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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Variance estimator that is optimal under absolute loss

Given a random i.i.d. sample from a population with a finite variance $\sigma^2<\infty$, what estimator of $\sigma^2$ is optimal under absolute loss? $$ \arg\min_{\hat\sigma^{2}\in F}\mathbb{E}(|\...
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5 views

Methods to Solve Stochastic Optimal Control Problem

I have a stochastic optimal control problem I want to solve numerically. The followings are the properties of the problem (or just something I knew and tried): The problem is not linear-quadratic so ...
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1answer
21 views

how to understand this math formula for bandwidth calculation?

I am reading a paper that uses the following equation to calculate the optimal bandwidth, however, I am confused about the position of "4" and "3" in the equation. is this a typo? or what does it mean?...
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28 views

Explain asymptotic optimality property of AIC (Akaike information criterion)

I am trying to use AIC in my research, I know how to apply it and how to interpret its value, but I do not understand the asymptotic optimality property in general and why AIC has this property. Can ...
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12 views

How to optimize a number of games with a limit set of second chances at guessing other options given probabilities?

Let's say I have a game with $M$ outcomes $\{O_i\}_{i=1}^{M}$ and I have a predictor $P$ that gives me probabilities for the outcomes of a game. I now have $N$ independent games that I want to ...
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1answer
38 views

D-optimal DOE suggest repeated samples

I tried to generate a D optimal design but the design output sounds very weird to me. I have a (real) process and I`d like to explore 3 factors, but the process have a lot of constraints so I provide ...
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18 views

Finding the 'optimal distribution' with one unknown variable

I am trying to find the optimal distribution (of ingredients) with one unknown parameter (skillset). Example, when baking a bread the ingredients are flour,salt,yeast and water. When I ask 100 ...
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19 views

How does caret package choose the optimal parameters when there are several such values?

First Example I fitted the following penalized logistic regression model using caret package as follows, ...
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41 views

Covariance matrix for a 2D state vector

I'm performing Optimal Interpolation (which in fact is a simplified Kalman filter with constant $\mathbf{K}$). My state variable is a 2D concentration field with a size of 370 x 400 on which I try to ...
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28 views

What is the best algorithm so far for probabilistic adaptive group testing?

I am checking https://en.wikipedia.org/wiki/Group_testing#Classification_of_group-testing_problems but could not figure out the answer. I am focusing on probabilistic adaptive group testing. The ...
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1answer
144 views

Rejection sampling for optimal $\lambda$ and $a$

Suppose $f(x) \propto \exp ({-(x-u)^2\over2\sigma^2}) I_{X>=a}$ and we cannot compute the normalizing constant. Consider rejection sampling using proposal density of a shifted exponential ...
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34 views

Hyper-parameters which minimize the variance of transformed multi-variate Guassian variable

Let $k < p$ be positive integers and $g: \mathbb R^k \rightarrow \mathbb R^p$ be a smooth Lipschitz continuous function. Let $y_1,\ldots, y_N \in \mathbb R^p$ and $a = (a_1,\ldots,a_N) \in \mathbb ...
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1answer
66 views

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 ...
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1answer
120 views

Is there a Fisher Information equivalent in MAP Empirical Bayes estimation?

Background The Fisher information for a linear Gaussian model is $\mathcal{I}_{\theta} = \frac{X X^T}{\sigma^2} $. This is used in optimal experiment design techniques, for example, maximisation of $|...
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39 views

At which rank should I reasonably stop selecting interview candidates?

It seems that my problem could be classified as an "optimal stopping" problem, but I am unable to make much of this information. It is an important problem for my organisation (an NGO), in order to ...
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2answers
268 views

What's the difference between “Optimal linear predictor” and “best unbiased linear estimator”?

Greene (econometric analysis 7th ed. p 53) states that OLS is the "optimal linear predictor": Then on the next page, he states that OLS is also the BLUE estimator (Gauss-Markov Theorem): I ...
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1answer
310 views

Estimator that is optimal under all sensible loss (evaluation) functions

Consider a probability distribution $D$ with a parameter $\theta$ and an i.i.d. sample $S$ from that distribution. I am interested in an estimator $\hat\theta(S)$ of $\theta$ that satisfies the ...
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32 views

Feature scaling and equivalence of the mean squared loss function optimal solutions

When applying the feature scaling, what is the mathematical proof that the optimum of the mean squared loss function is the same in both without-normalisation case and with-normalisation of the ...
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215 views

Does there exist an analogous statement to BLUE (Gauss-Markov) for GLMs?

I recall from my graduate school days that the Gauss-Markov (GM) theorem states that the Best Linear Unbiased Estimator (BLUE) in a linear regression is $\vec{\beta}=(X^TX)^{-1}X^T\vec{y}$. An amazing ...
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23 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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79 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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28 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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86 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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1answer
271 views

Questions about gradient descent and local optima

I'm starting out learning about gradient descent, and have a couple of conceptual questions. I noticed a common pitfall of gradient descent is getting stuck in the local optima. How can this be ...
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28 views

Existence of ε-optimal Borel measurable policies in stochastic control

I am reading the book "Stochastic Optimal Control: The Discrete Time Case", by Bertsekas and Shreve (hereafter called "the Book"), and I recently observed that a statement made in page 10 of the book (...
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1k views

Most efficient estimator of mean

Let's say we want to estimate a mean $E[F]$ for some unknown distribution $F$. If we only care about the MSE (and don't mind using biased and non-linear estimator), what theory is available to help us ...
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24 views

What type of statistical models would be useful for evaluating an athlete's performance?

Suppose a performance athlete wants to use statistics to design an optimal exercise regimen. The goal of the statistical model should be to essentially maximize the performance of the athlete by ...
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81 views

Probability of correct classification with optimal Bayes when increasing number of features

Consider the optimal Bayes classifier applied on a problem with N features. Let its probability of correct classification be $$P_N(corr)$$ Assume that we add an extra feature (so now we have N + 1 ...
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95 views

UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
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1answer
6k views

Chosing optimal k and optimal distance-metric for k-means [duplicate]

I have a data-set with roughly 20-dimensions and millions of points which I want to cluster. The goal is to find a set of clusters which: Are as distinct as possible from each other (minimum ...
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1answer
1k views

Nadaraya-Watson Optimal Bandwidth

I am currently working on a statistical project where I need to estimate a conditional expectation $E[Y|X=x_i]$ using the Nadaraya-Watson estimator. For doing that, I have the sample $(x_1,y_1),...,(...
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54 views

Verification of an optimal parameter from an empirical CDF

Suppose we have the following model for the variable $V_5$: $$V_5 = \prod_{k=1}^5(e^{\mu + 0.2X_k}+0.05e^{0.05Y_i - 0.00125}), X_i,Y_i\sim N(0,1)$$ What I wish to do is to solve the problem $\min_{\...
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2answers
438 views

Drawing numbered balls from an urn

PROBLEM There is an urn with a set of balls where each ball is labeled with a different integer. The numbers on the balls are known and are not a range of integers. For example the set of balls could ...
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86 views

A test for a binomial r.v. vs. sum of two binomial r.v.'s?

Suppose I draw a sample from one of the two possible random variables $X$ or $Y$ where $X\sim\text{Binomial}(p,N)$ and $Y=A+B$ with $A\sim\text{Binomial}(p,M)$ and $B\sim\text{Binomial}(q,N-M)$. ...
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1answer
320 views

Good, useful and characteristic experiments for (optimal) statistical design of experiments

There are more phenomena to which experimental design may be applied than there are alternative valid design strategies. This should be true, though there are many ways to properly design an ...