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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13 views

Proof of the unbiasedness of an estimate obtained by gradient search

As the solution to an MLE problem cannot be obtained analytically, I apply gradient search to find the solution $\hat{\theta}$ to satisfies \begin{align} \frac{\partial \mathcal{L}(\mathbf{y}\mid\...
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1answer
22 views

Question about MLE estimator for simple exercise

As I am working through my old university notes, due to excess of time lately, I've stumbled upon the following exercise: Let's consider a medical treatment in which the same drug is used twice. We ...
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the confidence intervals of an exponential distribution [closed]

Consider the random sample $X_1 \cdots X_n$ from a distribution with pdf $$f(x; \theta) = \dfrac {x}{2\theta^2}$$ if $0<x<2 \theta$. The most likelihood estimator of theta is given by: $\theta_{...
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20 views

Confidence Interval based on limiting distribution of the Maximum Likelihood Estimator Sigma and Sample Variance

Another statistics question I need some help with. Now, I am not too familiar with limiting distributions (had my course on it over a year ago). I have tried some things myself and ran some ...
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1answer
19 views

Two independent random variables X and Y are given. How can we compute Expectation minimum {X,Y} and Expectation maximum {X,Y}? [closed]

Two independent random variables X and Y are given. X is uniformly distributed in the interval [-2,5], Y is uniformly distributed in the interval [-4,2]. How can we compute Expectation minimum {X,Y} ...
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1answer
28 views

MLE for an undirected network degree distribution

I have an empirical undirected network. I assume, that a degree distribution is $ F(k) = 1 - e^{1 - \frac{k}{m}} $. and would like to estimate $m$. The only method I'm aware of for such task is MLE. ...
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1answer
35 views

Maximum likelihood estimator for 2 different samples [closed]

How to derive maximum likelihood estimator for /omega given by ml omega given 2 random samples (X1,Y1),......,(Xn,Yn). Where X is pulled from the gamma distribution and the Y comes from the standard ...
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26 views

A baksetball probability question using Neyman–Pearson lemma

It is known that the probability of a basketball player to make his first shot is $p=0.6$ A player argues that it does not matter if he made the previous shot or not his odds stays the same. We say if ...
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17 views

Realized variation and ML estimate

Assume that we have the following process: $$X_{t}=\int_0^t \mu \, ds+\int_0^t\sigma_s \, dW_s $$ And assume that we observe $$Z_i^n=\sqrt{n}\left(X_{\frac{i}{n}}-X_{\frac{i-1}{n}}\right) \sim N \...
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1answer
53 views

Mean square error of MLE

$\textbf{X}=(X_1,...,X_n)$ is a random sample from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \...
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2answers
26 views

Parameters estimation of a normal distribution

Given a statistical sample $X_1,\dots ,X_n$ from a population with density $\mathcal N \left(\mu, \sigma^2\right)$, supposing that we know the value of $\sigma^2$, I must calculate Cramer-Rao bound, ...
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1answer
39 views

Likelihood Function of Normal Variables

We know that for $i.i.d.$ random variable $Y \sim N(\mu, \sigma^2)$, $Var(Y) = \frac{1}{n}\sum_{i = 1}^n (y_i - \mu)^2$, and the likelihood function for $Y_1, Y_2, ..., Y_n$ is $$l(\mu, \sigma|Y) \...
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7 views

About a consistency of maximum likelihood estimation for conditioning

Consider independent random variables $N_i$ that obey $P_i(n_i|\theta_i)$ ($i=1,2,\dots,K$). Then the maximum likelihood estimator $\hat{\theta}$ is given by $\partial_{\theta_i}\log P(n|\theta)=\...
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1answer
29 views

Estimate size of population using hyper geometric distribution and maximum likelihood estimator

I saw the following in my notes, but I can’t quite remember why the argument holds... could someone help me please? Suppose there are N number of fishes in a lake and we want to estimate N. We catch ...
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1answer
42 views

Maximum likelihood estimator - Uniform distribution [closed]

Let X have an uniform distribution on the segment [a,b]. With the method of maximum likelihood, estimate the parameter d = b-a. Then check if the estimator is consistent. Hi, i tried solving this ...
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1answer
35 views

MLE with parameters constrained

I am wondering how I can solve this problem more elegantly: Let's say I have two set of i.i.d normal random variables with known variance $\sigma^2=1$: $$ X_1, X_2, ..., X_m \sim N(\mu_1, 1) $$ $$ ...
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1answer
26 views

Sample variance formula vs. Population variance formula usage

Just for sake of not having to write this equation out a few times, I will denote the residual sum of squares for a set of data points $X$ to be $$ RSS= \sum_{x\in X} \left(x - \overline{x}\right)^2 $$...
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11 views

Maximum Likelihood for Uniformally Distributed Errors (MA process)

I couldn't find similar questions to mine so here it goes. Consider the MA(1) model given by $$ y_t = e_t +be_{t-1} $$ with $e_{t}$ distributed uniformally over the unit interval [0,1]. Simulate ...
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31 views

How to prove that $H_0$ can be rejected using generalised likelihood ratio test?

