Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
1 answer
64 views

Proof for Asymptotic Normality for MLE with Multiple Parameters

I have a question about how I can prove asymptotic properties of multiple MLE estimators. Most resources you find online give the proof for the case, where only one parameter is estimated (e.g. here: ...
Red's user avatar
  • 315
1 vote
0 answers
71 views

Why the matrix normal distribution is not estimated with this "method of moments"?

According to the Wikipedia Page, a random matrix $\bf{X}\in \mathbb{R}^{p\times q}$ follows a matrix normal distribution $\cal{MN}(\bf M, \bf U, \bf V)$ means that $$ \text{vec}(\bf X) \sim \cal N ( \...
Miles N.'s user avatar
  • 185
4 votes
1 answer
122 views

How is Gaussian log likelihood value calculated in weighted LM, GLM and GLS?

I have examined these R functions glm.fit() gaussian()$aic stats:::logLik.glm() and ...
DrJerryTAO's user avatar
  • 2,373
0 votes
0 answers
41 views

Likelihood ratio test for samples from two different populations

This is basically exactly same as the post here. The answer given over there was not accurate as the null distribution specifically states $\mu_0=\mu_1=0.$ I have attempted it but am not sure if the ...
TryingHardToBecomeAGoodPrSlvr's user avatar
2 votes
1 answer
90 views

Real-world examples of conditional independence with marginal dependence in cross-sectional data

In a previous question, I explored the derivation of joint likelihood using only conditional independence assumptions - Derivation of Joint Likelihood with Only Conditional Independence Assumptions (...
spie227's user avatar
  • 321
3 votes
0 answers
41 views

How do you fit/learn a nonstationary Poisson process from data using maximum likelihood estimation?

I'm trying to understand how to fit/learn a nonstationary Poisson process from data. Through this question, I will explain how I understand the Poisson process, the nonstationary variation of it, and ...
stuz's user avatar
  • 131
2 votes
0 answers
38 views

Does MLE on more samples always have a lower risk?

Denote the MLE on $n$ i.i.d. samples with $\hat{\theta}_{n} = \arg\max_\theta \prod_{i=1}^n p(x_i; \theta)$. I wonder if adding more samples always makes the risk lower, or at least not higher. In ...
nalzok's user avatar
  • 1,817
6 votes
1 answer
226 views

MLE in stochastically increasing parametric family

Let $X$ have cumulative density function $F_{\theta}$, suppose this family is stochastically increasing in $\theta$, that is, for $\theta_1<\theta_2$, $F(x;\theta_2) \le F(x;\theta_1)$. We have one ...
Noppawee Apichonpongpan's user avatar
0 votes
0 answers
22 views

Hypothesis testing with unknown covariance matrix under the alternative hypothesis

I am trying to solve a binary hypothesis testing problem of the form: $H_1 : \mathbf{y} = \theta \cdot \mathbf{x} + \mathbf{n}$ $H_0 : \mathbf{y} = \mathbf{n}$ where $\theta \in \mathbb{R}$ under $H_1$...
burnedstudent's user avatar
3 votes
1 answer
50 views

Comparisions between REML and ML

I have two questions and really appreciate your answers. If Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) methods have the same fixed effects in a model, are the results from these ...
John's user avatar
  • 31
1 vote
1 answer
71 views

T-test when variance estimator is not sample variance [closed]

It is well known that if $x \sim \mathcal{N}(0, \sigma^2)$ and we have a sample size of $n$ observations of $x$, the distribution of $\frac{x}{\hat{\sigma}}$, where $\hat{\sigma}$ is the sample ...
Geo's user avatar
  • 57
11 votes
1 answer
326 views

Transforming Hessian to original parameters after centering and QR-transforming X

I'm starting to more routinely mean-center and QR-rotate design matrices $X$ when doing maximum likelihood estimation (MLE) or Bayesian posterior sampling. This helps with convergence and quickens ...
Frank Harrell's user avatar
3 votes
0 answers
47 views

Is there an analytical solution to the distribution of a sum of observations drawn from a Frechet distribution?

