# 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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### UMP two sided tests for exponential families

Consider a random variable $X$ with density $$f(x : θ) = C(θ)e^{η(θ)T(x)}h(x), θ ∈ Θ$$. Assume that $η(θ)$ is strictly increasing in $θ$ and that the family is full rank. Show that there will not be ...
• 207
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### D-Optimal value by hand

I'm trying to compute the D-optimal value by hand (on R) Based on this video I could get the D value by ...
• 479
1 vote
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### How to combine a noisy (but unbiased) estimate with a precise (but possibly biased) estimate in A/B tests?

Suppose I want to estimate some set of unknown quantities $\theta_1$, …, $\theta_N$. For each $i \in \{1, …, N\}$, I have two estimators: $\hat{\theta_i}_A$ and $\hat{\theta_i}_B$. The goal is to ...
• 1,357
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### Difference between Parsimonious model vs Optimal model

As per my understanding, parsimonious regression model is the model that has less variables but with those variables I can describe the data best. Is it so? Then ...
93 views

### Is there an optimality result for the two-sample Wilcoxon-Mann-Whitney test?

Is there any mathematical result that states that the Wilcoxon-Mann-Whitney (WMW) test is optimal in some sense, for a specific testing problem that is a subproblem of the general problem the WMW test ...
• 24.7k
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### General rule when max likelihood is suboptimal

Inspired by the question regarding Bessel's correction, I wonder whether there is a general rule regarding applicability of maximum likelihood for parameter estimation. My guess is that the parameters ...
• 9,238
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### Sliding parameters over a ML model

I'm analysing the performance of an algorithm based on neural networks and I have to tune two parameters( number of past days to consider for the turbidity and volume of water flowing) by hand. So ...
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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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### 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 ...
549 views

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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) ...
• 229
1 vote
517 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 $0$-$1$ ...
• 133
606 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-...
1 vote
126 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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234 views

### 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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### D-Optimality for regression of polynomial models in one variable with missing terms

Let's say I have a model that looks as follows: $$y = x + ax^3 + bx^5 + cx^7 + dx^9$$ Given $n$ free choices for x as input measurements how can I determine which $x$'s I should input to best ...
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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
94 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
900 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 ...
157 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
27 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