Questions tagged [parameterization]

For questions about how to parameterize some statistical model, or comparisons between different ways to parameterize.

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13 views

What is the definition of a scalar parameter?

I'm having trouble understanding what explicity is a scalar parameter. I understand what a location parameter and scale parameter represent but what exactly is the definition of a scalar parameter? ...
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13 views

Determine the MLE of parameter p for the negative binomial distribution [closed]

![Function of p, has a maximum value. This value can be found by taking the derivative of L with respect to p, and setting it equal to 0. I don`t know what the value of L will be in this case? ]1
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Reparameterize b(K,pi) in terms of theta

Question: $X_1, ... , X_n$ follows a binomial distribution with parameters K and 0 < $\pi\ <1.$ Use properties of Regular Exponential Class of distributions to show that the sample total $T = \...
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Delimit the area in that parameter space that contains 95.4% confidence

Given the equation $y = fc + fe\times \sin(2\pi(x-t_0)/12)$ Considering the two parameters of amplitude fc and fe simultaneously, delimit (using the χ2 variation method) the area in that parameter ...
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1answer
188 views

Normalize a periodic parameter

I am using inverse modelling software (PEST) to estimate a periodic parameter for the direction of anisotropy, $\hat{\theta}$, which is somewhere in $[0^{\circ}, 180^{\circ})$ (i.e., has a wavelength ...
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1answer
2k views

What is exponential family criterion to test the sufficiency and completeness of an estimator?

I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families: Let $X_{1},X_{2},...,X_{n}$ be iid observations from an ...
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33 views

Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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How to parameterize a bivariate Normal distribution output for a neural network?

For a neural net where the output is a Gaussian distribution, the output is usually parameterized as $(\mu=O_1, \sigma^2=e^{O_2})$. That is to say, the neural net will output the mean, and also output ...
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Why are these 2 ARIMA formulations equivalent?

In the "Understanding constants in R" section of his book, Hyndman & Athanasopoulos textbook "Forecasting: Principles and Practice" claims that the following AR processes equations are equivalent: ...
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2answers
79 views

Why can't algorithms avoid overfitting themselves?

So, I understand overfitting (bonus question: precise statistical definition of overfitting?). You don't want to match the noise in your sample. What I don't understand is why this requires a ...
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10 views

Alternative to plug-in estimation for log-tranformed linear model

I want to estimate a relationship of the form: $$y=ax^b\times\epsilon$$ If I log this model i get: $$\log(y)=\log(a)+b\log(x)+ \log(\epsilon)$$ If I then proceed and estimate this model using a ...
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29 views

Estimate distribution of aleatoric variable using Bayesian inference

Given a model as follows: $$y = cx + e$$ where y is the model output, x is the model input, c is an unknown variable and e is a Gaussian model error with zero mean: $$e \sim N(0,\sigma)$$ Data is ...
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1answer
27 views

Articles that work with covariates for mean, variance, and correlation simultaneously

Does anyone know of articles in which, in addition to modeling the mean parameter, are also modeled the variance and correlation parameters? I know the double generalized linear model, but they only ...
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20 views

Is this parametrization identifiable?

So I have this problem which I'm unsure of my answer. Any tip on how to treat it differently is more than welcome. X and Y are independent $\mathcal{N}(\mathcal{\mu_1},\sigma^2)$ and $\mathcal{N}(...
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14 views

Why are decision trees (especially ID3) non-parametric?

I was going through the definition of parametric and non-parametric models. So the parametric are the ones which have a fixed number of parameters that you are trying to learn and this number is ...
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State whether the model in question is parametric or non-parametric

The number of eggs laid by an insect follows a Poisson distribution with an unknown mean $\lambda$. Once laid, each egg has an unknown chance, $p$, of hatching and the hatching of one egg is ...
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1answer
25 views

Is there a formal relation between weight regularization and compression?

