Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

58 questions
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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 experiment. What are the best "problems" that truly…
EngrStudent
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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 aspect of the proof is that you do not need…
Lucas Roberts
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Justification for and optimality of $R^2_{adj.}$ as a model selection criterion

In a recent thread, use of adjusted $R^2$ ($R^2_{adj.}$) is mentioned in the context of model selection, e.g. The adjustment was invented as a solution to problems caused by variable selection Question: Is there any justification for using…
Richard Hardy
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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 can compute from the data $ \{y_j\}_{j=1,\ldots, L}$…
ArtificiallyIntelligence
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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 $(x_1,y_1),...,(x_n,y_n)$, where $n=14$, and I have chosen the…
JJFM
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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 be $B_k$ = {1,4,67,3,12}. There are two robots,…
Keith
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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 following condition: $$ \hat\theta(S) = \arg…
Richard Hardy
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Biased estimator obtained by optimal experiment design

I am using a model-based approach to infer the parameters of a given system. Namely, I represent my system by a model $\mathcal{M}$ with parameters $\theta$. To estimate the true value of $\theta$, I record the output $\mathcal{D}$ of my system to a…
Camille Gontier
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Is it possible to show that this estimator has minimum variance?

Doing some exercises I stumbled upon this tricky one: Suppose we have an independent random sample $(X_1, ... , X_n)$ with $X_i \sim Poisson(\lambda)$. Define $\theta = e^{-\lambda}$. Let $$ \delta_X = \begin{cases} 1 \quad if \quad X = 0 \\ 0…
Simone Vernengo
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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 understand the proof of the Gauss Markov theorem.…
user56834
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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 $\min_{\mu}\{F_{V_5}(1.2763) = 0.25\}$. By performing…
Emil
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Optimal prediction under squared percentage loss

I have to find an answer on the following question but I am struggling: Consider a leaf of a decision tree that consists of object-label pairs $(x_{1}, y_{1}), \dots, (x_{n}, y_{n})$. The prediction $\hat{y}$ of this leaf is defined to minimize the…
ghxk
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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 quantity, i.e. that it's a least favorable prior…
Maverick Meerkat
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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? the optimal value of h can be calculate (sic)…
flashing sweep
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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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