \frac{(\log(\theta))^n}{(\theta - 1)^n}\theta^{\sum_{i=1}^n x_i}$$ If they aren't what am I missing here?
statrat403
  • 77
1
vote
1 answer

ML estimation help with Poisson-like data

Suppose that you have $n$ observations, $y_1, y_2, ...y_n$ that are the result of the process $aX$ where $a$ is a real constant and $X$ is a random variable such that $X\sim\mathrm{Poisson}(-b/a)$. It is the case that $0>a>1$ and $b>0$. I am trying…
Ben S.
  • 505
1
vote
2 answers

MLE of a function of parameter

I am trying to get my head around the maximum likelihood estimator (MLE) of a function of a parameter. Say I have $X_i \sim \text{Poisson}(\theta)$ samples. I want to find the MLE of $\pi(\theta) = \exp(-\theta) = P_\theta(X = 0)$. MLE of the…
Brian Mark
  • 11
1
vote
1 answer

Sample mean and maximum likehood

I have a question considering the proof that "something" is the best maximum likehood estimator. I have no idea how to continue next with this problem, so far I just wrote the normal distribution, and tried to somehow rewrite it to the logaritmus…
user118331
  • 91
1
vote
1 answer

Maximum Likelihood Problems - exercise

I am working on a MLE question to find $\lambda$ where x has the distribution: $$\lambda^2e^{-2\lambda x^2}$$ This is what I have done: $$L(\lambda)=\prod{\lambda^2e^{-2\lambda x^2}}$$ $$=\lambda^{2n}e^{\sum-2\lambda x^2}$$ $$\log L(\lambda)=2n \log…
BigData
  • 109
1
vote
2 answers

Maximum Likelihood Estimate of a a discrete r.d - I spent more than 4 hours on this questions, help!!

Suppose $X$ is a discrete r.d with the following p.d.f: $$ \begin{array}{c|lc} X & \text{0} & \text{1} & \text{2} & \text{3} \\ \hline p(x) & 2\theta/2 & \theta/3 & 2(1-\theta)/3 & (1-\theta)/3 \end{array} $$ where $0\leq\theta\leq1$ was taken from…
BigData
  • 109
1
vote
1 answer

Maximum Likelihood Estimator Independent Exponentials

Here is the problem statement: Let $(X_i,Y_i)$ be a random sample from a distribution with pdf $$f(x,y;\theta)= \frac{1}{\theta^3}\exp\left(\frac{-x}{\theta}-\frac{y}{\theta^2}\right)\qquad 0
Hamilton
  • 11
0
votes
0 answers

Maximum Likelihood Estimator vs. Method of Moments Estimator

The question says: Let $X_1, X_2, ..., X_n$ be a random sample (I.I.D.) from the PDF: $$ f(x,\theta) = \frac{1}{\theta (1-x)^2} e^{\frac{-x}{\theta (1-x)}} $$ for $00$. Find the maximum likelihood estimator of $\theta$. Is this the…
Ram Zi
  • 67
0
votes
0 answers

MLE Estimation using Indicator functions

I was working on the task where it was asked to found the pdf of the sum of 2 rv having uniform distribution, using transformation theorem (representing bandwidths of routers, and the bandwidth of the system consisting of routers): $f_{S}(s) =…
Razmik Manukyan
  • 1
0
votes
2 answers

Maximum Likelihood estimator for negative binomial

Let $X$ be negatively binomially distributed to parameters $n \in \mathbb{N}$ and $p \in] 0,1[$, i.e. $$ \mathbb{P}(X=k)=\binom{n+k-1}{k} p^n(1-p)^k, \quad k \in \mathbb{N}_0. $$ Let $n \in \mathbb{N}$ be known and $p \in\, ]0,1[$ be unknown. Show…
calculatormathematical
  • 2,222
0
votes
1 answer

Finding the MLE estimator of a constant a for variable x + a that follows a log-normal

I will really apreaciate your help with this question. I have a variable y + a that follows a log-normal distribution. I want to find proof of whether the MLE estimator for a can or can't be found. However, I am having problems to do it. As far, I…
Jorge
  • 1
0
votes
1 answer

Maximum likelihood estimator - Partial derivative

I've trying to get the maximum likelihood estimator of $\theta_{MLE}$ but after doing the final derivate step. I've got -1. Am I doing the partial derivative wrongly? What is the MLE of $\theta$? $$ \begin{equation} f\left(x|\theta\right)= …
Hal
  • 105
  • 3
0
votes
2 answers

MLE for uniform distribution

Suppose we have a sample of $X_i = x_i$ for $i \in [1,n]$ where $\forall X_i$ they are iid with uniform distribution on the interval $[a,b]$ Now we want to find the MLE for the uniform distrubtion. We form the likelihood function: We get that $…
Tanamas
  • 1,837
0
votes
1 answer

Find the maximum value of given expression $S = \Sigma f_r.\log(P_r)$

With $\Sigma f_r=1$ and $\Sigma P_r=1$, $\forall f_r, P_r>0$. How to find the maximum value of $S$ where: $$ S = \Sigma f_r.\log(P_r) $$ Thanks for reading!
Monad
  • 101
0
votes
0 answers

Confusion about maximum likelihood estimation of exponential distribution

The probability density function of the exponential distribution is defined as $$ f(x;\lambda)=\begin{cases} \lambda e^{-\lambda x} &\text{if } x \geq 0 \\ 0 & \text{if } x<0 \end{cases} $$ We know that mle of the parameter $\lambda$ is $$\lambda =…
eet
  • 393
1
2
3 4