In my understanding, compression, strictly speaking, means that we diminish the amount of data required to describe something, such as a model. E.g. compressing an image file means to create a file ...
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1answer
66 views

The meaning of a parameterization of the logarithmic distribution

In calculus one learns that $$ p + \frac{p^2} 2 + \frac{p^3} 3 + \frac{p^4} 4 + \cdots = -\log(1-p). \tag 1 $$ Thus a discrete probability distribution on the set $\{1,2,3,\ldots\}$ is given by $$ \Pr(...
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Strategies for analyzing the functional relationship between two time series?

Suppose we have time-dependent survey data about name recognition for a political campaign. We're interested in learning how campaign spending effects that name recognition. My interest is in ...
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1answer
22 views

Flexible models and parameters

I just started reading Introduction to Statistical Learning with R and I am currently trying to work through the exercises. One of the questions is "What are the advantages and disadvantages of a ...
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826 views

gamma parameter in xgboost

I came across one comment in an xgboost tutorial. It says "Remember that gamma brings improvement when you want to use shallow (low max_depth) trees". My understanding is that higher gamma higher ...
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1answer
25 views

Expected value without complete sample space

The book way: Suppose, we have a bag with 8 balls numbered 1-8, we want to estimate the population parameter mean. we note down the entire sample space. (1,1)(1,2).. (8,8) calculate mean of each ...
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Hypothesis test practice question

" Prof. J conducts a hypothesis test on whether the proportion of all students who bike to school (denoted as p) equals 30%. Specifically, Prof. J has H0: p=0.3 versus HA: p≠0.3. He obtains a P-value ...
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Speed of transition parameter constraints

Given a logistic smooth transition regression \begin{equation} y_{t}=x_{t}^{\prime }\beta _{1}(1-{g}(z_{t};\gamma ,\delta ))+x_{t}^{\prime }\beta _{2}{g}(z_{t};\gamma ,\delta )+\varepsilon _{t}% \text{...
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How do I create error bounds after parameter calibration?

I have a power transform $f$ I am applying to an Ornstein-Uhlenbeck stochastic process $\{X(t), t\geq 0\}$: $$dX(t) = \kappa (\mu - X(t)) dt + \sigma dW_{t}.$$ From here, I was able to plug in my ...
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What to do when the meaning of a variable has changed over time?

I have dataset of a company with 2014 data with 15 variables then 2018 data with same 15 variables.I want to combine both the datasets however the meaning of 1 variable has changed meaning that ...
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1answer
214 views

Merits of reparameterizing the Gamma and inverse Gamma

Wikipedia states that the PDFs for the Gamma distribution is: $$ f(x|\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp(-\beta x) $$ However, in Rasmussen 2000, the pdf for the ...
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2answers
658 views

Justification of simulated annealing versus random search

I have a set of 16 integer parameters to optimize. The parameter space is too big for an exhaustive search, so I am using simulated annealing instead. I think my simulated annealing works - it finds ...
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1answer
34 views

What’s the difference between k-theta and alpha-beta parameterization for gamma distribution?

In my book “Mathematical statistics with Applications”, written by Wackerly, it’s stated that there are two methods for parameterization of gamma distribution. The first one is k-theta and the second :...
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61 views

Changing a conditional probability to a deterministic function

Suppose that we have a conditional density function $p(y|x;\theta^*)$, where $\theta^*$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this ...
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3answers
360 views

What is the difference between parameter and variable?

This is a question that I have in order to reconcile a difference in terminology. In the linear regression setting, we have $y=\beta x + \theta$. Here, we call $x$ a variable. When we are trying to ...
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1answer
358 views

Iterative parameter updates on student-t distribution (and approach for other distributions)

In a paper I found an iterative algorithm based on Bayes working with the following distribution and update criteria: In another source, I found the same update criteria in a whole different context. ...
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1answer
51 views

Different notions of over-parameterization

While reading a paper, I came across the statement This prediction function will be parameterized by a parameter vector $\theta$ in a parameter space $\Theta$. Often, this prediction function ...
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Confused by “mean” and “median” of $\alpha$ parameter in Lognormal Distribution

I read a book and find the following content (Fig.1). It is about lognormal distribution. What confused me is in the red box. In Fig.1, $\alpha$ is said "the mean of $z$ on the log scale". Then I ...
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1answer
305 views

Parameterization of Gamma Distribution

I have come upon different parameterizations of the Gamma Distribution, but not with regard to shape-scale or shape-rate. It is rather about the sign in the exponent. Wolfram lists the pdf as being ...
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1answer
261 views

Do I stick with the tuned model parameters even if they produce worse test scores?

