a method of estimating parameters of a statistical model by choosing the parameter value that optimizes the probability of observing the given sample.

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Help in finding the joint distribution

There is a metric $H$ defined as = $\sum_{i=1, j \neq i}^{N} \min |u_i - u_j| * ..*|u_{i+d-1} - u_{j+d-1}|$ where $u$ is a multi dimensional vector of dimension $d$ and $u_i,u_j$ $\in \mathcal{R}^d$ ...
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21 views

Maximum Liklehood estimator of Poisson

I am trying to solve the following problem, I was able to solve first three bits of the problem but after that i am stuck and do not have clue to solve the (iv) bit. The local government has ...
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12 views

asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
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29 views

What is relationship between Fisher Information and Variance in natural exponential Family?

I know that $Var(\hat\theta)\geq 1/I(\theta)$ where $I(\theta)$ is Fisher information. Let take an example of natural exponential family with density $f(x)=\lambda\exp(-\lambda x)$. In this case we ...
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10 views

Maximizing Log-Likelihood Estimation for Changepoint Detection

I'm trying to code the changepoint detection algo described here: ...
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1answer
33 views
+50

Calibrating Generalized Hyperbolic distribution in R - which parameters are valid and allow for a numerical calculation of absolute moments

I am using the R-package ghyp in order to calibrate and model. In fact my coding is based on this paper. I know that I could do quite a robust fit using ...
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69 views
+50

Help in maximum likelihood estimation derivation

The problem is estimating a metric $Vol(D)$ for the following situation : Given noisy observations the observed data are a random sample $Y_1,\ldots,Y_n$ where $Y_i \in \mathcal{R}^d$. The model for ...
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2answers
138 views

Gaussian noise model derivation

I have the following linear regression model, $y = f(x;w) + n$, where $y$ is the vector of true labels, $x$ is the observed data, $f(x;w) = w^Tx$, and $n$ ~ $N(0, \sigma^2)$ is the noise. Why then ...
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39 views

Estimating correlation hyperparameters of a Gaussian Process

I have an actual function that I need to simulate using a GP model. I've not done this before so I'm unclear of the steps. I have used the true function at different values of the inputs ($\vec X1, ...
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25 views

Fitting Multivariate Bernoulli distribution

I want to fit a model to a number of observations, each of them being a k-dimensional binary vector $(x_1, x_2, ..., x_k)$ where $x_i \in \{0,1\}$. Naturally I would like to fit a multivariate ...
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17 views

What is a convex support? (Bickel&Doksum, Mathematical Statistics, Basic ideas…Vol1)

Bickel&Doksum, Mathematical Statistics, Basic ideas...Vol1 page 122, just above Cor2.3.1, it says: Define the convex support of a probability P to be the smallest convex set C such that ...
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1answer
64 views

Power-law fitting and testing

I want to test the distribution that best fit a specific metric (that I call SD) extracted from the source code of systems. I have a guess that they follow a power-law behavior. My sample: 20 ...
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2answers
58 views

Maximum Likelihood estimator from sample distribution $N(0,\sigma^2x_i^2)$

Let independent random variable $Y_1,...,Y_n$ have respective distributions $N(0,\sigma^2x_i^2)$, where $i=1,2,...,n$ are known constants such that $x_i\neq 0$ for all $i=1,2,...,n$. Find the maximum ...
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2answers
61 views

Maximum Likelihood estimator of population variance and its derivation process

I have 2 questions about maximum likelihood and using it to calculate variance: Question #1: The question is about finding the derivative of the score function with respect to the parameter $ ...
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63 views

How to derive OLS through MLE? [duplicate]

I am just curious on finding about this derivation
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14 views

Determine if missing values are informative(due to treatment) or just noise.

