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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.

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Consistency of maximum likelihood estimator for negative binomial distribution

Let $X = (X_1,\ldots, X_n)$, where $X_1,\ldots,X_n$ are independent and have the same distribution: $$P_{\theta}(X_i = k) = \binom{k + r - 1}{k} \cdot \theta^r \cdot (1 - \theta)^k~,$$ with $k \in \...
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Maximum likelihood estimation for 2 gaussian curve

math in general is not exactly my strong suit, i recently found myself staring at this equation. $y = p(\frac{1}{\sigma_1\sqrt{2\pi}})exp (-\frac{(x-\mu_1)^2}{2\sigma_1^2}) +(1-p) (\frac{1}{\sigma_2\...
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Optimising unrounding using maximum likelihood

I have a bunch of rounded random independent numbers. I want to replace them with unrounded numbers such that the unrounded numbers are 'most likely' to have been generated by some (continuous) ...
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How to show that the multivariate normal depends on the data only through $\Sigma x_n$ and $\Sigma x_n x^T$

I've shown something similar for the 1-dimensional case. That $\Sigma x_n$ is the sufficient statistic of the gaussian mean and $\Sigma x_n$,$\Sigma x_n^2$ are the sufficient statistics of the ...
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Calculating Maximum Likelihood

Suppose that $Y_1,\dots,Y_n$ is a random sample from the density function given by $$f(y|\theta)=\begin{cases}\frac1\theta, &y\in(0,\theta), \\ 0, &\mathrm{otherwise.}\end{cases}$$ ...
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How to show the maximum likelihood of $\theta$?

Let $x$ have a uniform density $f_x(x\mid\theta) \sim U(0,\theta)=\left\{ \begin{array}{ll} \frac{1}{\theta} & 0 \leq x \leq \theta \\ 0 & \text{otherwise} \end{array} ...
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Relation between MLE and Cramer

Suppose that X1,X2,... are i.i.d Bernoulli random variables with unknown success probability θ∈[0,1]. How could you show that the MLE of θ attains the Cramér-Rao lower bound and is therefore the best ...
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Relation between Maximum Likelihood Estimator (MLE) and Cramér-Rao Lower Bound [on hold]

Suppose that $X_1,X_2, ...$ are i.i.d Bernoulli random variables with unknown success probability $\theta∈[0,1]$. Show that the MLE of $\theta$ attains the Cramér-Rao lower bound and is therefore the ...
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Bayesian Posterior Mode VS MLE [on hold]

Suppose that $X_1, X_2, \dots $ are i.i.d Bernoulli random variables with success probability equal to an unknown probability $θ∈[0,1]$. Considering the example of flipping a coin with $n=1$, provide ...
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Does MLE really care about PDF?

I am wondering, whether MLE really cares whether it operates on proper distributions. Lets take a look at the following situation: likelihood: $$L(\theta \mid x) = \prod_{n}^{N}{f(x_n \mid \theta)}$$ ...
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MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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Likelihood interval for binomial counts

I have an assignment question regarding a "likelihood interval" that I don't really understand. The question asks to consider counts of $X_i$, with $i\in \{1,...,N\}$, modelled as independent ...
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How do I show that the MLE for all sides of a die is the same as MLE for 5/6 sides of a die

I have two sets of hypotheses: $H_0$: each of the six sides of a die has probability 1/6 of being rolled, versus $H_a$ being not $H_0$ $H_0$: 5 sides of a six-sided die each has probability 1/6 of ...
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Phase marginal for multivariate complex Gaussian density

The following is a cross-post from stats.stackexchange, which I am including here since it is mostly about a hard integral. Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and ...
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Computing Maximum likelihood and least squares estimators

I am having trouble with the following question, particularly the first part. Doesn't least squares require that errors be the same across the RV drawn from a distribution. However the variance ...
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Exponential distribution MLE with lifetime and frequency table

\begin{array}{c|c} \hline \text{Lifetime (months)} & \text{Observed frequency} \\ \hline 0-2 & 50 \\ \hline 2-4 & 35 \\ \hline 4-6 & 25 \\ \hline 6-8 & 15 \\ \hline 8-10 & 5 ...
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Strong convexity of Loss function in Logistic Regression

I wonder if the Loss function of a Logistic regression can have strong convexity when the explanatory variables are linearly independent. From a theoretical point of view, if I have a sample of p ...
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Proof explanation: Sufficiency and maximum likelihood

I was reading a book on mathematical statistics and came across the following statement. Suppose that the statistic $T(X)$ is sufficient for the parameter $\theta$. If $\hat\theta(X)$ is a maximum ...
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Find the $P(\theta - 0.1 \leq MLE \leq \theta + 0.1)$

I have a continuous random variable with density: $f(x|\theta) = \frac{\theta}{x^{\theta+1}}$. I have calculated the MLE($\hat\theta$) as: $\frac{n}{\sum_{i}^{n}log(x_i)}$ I am stuck on figuring out ...
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Defining a custom probability density function for Maximum likelihood in matlab

I am trying to minimise a likelihood function and estimate the parameter value of $\lambda$ by fitting to the following data. $t$ is the time and $N(t)$ si the population measured at those specific ...
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How is L2 regularization derived?