The shorter and more general version of this question: If tuning a model via cross-validation (within training set) produces worse results on the test set than my previous default/baseline model, do I ...
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1answer
746 views

parameter tuning using nested cross validation

Parameter tuning in SVM has been performed using a nested cross-validation(CV) approach with 45 folds(outer loop) and 13 folds(inner loop). In this process, the outer loop will have 45 prediction ...
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23 views

Estimation of covariance over a range of independent variable

I have a set of data that comprise 2 dependent variables (let's call them $x_1$ and $x_2$) evaluated at different temperatures, T. There is an assumption that for a range of T ($T_0<T<T_1$) ...
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25 views

Mixed parameterization of sample from normal distribution

I am studying exponential families and mixed parameterizations. Now, I am told that $$ \mathbf{\theta} = \begin{bmatrix}\mu\\ -\frac{1}{2\sigma^2}\end{bmatrix} $$ is the parameter in a variation-...
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3answers
257 views

Normal distribution parametrization

I have the following hierarchical model: $y_{i} = \alpha + \beta_{i}x_{i} + \varepsilon_{i} $ where $\varepsilon \sim N(0,\sigma^{2})$. $\beta_{i} \sim N(\gamma x_{i},\sigma^{2}x_{i}^{-2})$ ...
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What is the point of graphical models?

I spent the day learning about the bnlearn package in R only to discover that Bayesian models do not work with undirected graphs. I'm trying to learn about the Markov Random Field Network, and so far ...
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1answer
25 views

How can I write an asymmetric-BEKK(1,1,1) model

To write a BEKK(1,1) model, I would write something like this, $$H_t=C^*C^{*'}+A_{11}\varepsilon_{t-1}\varepsilon_{t-1}'A_{11}'+ B_{11}H_{t-1}B_{11}' $$ How could I extend this to write the BEKK(1,...
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1answer
217 views

Weibull Parameter Estimation

I am doing a project in which I need to estimate Weibull parameters for car part failures (I know the data follow Weibull). I have data for 1000 cars (part failure data). Now the problem is suppose ...
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1answer
220 views

Understanding the definition of a location parameter

In some probability distributions, like normal or (non-standard) t distributions etc, there are location parameters such that a change to this parameter leads to the distribution moving rigidly to the ...
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1answer
83 views

good terminology for the parameters of a lognormal distribution?

Is there any good short terminology for the two parameters of a lognormal distribution? I have been using mean-log for $\mu$ and volatility for $\sigma$, where the lognormal variable $X$ has $\ln(X)$ ...
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2answers
270 views

Finding a correspondence between time-series elements

My problem deals in particular with time-series data about server performance, but the solution is sure to be applicable to many types of data sets. Pardon me if the answer is well-known; I don't know ...
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1answer
89 views

Formula for cross-classified (a.k.a., crossed random factors) mixed effects model with interaction between two “second level” variables

I have a crossed-classified (Hox, 2010) mixed effects model—also known as crossed random factors (West, Welch, & Galecki, 2015), but I am struggling with how to write the formula for an ...
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1answer
355 views

What does mean the “B” in a GLM result?

       
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Convert regression parameter standard error estimates to standard deviation estimates

Lets say I fit a linear model (in R), of y ~ x: x <- runif(100,0,5) y <- x*0.5 + rnorm(length(x)) summary(lm(y~x)) The summary output returned is: ...
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What's in a name: hyperparameters

So in a normal distribution, we have two parameters: mean $\mu$ and variance $\sigma^2$. In the book Pattern Recognition and Machine Learning, there suddenly appears a hyperparameter $\lambda$ in the ...