All Questions
36 questions
7
votes
1
answer
411
views
Estimation of a uniform distribution corrupted by Gaussian noise
Problem definition
I have a dataset composed by $m$ observations $y^{(1)},\dots,y^{(m)} \in \mathbb{R}^2$ generated as follow
\begin{equation*}\begin{aligned}
y &= z + v \newline
z & \sim\...
- 473
0
votes
0
answers
25
views
Separating components of a likelihood maximization
Apologies for the naive question, but I have a problem I would like to solve.
Suppose I have a two dimensional likelihood of the form
$L \propto \exp\{-\frac{1}{2}\} \begin{bmatrix}x & y\end{...
- 1
0
votes
1
answer
198
views
Fitting Gumbel distribution based the maximal observation
Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution.
Question: Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (...
- 747
1
vote
1
answer
219
views
How do I use MLE for non-iid actual data?
In this paper, the author try to fit the Gumbel distribution based on the r largest value of each year using the maximal likelihood estimators: the likelihood function for r largest values $X_{n1},\...
- 747
4
votes
1
answer
540
views
Can we fit extreme value distribution by build-in package?
I try to find a package in R to fit Gumbel distribution by Block Maxima Approach using maximal likelihood function (see here)
$$
G(x; \mu , \sigma)=\exp[-e^{-\frac{x-\mu}{\sigma}}].
$$
The block ...
- 747
0
votes
0
answers
212
views
Weighing Maximum Likelihood Estimations
I'm trying to arrive at a time series of optimized parameter values $Z_t$ that maximizes the likelihood of occurrence of a specific time series $Y_t$. There is a subsample within the sample that ...
0
votes
0
answers
162
views
R: Getting Wrong Profile Likelihood Confidence Interval Estimates
I am trying to estimate the profile likelihood confidence interval (CI) of the parameters ($\xi$, $\sigma$) of the Generalized Pareto Distribution (GPD). However, the lower estimate (left CI) of $\xi$ ...
- 750
2
votes
0
answers
49
views
MLE for the number of samples given $k$ largest values
I have the views on the top 100 videos using a tag in TikTok and want to estimate the total number of videos in that tag. I know the distribution for other tags so I can make a guess as to what it is ...
- 2,608
5
votes
1
answer
303
views
MLE for the maximum of n values that are observed only with noise
Suppose $x_1, ..., x_n$ is a fixed set of real numbers. Let $\epsilon_1, ..., \epsilon_n \sim N(0, \sigma^2)$ be i.i.d. with known $\sigma^2$, and suppose we get to observe only $z_i = x_i + \...
- 2,160
0
votes
1
answer
109
views
Compare return levels of fitted GPD using MLE in different R packages
This question is related to this post: Different quantiles of a fitted GPD in different R packages?
I want to constraint "potvalues" data to be in a period of 6 years, this is, 16 observations per ...
- 1
5
votes
1
answer
5k
views
Asymtotic distribution of the MLE of a Uniform
A property of the Maximum Likelihood Estimator is, that it
asymptotically follows a normal distribution if the solution is unique.
In case of a continuous Uniform distribution, the Maximum Likelihood ...
9
votes
1
answer
6k
views
MAP estimation as regularisation of MLE
Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this:
It is closely related to the method of maximum likelihood (ML) estimation, but employs ...
- 1,059
1
vote
0
answers
371
views
Statistics of Extremes: Fitting the GEV distribution with MLE vs L-moments
I created a synthetic series that is supposed to simulate a series of peak discharges in blocks of years in arid catchments. The magnitudes were simulated via the Lnorm dist.:
...
- 123
5
votes
1
answer
508
views
Maximum Likeilhood estimate of shape parameter of GPD is negative, even though exceedances are positively skewed
I am looking at fitting a Generalized Pareto Distribution (GPD) to extreme events which exceed a certain value threshold for Bilbao waves data.
Selecting threshold at c=7.5, resulting in 154 ...
- 51
5
votes
2
answers
869
views
Maximum likelihood and Gumbel distribution. Does the likelihood have a global maximum?
It appears to me that if I move the mode $u$ more to the negative and increase the scale parameter $\alpha$, one can get always a higher likelihood. If this is true, is there a limit of the likelihood?...
- 404
5
votes
1
answer
73
views
Bayesian and frequency tail estimation
The tail probability can be estimated by two methods:
In Bayesian method:
$$P_B(X>a)=\int^{\infty}_{-\infty}\pi(\theta|x)[1-F(a|\theta)]d\theta$$
In Plug-in frequency method:
$$P_F(X>a)=1-F(a|\...
- 669
0
votes
1
answer
170
views
maximising a linear model function with unknowns
If i have this linear model
$$Y_{i,t}=\gamma_t(x_i)+v_{i,t}, v_{i,t} \stackrel{iid}{\sim}N(0,\sigma^2), i=1,\ldots,m.$$
$$\gamma_t(x)=\beta_{1,t}+\beta_{2,t}\frac{1-e^{-\lambda x}}{ \lambda x}+ \beta_{...
- 1
25
votes
2
answers
1k
views
Fitting custom distributions by MLE
My question relates to fitting custom distributions in R but I feel it has enough of a probability element to remain on CV.
I have an interesting set of data which has the following characteristics:
...
- 2,911
3
votes
0
answers
685
views
What's the use of the expected fisher information matrix over the hessian in the Newton Raphson approach to finding the MLE?