I just proved to myself why the regularization is added rather than multiplied to loss function. I did so by taking the MLE formula... $$argmax\sum log(P(x_{i}|\Theta ))$$ and since we know that ...
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How to find $E\left[\frac{1}{\bar{X}}\right]$

Given $X_1, X_2, ..., X_n \sim X$ where $$f_X(x;\theta) = \theta^2 xe^{-\theta x}, \quad x>0,\quad\theta >0$$ I was able to find the expectation $E[X]=\frac{2}{\theta }$ and that the MLE is $\...
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Simplify maximum a posteriori

I have the following example of a maximum a posteriori... $$\prod_{n=1}^N \mathcal{N}(y_n|\beta x_n,\sigma^2) \mathcal{N}(\beta|0,\lambda^{-1})$$ Where I am multiplying the prior by the likelihood ...
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likelihood of Gamma distribution confusion

Gamma is given by $$f_X{(x)} = \frac{\lambda^a x^{a-1}e^{-\lambda x}}{\Gamma(a)}$$ I can remove the constant being $\frac{x^{a-1}}{\Gamma(a)}$ $$L(\theta | x_{1}, x_2, \cdots, x_n = \bar{x}) = \...
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How to derive the MLE of Bernoulli?

Likelihood is given by $$L(\theta | x_1, x_2, \dots, x_n = \bar{x}) = (\theta)^{n \bar{x}} (1-\theta)^{n(1-\bar{x})}$$ log likelihood is $ln (L) = n \bar{x} \ln(\theta) + n(1-\bar{x})\ln(1-\theta)$...
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finding the maximum likelihood and probability distribution function to fit to data

There are data on a pathogen population $N(t)$ measured over time (t). I am using the ODE of, ${dN\over dt}=-\lambda N$ to model its decline over time. So, $ N(t)=N_0 e^{- \lambda t}$. However, ...
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Understanding Maximum Likelihood Estimation

I'm having a hard time understanding the use of MLE. I understand what it does, it is intended to give the best parameters for a particular model to represent a population. My question is, is MLE ...
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Compute the bias of $\frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}}$

I found the maximum likelihood estimator of $x$. Now, how to compute the bias of: $$ \hat{x}_{ML} = \frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}} $$ Where $\hat{x}_{ML}$ ...
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analytical distribution of the maximum likelihood estimator for a uniform distribution

Obviously the MLE of $\theta$ for a distribution $X_1, X_2, \dots, X_n \sim Uniform(0,\theta)$ is $\hat{\theta} = max(X_1, X_2,\dots,X_n)$ Now, assume $\theta = 1$. If you take repeated samples with ...
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maximum likelihood of a functional sequence

For example, we know maximum an example log-likelihood is $$\frac{\partial \log p(x|A_1,\cdots,A_k,\cdots,A_T)}{\partial A_k}=x_k$$ Also, $$A_k=E\sin(k)$$ How to calculate $$\frac{\partial \log p(x|...
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Demonstrate that $m_n = max(x_1,…,x_n)$ is a sufficient statistic .

Let $X1, . . . , Xn$ be independent and identically distributed with density $P_θ(x) = $ \begin{cases} 2x/θ^2 & \text{for }0\le x<θ\\ 0 & \text{else} \end{cases} Demonstrate that $...
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For a Maximum Likelihood Estimation with events that implicate each other, how should the likelihood function be constructed?

The probability of a student with a skill parameter of "s" to obtain at least a score of "k" in a certain test is defined as: $$\frac{1}{e^{b_k-s}+1}$$ Where $b_k$ is a difficulty parameter of ...
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How to define a maximum likelihood objective for a variable assignment?

Suppose that we have a set of candidate points $c_1<...<c_k$ and a set of variables $x_1<...<x_n$ where $k \neq n$. Furthermore, we have the following distributions: $p(x_1), p(x_2), ..., ...
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Given an $X_1,…,X_n \sim Unif[0,1]$ i.i.d. sample, after ordering them, one of $X_k^*=x$, give a maximum likelihood estimation for $k$ using $x$.

