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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 $\widehat{\mu}$.

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Maximum-likelihood-estimator of $f_a: \mathbb{R} \to [0, \infty), f_a(x) = (a + 1) \cdot x^a \cdot 1_{[0, 1]}(x)$ with $a > 0$

Given the continuous density function $f_a: \mathbb{R} \to [0, \infty), f_a(x) = (a + 1) \cdot x^a \cdot 1_{[0, 1]}(x)$ with $a > 0$, what is the maximum-likelihood-estimator for a? I am not sure about my current solution and would be grateful if…
curiouscupcake
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Maximum Likelihood Estimation on constant probability

The given graph represents users' friend relationship in SNS. By assuming the link probability is a constant, then how can I perform maximum likelihood estimation on the link probability? In different case, if I cluster '2' and '4' to group A, and…
dkssud
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Maximum-likelihood estimation of parameter of a distribution function

My Problen: A Random variable $X$ has the following distribution function: $$ F_X(x)=% \begin{cases} x^\theta, &x\in[0,1]\\ 0, &\text{otherwise}. \end{cases} $$ We have gotten x1 = 0.40 x2 = 0.75 x3 = 0.95 from three independent trials. Determine…
August Jelemson
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Likelihood function ARCH models

Suppose that the values of {$\epsilon_t$} are drawn from a normal distribution having a mean of zero and a constant variance $\sigma^2$, then the likelihood function is: $$L_t=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-\epsilon^2_t}{2\sigma^2}}$$ My…
user742138
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Without using calculus determine the MLE of $(\theta_1,\theta_2)$

If $(x_1,\cdots,x_n)$ is a sample from a $\mathcal{U}(\theta_1,\theta_2)$ (uniform) distribution with $$\Omega=\{(\theta_1,\theta_2)\in\mathbb{R}^2:\theta_1<\theta_2\}$$ determine the MLE of $(\theta_1,\theta_2)$ without using calculus. OK, I know…
falamiw
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MLE: given density $p_{\theta}(x) = \frac{1}{\theta}, \text{where } 0 \leq x \leq \theta.$

Given: $\theta>0$, $X_1,...,X_n$ i.i.d. RV with $p_{\theta}(x) = \frac{1}{\theta}, \text{where } 0 \leq x \leq \theta$ and $p_{\theta}(x)=0, \text{else.}$ In the given exercise the MLE solution $L_X(\sigma)$ is…
Onerock
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Injective function of an MLE is an MLE

I'm told that if $\theta_{ML}$ is the MLE of parameter $\theta\in \mathbb{R}^n$ and $g:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is injective then $g(\theta_{ML})$ is the MLE of $g(\theta)$. This doesn't exactly make sense to me though. If we required…
Addem
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Maximum Likelihood Estimation - Symbol Change

I'm trying to understand why when simplifying the exponential likelihood function, the symbol changed from a product of sequences to summation and why isn't the theta in the exponential to the power of n?. $$ \ p(y | \theta) = \prod_{i=1}^{n}…
steve
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Why can't we use calculus in finding the M.L.E. of Uniform(-theta, theta)?

I've understood that we use maximum/minimum of x's as MLE of theta. But no one so far has explained the reason why differentiation won't work. Please explain.
Aboli Barhate
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Maximum Likelihood Estimation of P(x|theta) = 1/(1-theta)

I want to calculate the maximum likelihood estimation of $P(x|\theta) = 1/(1-\theta)$ for $\theta<= x <=1$. I end up with $log1 - nlog(1-\theta)$ and when I want to take the derivitie I end up with $-n*-1/(1-\theta)$ how should I proceed? Because I…
Mona Jalal
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Find maximum-likehood estimator

Say we have $n$ identically distributed random variables with marginal density function given by $$ p_\theta(x)=\dfrac{2\theta}{9^\theta}x^{2\theta-1} $$ for $x\in[0,3]$. It is given that $\theta>0$ (the unknown parameter). I want to find the…
Sha Vuklia
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Likelihood function binomial

I have trouble finding the likelihood function in an applied problem involving the binomial distribution. Given are $N$ independent random variables having identical binomial distributions with the parameters $\theta$ and $n = 3$ where $n_0$ of them…
Anna D.
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Maximum Likelihood Function

Suppose the random variables $X_1, X_2, . . .X_n$ are independent each with the distribution $$f(x;\theta) = \frac{\theta \ x^{\theta-1}}{3^\theta} \ \ \text{ for} \ \ 0 \leq x \leq 3.$$ Find the Maximum Likelihood estimate for $\theta$.
user3026388
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Maximum likelihood estimator(1)

I've been looking at some statistical distribution work and found a question that I can't solve through standard derivation of an MLE. I've been told to maybe look at an indicator function but I've no idea how to use those for these. The question is…
kvj121227
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How to find the MLE and asymptotic variance for a piecewise function?

The following model is proposed for the distribution of family size in a large population: P(k children in family;$\theta$) = $\theta^{k}$, for $k = 1, 2, ...$ P(0 children in family;$\theta$) = $\frac{1-2\theta}{1-\theta}$. I tried to multiply…
Lawrence
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