Let $X_i$ be an iid draw from a Frechet distribution. Let $\alpha_i \in \mathbb{R}$. Is there an analytical expression of the distribution of $\alpha_1X_1 + \alpha_2X_2 + \alpha_3X_3$? That is, can I ...
John Go's user avatar
  • 31
2 votes
0 answers
65 views

Doing "maximum p-value estimation" instead of maximum likelihood

Whenever we do maximum likelihood estimation, we look for the parameters that maximize the probability density of the data. On the other hand, when we compute p-values, we look at the tail probability ...
Mike Battaglia's user avatar
1 vote
0 answers
33 views

What makes a curve a good fit in the context of logistic regression

As I wanted to gain a better intuition between why separation is a problem in the context of logistic regression, I did create in R two models, one where y is perfectly separated at $x=5$, and one ...
She Wonders's user avatar
2 votes
1 answer
154 views

Derivation of Joint Likelihood with Only Conditional Independence Assumptions (Without Marginal Independence)

I'm working through a derivation of the joint likelihood for a dataset $(\mathbf{X}_i, \mathbf{y}_i)$ under the assumption of conditional independence of $\mathbf{y}_i$ given $\mathbf{X}_i$, without ...
spie227's user avatar
  • 321
0 votes
0 answers
23 views

Maximum likelihood estimation of binned data

I am currently exploring methods for maximum likelihood estimation with binned data and have observed that many approaches utilize either the multinomial or Poisson distribution to calculate the ...
Ahmed Abdullah's user avatar
1 vote
1 answer
43 views

Unbiased Variance MLE Distribution

If you take $10000$ samples from a normal distribution, the unbiased variance MLE (with Bessel's correction) is $$\hat{\sigma}^2 = \frac{1}{9999}\sum_i (x_i - \hat{\mu})$$ Apparently the distribution ...
Trajan's user avatar
  • 503
1 vote
0 answers
17 views

Program for maximum partial marginal likelihood estimator (MPMLE) of regression parameters

I have a system of equations of probabilities which looks like where . And I would like to solve the above system for U_1, U_2, U_3. I would like to ask is there any package in R or python for these ...
Ishigami's user avatar
  • 195
0 votes
0 answers
22 views

Learning a probability distribution from samples drawn from unknown function

I am wanting to learn some probability distribution $p$ from data (using e.g., Kernel Density Estimation, a Normalizing Flow, whatever your favourite machine learning model is). If I had a dataset $D =...
Craig Innes's user avatar
5 votes
2 answers
129 views

What does one do when the Hessian matrix is too sensitive?

I used the survival package in R to calculate the maximum likelihood estimates of Weibull regression co-efficients in R. I then tried to replicate the results by writing the log likelihood function ...
shishir rao's user avatar
6 votes
1 answer
245 views

Variance of MLE's in mixture distribution

I am studying mixture models, and I am interested in calculating the variance of the estimators using maximum likelihood. How is the variance calculated in this case? I already implemented the EM ...
daniel's user avatar
  • 281
1 vote
0 answers
22 views

Convert units, get different results when fitting extreme value distribution with extRemes

I am using the fevd() and lr.test() functions to examine precipitation using the extRemes R ...
shaider's user avatar
  • 11
4 votes
1 answer
236 views

Finding the MLE using a contour plot

When finding the MLE of something using a log-likelihood contour plot, is the MLE in the middle of the middle ring?
user438310's user avatar
0 votes
0 answers
25 views

How to compare models with different predictors if maximum likelihood estimates of models render false convergence?

I have one full model and several partial GLS models that I want to compare using AIC or BIC to select which factors to include in the final model. I use ML to fit each model before comparing their ...
Agus Camacho's user avatar
1 vote
1 answer
130 views

Wilks' theorem, confidence regions on a projected subspace?