I'm not sure if this is those appropriate way to phrase this question. I have two populations of measurements. In my control population I have a quantitative measure of a proteins abundance. I expect ...
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1answer
48 views

Constrained MLE of multivariate normal

this might be obvious one but I have spent much time without gaining anything. If $\underline{X}$~$N_p(\underline{\mu},\sigma^2 I)$, where $\mu$ is known to lie on the unit sphere ($\mu^T\mu$), show ...
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1answer
15 views

confidence level in non nested models

I have 7 non nested models: I use the Akaike method to see which one best describes the data. If the models were nested I would use the likelihood ratio test: this woluld also automatically give me ...
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19 views

How to calculate the volume, area and other measurements of a human body from picture? [closed]

Picture 1 shows that there are different sizes, shoulders (if body-builder), etc. Picture 2 illustrates this in real life animation and side view as this would be key to work out the ...
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1answer
20 views

How to validate a lognormal random walk for time series data

I am currently working on a project where I need to simulate the prices of a set of $D$ substitutable commodities over time. I was hoping to do this using the following $D$-dimensional lognormal ...
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27 views

Finding MLE when there's constraint that parameters sum to 1

In a total population of $n$, we have $k$ types of people, with $n_1, \dots, n_k$ of them in each corresponding group. The probability of a person belonging in a group is $\theta_1, \dots, \theta_k$, ...
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31 views

Help in estimation : MLE formulation

The intrinsic dimension (ID) of a data set generated from dynamical system is the minimum number of free variables needed to generate the data. Given a time series expressed in $m$ dimension (ID) we ...
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42 views

Conditional vs. Unconditional Maximum Likelihood

I have some questions on the difference between conditional MLE (CMLE) and unconditional MLE (UMLE) in practice. In what follows I will only talk about the unconditional and conditional mean and leave ...
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12 views

Appropriate relative standard error interpretation/calculation

To estimate the proportional change from a reference value for a given covariate, I am trying to understand which method is more appropriate from an interpretation of the precision of the estimate. ...
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1answer
48 views

Comparing coefficients of linear “ Stochastic Frontier Production and Cost Functions” in R

I am using the function frontier::sfa() in R to obtain the Stochastic Frontier Production and Cost Function as followed: ...
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2answers
41 views

Maximum likelihood + Kernel Density Estimation

Here my problem. I have a limited data sets of random variables $x: x_1, x_2, ..., x_N$. I can estimate the probability density function of $x$ by mean of Kernel Density Estimation method. (It works ...
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1answer
25 views

Copula estimation

I want to fit a copula distribution. My question is: Is it equivalent to estimate the marginal distributions using marginal samples and later estimate the parameters of a copula to estimating all the ...
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48 views

Newton Raphson Over-estimates Parameters

I have implemented an almost plain vanilla algorithm to find the MLE estimates of 3 parameters in a log-likelihood function (in R.) When I test my algorithm with some simulated data it does pretty ...
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2answers
84 views

REML or ML to compare two mixed effects models with differing fixed effects, but with the same random effect?

Background: Note: My dataset and r-code are included below text I wish to use AIC to compare two mixed effects models generated using the lme4 package in R. Each model has one fixed effect and one ...
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42 views

Is this problem Bayesian? And can I use variational approximation?

Suppose there are $N$ samples of observations $\mathbf X(n)$ ($n=1,\cdots,N$), which are given by probability distribution $p(\mathbf X(n)|\mathbf Z(n))$ with their conditions are given by hidden ...
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1answer
51 views

How do I use the Hessian matrix for maximum likelihood estimation?

I am trying to teach myself maximum likelihood estimation using the Newton-Raphson method and related iterative methods. I don't understand the link between the hessian, the expected value of the ...
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18 views

Marginal pseudo-likelihood and consistency?

For a given set of random variables, $X_1,...,X_n$ we know that in many cases finding the maximum of the pseudo likelihood: $$PL(x_1,\ldots,x_n) = \prod_{i=1}^n p(x_i | ...
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15 views

How to compare two matrices (A given matrix and a scaled up one)?