Given distribution function: $f(x) = \theta x^{\theta - 1}, \text{ if } 0<x<1 \text{ and } 0 \text{ otherwise}$. 1) Test hypothesis $H_0 = \theta = \theta_0 $ with generalised likelihood ratio ...
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How would I show this turning point indeed has the second derivative $<0$. Showing the turning point is indeed the Maximum Likelihood Estimator

This is a question from Statistics but it really boils down to some algebra that I cannot seem to get my head around. Consider $(X_i)_{i=1,2,...,n}$ iid such that each is $N(u, u^2)$. Then the log-...
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39 views

What are sufficient conditions such that consistency of ML estimate implies consistency of MAP estimate?

I am interested in under what conditions the frequentist consistency of a Maximum-Likelihood estimator is enough to give the consistency of a maximum-a-posteriori point estimate, with the further ...
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Establishing connection between ERM (Empirical Risk Management) and MLE

My question is specifically about section 2.3.5 "Connection to maximum likelihood estimation" from Nielsen (2016) where the connection of Empirical Risk Management (ERM) to Maximum likelihood ...
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35 views

Maximum Likelihood Estimator and MVUE for theta-squared [duplicate]

Let $Y_1,\ldots,Y_n$ denote a random sample of size $n\ge4$ from Bernoulli ($\theta$) distribution where $$0 < \theta < 1$$ is the success probability. How would I derive the maximum likelihood ...
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39 views

Find the Maximum Likelihood Estimate for the Population [duplicate]

I have been presented with this problem and I am not sure if I am achieving the right conclusions, hopefully you can help me! Considering that I am watching a game with an unknown number of players ($...
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1answer
29 views

MLE for normal distribution with mean and variance unknown, consistency and histograms

Consider $X_1, X_2, \ldots, X_n$ i.i.d $N(\mu, \sigma^2),$ where both parameters are unknown, and consider estimation of $\sigma^2.$ Consider the MLE for $\sigma^2.$ We know it is, $$\hat{\sigma^2_n} ...
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1answer
66 views

How to find MLE from a cumulative distribution function?

I'm new to probability. Given the cumulative distribution function $f_Y(y)=\theta e^{-y\theta}$ defined from 0 to infinity, I would like to find the parameter $\theta$ such that it maximizes the ...
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47 views

Bias and variance of maximum likelihood estimator

An exponential distribution has a probability density function: $$p(y|v) = exp(−e^{−v}y − v)$$ It has one parameter: a log-scale parameter $v$. If a random variable follows a gamma distribution with ...
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28 views

Find the asymptotic distribution of the MLE of $f(x\mid x_{0},\theta)= \theta \ x_{0}^{\theta}\ x^{-\theta-1}$

To find MLE of $f(x\mid x_{0},\theta)= \theta \ x_{0}^{\theta}\ x^{-\theta-1}$ $x>=x_{0}, \theta>1 $ $l(\theta)=n\ln \theta +n\ln x_{0} + (-\theta-1)\sum_{i=1}^{b}\ln(-x_{i})$ $\hat{\theta}=\...
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1answer
12 views

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 ...
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1answer
62 views

Maximum likelihood estimation of parameter $N$

Every competitor in a marathon has a unique number on their shirt, from 1 to N. N is unknown. The observation is $n_1, \ldots ,n_K$, which are randomly sampled from the $N$ competitors with equal ...
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1answer
37 views

Finding maximum likelihood estimator from pdf $(\theta +1)x^\theta$ for $0<x<2$

Given that $$F(x)= (\theta +1)x^\theta\qquad \text{for} \qquad 0<x<2$$ find maximum likelihood estimator. My progress: $L(\theta) = \prod_{i=1}^{n}(\theta +1)(x_i)^\theta$ $L(\theta) = (\...
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1answer
38 views

What is the Probability density function of MLE

Given a random sample of size N from a population with probability density function $f(x)$ that depends on a paramter $\hat{A}$. Its MLE is the minimun of the random sample. Now, I have been asked to ...
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8 views

How the variance impact the ML estimation

I'm estimating with an ML method the position of an agent I know the position (x,y) of 4 anchors and the distribution function of each anchor in regarding the ranging to the agent ( $N\sim(d_i,\sigma^...
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1answer
34 views

Maximum likelihood estimators of the parameter of an aggregate loss (Poisson frequency, exponential loss)

Question: $$N\sim Poisson(\theta), X\sim exp(\theta)$$ $$S = X_1 + X_2 + ... + X_N$$ With $4$ observed aggregate loss $s_1, s_2, s_3, s_4$. What's the maximum likelihood estimator of $\theta$? My ...
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How to compute the stably adjusted profile likelihood given in Barndorff-Nielsen and Cox (1994)?

