# Questions tagged [optimal]

For questions about optimality properties of statistical methods, such as optimal parameter estimation or optimal testing. Both for questions about optimality theory in general, and for questions about optimality properties of specific procedures.

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### From inputs, updating $f\left(X_{i},Y_{i}\right)$ and $g\left(X_{i},Y_{i}\right)$ much other than $\overline{X_{i}}$ and $\overline{Y_{i}}$ to compare [closed]

(Premier League $2019$/$20$). The season was affected by the COVID–$19$ Pandemic while each team had a quarter of their schedule left (I call it $1/4$ because $1/2$ teams had $4$ away matches, and $5$ ...
14 views

### Optimal MSPE predictor, estimating equations

I am reading through Bickels and Doksum's Mathematical Statistics book, and in the second chapter they give justification for the use of a minimum contrast estimator: However, I do not quite ...
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1 vote
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### What does it mean for the Bayes Classifier to be optimal? [closed]

The Bayes classifier is always called the 'optimal' classifier. What does this actually mean? In particular does optimal mean the Bayes classifier will never make a mistake when predicting the label ...
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### Optimality conditions for the LASSO

In this paper, on page 1122, it states that the optimality conditions for the LASSO give $\hat{\beta} = n_{\lambda}(\hat{\beta} - X^T(X\hat{\beta} - y))$, where $n_\lambda$ is the soft-thresholding ...
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1 vote
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### What is the theoretical justification for alternatives to MSE minimisation?

I'm trying to wrap my head around the connection between statistical regression and its probability theoretical justification. In many books on statistics/machine learning, one is introduced to the ...
7 views

### Method to find optimal settings given an output metric

I am trying to optimize a screening problem that is usual done through brute force. I am trying to expand my skill set but I don't know where to start. I have built a visualization tool that helps ...
115 views

1 vote
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### Optimal Feature Engeneering creation: best optimization method?

basically I would like to solve this problem: (1) say I have N features that I want to transform with a generic f(x, theta) ...
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1 vote
298 views

### Question on Optimal predictors for the 0-1 loss function

The input $X \in \{0, 1\}$ and label $T \in \{0,1\}$ are binary random variables, and the set of predictors that we consider are the functions $y : \{0, 1\} \rightarrow \{0, 1\}$. Recall the $0$-$1$ ...
• 143
190 views

### UMP test and non-decreasing power function

Let $\phi$ be a UMP test for $H_o: \theta \leq \theta_0$ and $H_1: \theta > \theta_0$. Let its power function, $E_\theta(\phi)$, be differentiable w.r.t. $\theta$. Show the power function is non-...
15 views

### Help needed in the following Most Powerful Test Problem

I have solved this problem. I am just not sure if it is correct or not. Let me explain my approach. To compute the size: $\alpha = E_0(\phi(x)) = P_0(1) + P_0(2) + 0.25 P_0(3) = 0.0225$. Similarly, ...
1 vote
50 views

### Optimality in Clustering algorithms and an Optimality guaranteed clustering algorithm

So there are lots of clustering algorithms with different characteristics. What I am interested in now is a clustering algorithm which guarantees to find the optimal clustering result (if exists). And ...
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### Most powerful test for Hardy–Weinberg proportions

Consider a population with three kinds of individuals labeled $1, 2$, and $3$ occuring in the Hardy–Weinberg proportions $f(1,\theta)=\theta^2,f(2,\theta)=2\theta(1−\theta),f(3,\theta)=(1−\theta)^2$. ...
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### Uniformly most powerful test does not exists

I am having tough time understanding this concept The book says: “We caution the reader that UMP tests for testing H0 : θ1 ≤ θ ≤ θ2 and H0′ : θ = θ0 for the one-parameter exponential family do not ...
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### Optimal cut-off value for continuous covariate with binary outcome [duplicate]

Research question: What is the optimal cut-off value for a continuous covariate (age) with a binary outcome (suitable or not for a certain treatment). I realize that we lose information by ...
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### Least favorable prior - Find the distribution that maximizes the Bayes risk

Suppose I've found that the Bayes risk is of the form $$r(\theta) = \int_{-a}^a \theta^2 \pi(\theta)d\theta$$ I want to show that the following distribution, $\pi(a)=\pi(-a)=0.5$, maximizes this ...
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1 vote
68 views

### D-optimal design with nuisance parameters

I am a mechanical engineer trying to develope an optimal design of experiments in a problem with nuisance parameters. I would like to calculate the parameters $\mathbf{d}$ to optimally estimate ...
1 vote
484 views

### Can we always get an optimal $k$-means cluster arrangement?

I am currently studying $k$-means clustering. An optimal $k$-cluster arrangement is defined as follows: Fix a distance $\Delta$ and $k < n$. Assume $\mathbb{X}$ have been partitioned into $k$ ...
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1 vote
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### how to sample data for regression that is the most informative?

Background I have a unknown function $$f(x_1, x_2)$$ But I have access to evaluate this function finite $L$ times, $$y_j = f(x_1^j, x_2^j), j=1,\ldots,L$$ Then I have a model $\hat{f}$ which I ...
139 views

### Is there a UMVUE for arbitrary distribution with density and variance?

Let F be the family of all distributions with probability density and finite variance, and $X_1, ..., X_n$ be random samples from F. Does UMVUE for variance exists for this situation?
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1 vote
26 views

### Relationship between 4SID and 4DVAR

Background: The "4SID" method allows one to back analytic models from data using a form similar to that of a Kalman filter. (1) The 4DVAR is an approach to use data with an analytic model for weather ...
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1 vote
88 views

### D-optimal mixture design- poor evaluation of design

My problem is related to the design evaluation of my set of experiments. I am trying to design a mixture of 4 materials, each having certain minimum and maximum dosage constraints. To come up with a ...
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1 vote
31 views

### Bayes decision boundary naming [duplicate]

Why the optimal classifier is called "Bayes". I do not see the connection with being Bayesian or so. Edit: This question asks about naming. Not about math. Another phrasing of the question: Given ...
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1 vote
110 views

### What parameter to use when testing a SVM classifier on an independent test dataset?

I am trying to make a classifier that effectively distinguishes between control group and patient group, and then I want to use that classifier to distinguish high-risk patients who convert to patient ...
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1 vote