This may be a naive question, but I'm looking at the Newton Raphson iterative approach ( i.e. using the formula $\boldsymbol{\theta }^{(j+1)} = \boldsymbol{\theta }^{(j)} + \textrm{Hess}_{-\ell}(\...
- 31
2
votes
0
answers
30
views
Optimizing while collecting data - optimization in a real world problem
I want to conduct a soil analysis using a different mix of let says Nutrition A, Nutrition B and Nutrition C.
Since I can put for each nutrition multiple values, I cannot try out all the possible ...
- 547
3
votes
2
answers
351
views
Uniform distribution MLE
Just a quick question:
I know a $U(0, A)$ with density of $1/A$ has as MLE of $X_{max}$, but would a $U(1,1+A)$ have the same MLE that of $X_{max}$?
I'm assuming so but just for clarity.
- 33
13
votes
2
answers
9k
views
Markov chain Monte Carlo (MCMC) for Maximum Likelihood Estimation (MLE)
I am reading a 1991 conference paper by Geyer which is linked below. In it he seems to elude to a method that can use MCMC for MLE parameter estimation
This excites me since, I have coded BFGS ...
1
vote
0
answers
429
views
What is the difference between Restricted Maximum Likelihood (REML) and Maximum Likelihood (ML)? [duplicate]
I am a first year graduate student in biostatistics, and I have somewhat of an idea of the difference between REML and ML. However, I want a more in-depth understanding of each estimation method, ...
2
votes
1
answer
455
views
Likelihood for dependent data above a threshold
Let $(Y_t)$ a real-valued stationary Markov chain and $u$ some positive threshold. We assume that for $y>u$,
$$Y_{t+1}|\{Y_t=y\}\sim\mathcal{N}(\alpha y+\mu y^\beta,\sigma^2 y^{2\beta})$$
I want ...
- 233
3
votes
1
answer
4k
views
Have MLE estimators for Generalized Pareto Distribution. Given a known value of $c$, how do I calculate $a$ and $b$ using the provided estimators?
I am doing research into the three parameter Generalized Pareto Distribution
$$
f(x|a,b,c) = \frac 1 b\left(1+a\left(\frac{x-c}{b}\right)\right)^{\big(-1-\frac 1 a\big)}
$$
for finding VaR and CVaR. ...
- 175
1
vote
1
answer
485
views
Parameter estimation problem: maximum likelihood [duplicate]
Suppose I have some observations $x_{1}, x_{2}, \dots, x_{n}$. I also have a probability density function with one unknown parameter $\theta$. I would like to find such $\theta$, which would give the ...
- 21
7
votes
2
answers
4k
views
Finding the maximum point of probability density function
I'm curious about why we always find mle using the first (partial) derivative without checking the end points or singular point or the second (partial) derivative? Thx a lot!
- 463
3
votes
1
answer
962
views
How to find $\arg\max$ of a neural network?
Let's say I have a neural network $f$ that takes input $\vec x \in \mathbb {R}^n$ and produces output $f(\vec x) \in \mathbb{R}$.
How can I find $\hat x = \underset{\vec x}{\arg\max} \; f(\vec x)$?
- 3,132
2
votes
1
answer
3k
views
Different quantiles of a fitted GPD in different R packages?
I am performing an extreme value analysis for meteorological data, to be precise for precipitation data available in mm/d. I am using a threshold excess approach for estimating the parameters of a ...
- 33
10
votes
1
answer
3k
views
Maximum likelihood estimator for minimum of exponential distributions
I am stuck on how to solve this problem.
So, we have two sequences of random variables, $X_i$ and $Y_i$ for $i=1,...,n$. Now, $X$ and $Y$ are independent exponential distributions with parameters $\...
- 1,903
0
votes
0
answers
30
views
Accounting for minimum dependent measure in data when fitting a distribution
I have what is possible a naive question. I am current comparing various models (i.e. distributions). And the comparisons do not involve different distributions but rather how the model is fed the ...
- 43
1
vote
0
answers
74
views
Mode of Joint Posterior - Maximization Problems
I have a problem whereby I get two different answers if I try to maximize a function.
let
$ \begin{bmatrix} Y_{o}\\ Y_{a} \end{bmatrix}|\phi\sim N (0,\phi^{-1}A^{-1}) $
$\pi(\phi)=\frac{1}{\phi}$,
...
- 121
2
votes
2
answers
2k
views
Most suitable algorithm for optimizing Maximum likelihood function
What is the most suitable optimization algorithm for optimizing maximum likelihood estimator? In excel I used GRG non linear optimization algorithm, is that good enough?
I want to write my own code ...
- 205
5
votes
1
answer
3k
views
What do I need to consider when using the Hessian to compute S.E.'s?
I use optim() in R to do a lot of MLE. I've used the approach for a lot of problems, but the one I'm working on right now consists of fitting the parameters of the generalized extreme value ...
- 958
1
vote
1
answer
5k
views
Fitting GEV to non-stationary time series of extremes (general stationarity question?)
I'm fitting the generalized extreme value distribution (GEV) to a series of annual maxima of variable $X$. $X$ exhibits a linear trend.
When I fit the GEV to $X$, I think I have the choice to
Use ...
- 958
2
votes
2
answers
618
views
Predicting a maximum value with little data
My problem is i'm trying to figure out how many servers might be required to handle a theoretical maximal load of data requests. To do that I need to know what the maximum number of requests in a ...
- 123