Background My context is I am thinking about maximum likelihood estimation of a parameter $\theta\in \mathbb{R}^r$ by drawing samples $\{x_i\}$ from a distribution which depends on $\theta$. I know ...
Jagerber48's user avatar
1 vote
0 answers
19 views

Is there a simple estimate for power transform (Box-Cox, Yeo-Yohnson) transform, that can be computed in O(n)? [closed]

So the power transform defines some function $f(\lambda,y)$ and then we're trying to find the $\lambda$ assuming that $f(\lambda|\mathbf{y}) \sim Normal(\mu,\sigma)$. This is usually done via MLE ...
user1747134's user avatar
2 votes
1 answer
132 views

Widespread inconsistency in maximum likelihood estimation approach to logistic regression

I have found some widespread inconsistency in how the loss function of logistic regression is derived through the maximum likelihood estimation approach. In the logistic regression model, we assume ...
Fraïssé's user avatar
  • 1,630
1 vote
0 answers
37 views

Max Likelihood of GBM with 2 Markov States

Consider the stochastic process $$dX_t = \mu_{\epsilon_t}X_tdt + \sigma_{\epsilon_t}X_tdW_t$$ where $W_t$ is a standard Brownian motion. The process $X_t$ is a geometric Brownian motion (GBM) whose ...
Alex's user avatar
  • 387
0 votes
0 answers
54 views

Estimate of mean in semiparametric model. Box-Cox fails for negative mean

I have a time series of positive values $X_t \geq 0$ satisfying the following model: $$\begin{cases}f_*(X_{t}) = f_*(X_{t-1}) + \mu + \varepsilon_{t}, &\forall t \in \{1,2,\dots,T\},\\ X_0 = 1, &...
Uomond's user avatar
  • 1
0 votes
0 answers
59 views

Terminology for types of errors and uncertainty: intrinsic? fitting?

Say I have a sequence of given variables $x_i$, $i=1,\ldots,n-1$ and the response $y_i$ and we explore the model $y_i\sim \text{Normal}(\alpha x_i,\sigma^2)$ with $y_i$ independent of $y_j$ for $i\neq ...
Enredanrestos's user avatar
2 votes
1 answer
92 views

Optimizing loglikelihood (for Vasicek loss model with autocorrelated systemic risk)

Please find the derivation of this optimization problem below; however, if too long, only the optimization problem is relevant. Optimization problem For $T$ observations $x_{1},...,x_{T}\in(0,1)$ and ...
user avatar
1 vote
1 answer
85 views

Asymptotic variance of MLE estimator and Fisher Information

I am trying to figure out what I am getting wrong about the following: Consider an iid Gaussian random variable $X$ with mean $\mu$ and variance $\sigma^2$. Suppose we know $\sigma^2$ and we are ...
Residual Claimant 's user avatar
1 vote
0 answers
48 views

Hessian for log likelihood of regression with respect to covariance matrix

I am interested in the residual covariance of a multivariate regression model. The regression is $$ Y_t = X_t \beta + \varepsilon_t $$ and I have a log likelihood as follows $$ \mathcal{L}(\Sigma) = \...
Ivan's user avatar
  • 11
3 votes
2 answers
100 views

Likelihood from forecast::Arima vs. manual replication

I am trying to replicate some results from forecast::Arima in R. I am particularly interested in the likelihood that I would like to use for some likelihood ratio ...
Richard Hardy's user avatar
2 votes
1 answer
128 views

Linear regression MLE of slope; OLS and bivariate MLE

Assume the variables X and Y, sampled from a bivariate normal distribution with mean vector $(\mu, \mu)$,and both having the same population variance $V$, which is known. The only unknown parameter is ...
supun's user avatar
  • 21
1 vote
1 answer
78 views

Expectation of the minimum of random variables (Exponential + Erlang)

Consider the following random variable $$ Z=\min_i\{X_i+Y_i\} $$ for $-n\leq i\leq n$, where $X_i\overset{\mathrm{iid}}{\sim}\text{Exp}(\lambda)$, $Y_i\overset{\mathrm{iid}}{\sim}\text{Erlang}(|i|,\...
sam wolfe's user avatar
  • 150
3 votes
1 answer
87 views

Maximum of two independent gamma variables

Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$. Are there known results for the distribution of $X$? Actually I only need to know $\...
Luis Mendo's user avatar
  • 1,191
1 vote
0 answers
12 views