I'm interested in computing a modified profile likelihood and came across the stably adjusted profile likelihood proposed by Barndorff-Nielsen and Cox in their book 'Inference and Asymptotics' which ...
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42 views

Most Powerful Test For Two Binomial Distributions

Let $X \sim Binomial(n_x,p_x)$ and $Y \sim Binomial(n_y,p_y)$ independent with known sample sizes. Then let $p_x \leq p_y$, Compute MLE for $p_x$ and $p_y$ Find the most powerful level 0.05 test of $...
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Distribution of sample mean squared

I have an iid sample of n Poisson($\theta)$ RVs. I have derived that the MLE for $\phi = \theta^2$ is $\bar{x}^2$. I need to show this estimator is consistent. To show consistency I have to show $Var(...
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Maximum Likelihood and Method of Method of Moment

Maximum Likelihood Hi all, I am stuck in the question (posted above). I am attaching the image of what i tried solving. I am not sure if the logic I followed is correct. Especially with the method ...
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0answers
18 views

Estimate the position of an object (x,y coordinates) via MLE

I'm trying to estimate the coordinate in a 2D space of an object. I know that there are 4 anchors which distance from my object can be modeled as a normal distribution $N\sim(d_i,\sigma)$ where I ...
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1answer
23 views

Maximum Likelihood Estimation: difference between probability and likelihood

In Maximum Likelihood Estimation (MLE), we want to maximize the probability $P(x_1, x_2, x_3,.. x_n | \theta)$, where $x_1, x_2, x_3,.. x_n$ are the datapoints and $\theta$ is the parameter vector. ...
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1answer
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Why has the MLE for a Gaussian Distribution only one solution although not being “jointly” convex in mean and variance

I am currently looking into the Maximum Likelihood Estimate (MLE) for the mean $\mu$ and $\sigma^2$ of a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$ for a given set of samples $$\left\{x_i \ \...
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16 views

Variance of the MLE estimator of a function of bernoulli parameter

I have a bernoulli RV $X$ defined as below: $pr(X=1)=1-(1-p)^{k}$ $pr(X=0)=(1-p)^{k}$ Assume I have $N$ independent observations on the RV $X$. I am interested in finding the MLE estimator of $k$ ...
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3answers
84 views

Exponential Likelihood Function

Suppose $X_1, ..., X_n \stackrel{iid}{\sim}$ Exponential(rate = $\lambda$) independent of $Y_1, ..., Y_n \stackrel{iid}{\sim}$ Exponential$(1)$. Define $Z_i \equiv \min\{X_i, Y_i\}$ I want to find ...
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0answers
6 views

Minor details in verifying MLEs of bivariate normal

I am working through Chapter 9, Section 1, Exercises 1.12-1.14 on pages 294-300 of the book An Introduction to Probability and Statistical Inference, Second Edition, written by George Roussas. The ...
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1answer
92 views

Completeness, UMVUE, MLE in uniform $(-\theta,2\theta)$ distribution

Let $\theta >0$ be a parameter and let $X_1,X_2,\ldots,X_n$ be a random sample with pdf $f(x\mid\theta)=\frac{1}{3\theta}$ if $-\theta \leq x\leq 2\theta$ and $0$ otherwise. a) Find the MLE of $\...
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Asymptotic Distribution of MLE With Infinite Number of Solutions

I have a problem with infinite number of solutions to maximal likelihood estimation. An example could be found from here Example 8.3. I will rewrite this example in the following, Suppose that $Y \...
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1answer
15 views

Covariance selection does not converge

I am new to covariance selection, and trying to fit a set of high-dimensional data with undirected Gaussian graphical model. The graph structure of multivariate normal distribution has been given. I ...
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0answers
24 views

Finding the MLE of $\theta$

Working through this given problem on maximum likelihood estimation (MLE). The density is given as $$f(x \mid \theta) = \frac{1}{2\theta} e^{-|x|/\theta}, -\infty < x <\infty $$ I get $$L(\...
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1answer
42 views

What is the maximum likelihood estimate of $\theta$?

A random sample of size $7$ is drawn from a distribution with p.d.f $$f_{\theta}(x)=\frac{1+x^2}{3\theta(1+\theta^2)}, -2\theta \le x \le \theta,\;x>0 \;\text{and otherwise zero}$$ and the ...
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65 views

How to find the MVUE for the Var$(x_i)$ of a Geometric Distribution?

Suppose that $X_1, \ldots ,X_n$ form a random sample from a geometric distribution with parameter $0<\theta<1$ whose PMF is $p(x) = θ(1−θ)^x$,for $x \in \{0,1,2,\ldots\}$. It is known that this ...

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