The problem: The title sums up the problem pretty well, we have a sample from an independent idential (uniform) distribution (i.i.d.): $X_1, ..., X_n \thicksim Unif[0,1]$, and we only know one ...
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How to understand that the solution to least squares problem transformed with Box-Cox Transformation, is a generalized mean with $h(x)=x^\lambda$?

The least squares problem $\min_a \sum_i^n (x_i-a)^2$ is sometimes solved using transformed variables, that is, solving $\min_a \sum_i^n [h(x_i)-h(a)]^2$. The solution to this latter problem is $a=h^{-...
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How to find the MLE of these parameters given distribution?

Let $X$ and $Y$ be independent exponential random variables, with $$f(x\mid\lambda)=\frac{1}{\lambda}\exp{\left(-\frac{x}{\lambda}\right)},\,x>0\,, \qquad f(y\mid\mu)=\frac{1}{\mu}\exp{\left(-\...
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Conditional probability with MLE of Poisson variable

I'm having some trouble with this study question and would appreciate any help. This may be a duplicate but I have not been able to find any others. Question: Leaves of plants are examined for bugs. ...
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What is a good motivation or analogy for the maximum likelihood estimation problem?

I am trying to think of a good motivation for maximum likelihood estimation. Given a set of random variables $X_1, \ldots, X_n \sim f_X(x_1, \ldots, x_n |\theta)$, the maxmum likelhood estimation ...
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Fusing Sensor Data from Identical Source

I'm trying to find info on methods for fusing multiple readings of the same binary data, given some performance metric such as 'rate of success'. E.g. if N motion detecting sensors are monitoring the ...
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likelihood of poisson distribution

If $(x_1,\cdots, x_n)$ is a sample from a Poisson(θ) distribution, where $θ ∈ (0,∞)$ is unknown, then determine the $MLE$ of $θ$. my attempt: so the probability density of poisson is $$p_{\theta}(x)...
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Intuition behind MLE - Why is the MLE of inverse theta function the maximum X and not the minimum.

So I'm trying to understand why the MLE of $\theta$ for $$f_X(x) =\frac{1}{\theta}, \ \ 0\leq x\leq\theta$$ intuitively. For reference, I am using Example 5 from this paper that I found online. The ...
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Likelihood-function of a standardized beta-distribution

Suppose $X_1,..X_n$ are i.i.d. RV and we also have a Distribution-function: $F(x|p_o,p_1,\gamma) = p_oI(x\geq0) + p_1I(x\geq1) + (1-p_o-p_1)F_o(x|\gamma)$ Where $F_0$ is defined on the intervall $(0,...
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What is the maximum likelihood estimator for $e^{-\theta}( (P(X_i = 0))$?

Suppose $X_i$ is $iid$ $Poisson(\theta)$ What is the maximum likelihood estimator for $e^{-\theta}(= P(Xi = 0))$? I already found the MLE for the $\theta$. how do you then find the MLE of $e^{-\...
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Log-likelihood problem of frequency statistics

I encounter the following problem during solving the following problem: the population $n=237$ diseased people are required to perform a 6-successive-day diagnostic test on cancer. And the variables ...
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Finding Fisher information

Let $X$ distribution belongs for the family $\mathcal{P}\{P_{\theta}, \theta \in \Theta \}$. We need to find Fisher information $I(\theta)$ according $n$ simple sample, when $P_{\theta}$ is $N(\mu,\...
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How do we find the asymptotic variance for the maximum likelihood estimator from the Rao-Cramer lower bound?

How do we find the asymptotic variance for the maximum likelihood estimator from the Rao-Cramer lower bound? As a concrete example, I have found that the Rao-Cramer lower bound for $$f(x;\theta)=\...
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MLE : Effect of incorrect variance on the mean for a normal distribution

Consider we have univariate samples, $x_k$, belonging to a category $\omega$ drawn from a dataset D according to an assumed distribution $p(x|\omega)$ $\sim N(\mu, 1)$. However, let the true ...
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A 'Kinda' Normal Distribution Over the Unit Interval? (the ladybug won't die)

Update: I added the stochastic-processes tag. For my non-theoretical computer model/application, I'm trying to approximate a Bernoulli random variable. So we are 'watching the ladybug' to get a real-...
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Issues Computing an Integral for Statistics Problem

I have a statistics midterm coming up, and while studying practice midterms, I came across an integral I was unable to solve. I tried using the basic integral rules, as well as integration by parts, ...
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Confusion with maximizing likelihood of binomial parameter $p$

Consider the following data from a sample of a binomial distribution $X$ with $n=2$ and unknown parameter $p$: $$P(X=0)=.175,\ \ P(X=1)=.45,\ \ \text{and} \ \ P(X=2)=.375.$$ My goal is to find the ...