Analysis of the relative likelihood with parametric bootstrapping

Using parametric bootstrapping, I find that the relative likelihood of model A is $l_A$ and of model B is $l_B$. Repeating the analysis several times yields a distribution of likelihood values for ...
Medical physicist's user avatar
0 votes
0 answers
25 views

Linearity of and pointwise equality in expectation of min() function

Consider the expressions $f = c + s*E[min(a/s, X)]$ and $g = E[min(c + a, c+sX)]$ where c >= 0 0 < s <= 1 a >= 0 X ~ Poisson($\lambda$/s) I'd like to think that $f = g$, reasoning as ...
BeechAndBirch's user avatar
3 votes
1 answer
42 views

Maximum Likelihood Estimation for Pairs of Observations

I have $n$ pairs of observations $(x_i,y_i)$, where each $y_i$ is distributed according to $\text{Pois}(\theta x_i)$, and I wish to do a maximum likelihood estimation for $\theta$ only based on this ...
whiteboardmarker's user avatar
5 votes
1 answer
611 views

Why is so much of likelihood theory focused on solving the score function equation?

When I first learnt about maximum likelihood estimation, I was told that I should check that the solution of $\frac{d}{d\theta}\text{log}(L(\theta;x_1,...,x_n)) = 0$ was really the maximum likelihood ...
Noppawee Apichonpongpan's user avatar
0 votes
1 answer
42 views

Is there an "observation noise" formulation for logistic regression?

In formulating linear regression as the solution to maximum likelihood problem, we need the assumption that the data $X$ and label $Y$ are related by $$Y = w^TX + \epsilon$$ where $\epsilon$ is a ...
Shamisen Expert's user avatar
0 votes
0 answers
16 views

Bishop gradient calculation

In section 3.1.1 of Pattern Recognition and Machine Learning by Christopher Bishop, it is written that $$\ln p(\mathbf{t} | \mathbf{w}, \beta) = \frac{N}{2} \ln \beta - \frac{N}{2} \ln (2 \pi) - \beta ...
Rahul Yadav's user avatar
1 vote
0 answers
16 views

KS test contradicts maximul likely hood fitting

I have two data sets (let call them L and G) that represent a measured physical quantity $\Gamma_n^0(\delta\Gamma_n^0)$. I am performing a Kolmogorov-Smirnov test in ...
Thanos's user avatar
  • 151
0 votes
0 answers
8 views

Source of the relationship between the number of parameters and data dimensionality

It is commonly understood that, in classical statistical models, the number of observations should be at least equal to the number of parameters in order to ensure the estimability of those parameters,...
Lydia2kkx's user avatar
3 votes
0 answers
64 views

Is this a mistake on Wikipedia on Standard Deviation?

On the Wikipedia page about standard deviation, in the section `Estimation', it says Unlike in the case of estimating the population mean, for which the sample mean is a simple estimator with many ...
Riemann's user avatar
  • 181
0 votes
0 answers
34 views

How do we derive the standard cross entropy loss from negative log likelihood in a supervised (conditional) learning setting?

I know that when optimizing neural networks (supervised) that cross entropy loss is equivalent to negative log likelihood is euivalent to MLE but I can't get all the math together. I am trying to ...
Meem12's user avatar
  • 11
0 votes
0 answers
10 views

Is the significance test in GMM more reliable than it in ML, especially when the regularity conditions are not met?

Maximum likelihood vs generalized method of moments From the answer to the above question, we can draw a conclusion that GMM and ML can get the same parameters' estimators when the model is exactly ...
Xu  Yang's user avatar
  • 41
0 votes
0 answers
54 views

Is Arithmetic Mean from the definition of Maximum Likelihood Estimation? [closed]

I am wondering if the arithmetic mean is derived from Maximum Likelihood Estimation or not. While I am studying Maximum Likelihood Estimation for the mean of a binary distribution, the mean was ...
xabzakabecd's user avatar
  • 3,585

1
2 3 4 5
81