I have a matrix with 0.25 million rows and 50 columns. I have scaled up this matrix to 1.5 million rows and 50 columns using a Method A. I would like to measure the quality of the method I have ...
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30 views

Complete Logistic Regression framework using K-Cross validation

I'm implementing a logistic regression model in a low event rate data. I have gone through many webpages (including stackoverflow, including my questions) but none answer or describe the end-to-end ...
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2answers
75 views

Show that the value is, indeed, the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
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1answer
29 views

Find MLE by combining two different experiments

I am trying to solve the following problem on an assignment. Anna has two dice. Die $F$ is fair: it rolls $1,2,3,4,5$ or $6$ with equal probability $\frac{1}{6}$. Die $D$ is dodgy: Anna suspects ...
2
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1answer
112 views

maximum-likelihood of a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...
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31 views

How to find the significantly different groups in Random Effects

I am using a mixed effects model as created here (using dummy data for now) in this R script ...
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1answer
43 views

MLE vs MAP vs conditional MLE with regards to logistic regression

We have some set of iid RV's: $(X_i, Y_i), \; i=1,\ldots n$. We believe each to be distributed as $P(X_i, Y_i | \theta)$. So that $$ P(X,Y | \theta) = \prod_i P_i(X_i, Y_i | \theta) $$ Now using ...
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47 views

ML estimation of parameters that do not completely specify the model

I was wondering how ML is defined when the parameter does not completely specify the model. More concretely, suppose $X_1, X_2, \cdots, X_n$ are drawn iid such that $P(X_1=i)=\theta_i$, $ 1 \leq i ...
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6 views

Segregation Analysis for predicting age-specific cancer risk

I am relatively new to the worlds of bioinformatics and genetics research. I have been tasked with presenting to my lab the potential value of a paper that uses Complex Segregation Analysis for a risk ...
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1answer
26 views

Standard errors of the MLEs

Can anybody tell me how to find numerical values for standard errors of the MLEs of Weibull distribution using the uncensored real data set on the breaking stress of carbon fibres (in Gba) reported by ...
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71 views

What properties of a likelihood function are required for quasi-likelihood estimation?

Quasi-likelihood seems like a great way to use Iteratively Weighted Least Squares to fit linear linear models with a very general class of likelihoods. But what is that class? Obviously the ...
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1answer
47 views

Does the log likelihood become unimodal when the sample size goes to infinity?

I know that, under the usual regularity conditions, the MLE converges to the true parameter values as the sample size gets large. And the scaled MLE tends to being normally distributed. However, in a ...
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29 views

Question on MLE

I apologize for the basic question. If $\{p_\theta(x): \theta\in K\subseteq\mathbb{R}\}$ is a smooth family of distributions, then the MLE $\hat{\theta}_n,$ under suitable regularity conditions ...
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71 views

Does least squares regression imply normality of errors?

For the linear model $$y_i=\beta_0 +\sum_{k=1}^{n}\beta_k x_{ik} + \epsilon_i$$ the parameter estimates are the same for the maximum likelihood method and the least square method (minimizing ...
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95 views

Maximum likelihood estimator for $\theta$ and $E[X]$

Let $X_1,..., X_n $ be a random sample of a variable with PDF: $$f(x|\theta)=\frac{\theta}{x^2} I_{(\theta, \infty)}(x), \theta >0$$ Find the maximum likelihood estimator for $\theta$ and $ E[X]$ ...
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27 views

Parameter Estimation for Naive Bayes - Maximum a posteriori and Maximum Likelihood

I am wondering if I understand those terms correctly. To summarize my thoughts: In naive Bayes, our decision rule is basically the Maximum a posteriori (MAP) estimate of our hypothesis. We assign an ...
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2answers
138 views

Under what circumstances is the log likelihood function of a point process concave?

I am trying to understand under what circumstances the log likelihood function of a point process concave. Assume that the process can be defined by a conditional intensity function and that the log ...
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83 views

How to compute (or numerically estimate) the standard error of the MLE

I have a model for which I know the log likelihood function, the gradient of the log likelihood and the Hessian of the log likelihood. For given data I can compute the MLE using a generic optimizer ...