Questions tagged [estimation]

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Where does the linear regression assumption that the errors are uncorrelated enter into the proof of Gauss Markov and that Least Squares is BLUE?

I often see that the "Spherical Error" assumption is invoked for Gauss Markov. One of the parts of the assumption is that the variance is constant given $X$. The other is usually that the ...
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Expected magnitude of cosine similarity among $B$ Gaussian vectors?

Take an IID sample of $B$ vectors $x$ drawn from a Gaussian with mean $\mu$ and covariance $\Sigma$. Define the following: $$S=\frac{1}{B^2}\left\|\sum_i^B x_i x_i^T\right\|^2_F$$ $S$ gives the ...
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Why has the admissibility of the Graybill–Deal estimator eluded a proof for so long?

The Graybill–Deal estimator is used to estimate the shared mean of two normals with unknown variances. I understand the literature has proven it is unbiased, but a proof of admissibility has not yet ...
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95% Confidence Interval should mean 95% probability that the interval contains the true parameter [closed]

What the image below represents is true. Black intervals belong to samples which properties allow produce CI that contain the true parameter, and purple intervals belong to samples which properties ...
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+50

Unbiasing estimator of $\|\Sigma\|_F^2$

I have access to samples of some distribution with second-moment matrix $\Sigma=E[xx^T]$ and need an estimate of $\|\Sigma\|_F^2$ (which can be used to set optimal size for LMS) We can use Frobenius ...
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Can there be any situations where MLE performs better than MPS in terms of MSE or Bias?

Cheng and Amin (1983) proposed the maximum product of spacing estimation method as an alternative to maximum likelihood estimation. They stated that MPS behaves better in small sample cases than MLE ...
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Ratio of first two moments of sample covariance eigenvalues

Suppose we form $X$ by stacking $b$ examples drawn from some distribution in $n$ dimensions as rows and look at $\{\lambda_i\}$, the eigenvalues of $\Sigma=X^TX$. Consider the following random ...
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5 votes
1 answer
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How can I fit distribution for data which "almost fits"?

I have a sample for events occurring at certain continuous distances (kilometers), let's suppose emergency calls to hospitals. I have 200k observations, coming from 500 hospitals for an entire month. ...
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Fixed effects or independent models?

Suppose we are running a model, let's say a linear regression model or a logistic regression. Suppose also that we have gathered data for three cities: City A (4000 surveys), City B (5000 surveys) and ...
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1 answer
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Can Calder 2022 find boundaries of multiple connected components?

Calder et al. 2022 (version 2) shows an interesting method for estimating the boundary from a point cloud. One of the impressive aspects of this is the algorithm should be able to find non-convex ...
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Estimating the distribution of attributes over a city

I have various shapefiles of attributes of a particular city, like the location of crimes, traffic accidents and greenspace. I'd like to plot the distribution of these attributes over the whole city, ...
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1 answer
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Parameter estimation in a drifting normal distribution

Suppose we observe $n$ pairs of points $(a_1,b_1),~(a_2,b_3)~..~(a_n,b_n)$. The underlying data generating process is known to be as follows: $u_i \sim N(0,1)$ and $a_i \sim N(u_i,1)$ $b_i \sim N(u_i ...
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Guidelines in selecting weight in weighted squared error loss function [duplicate]

I am new to loss functions in Bayesian analysis and from what I understand are: Squared error loss function - the estimator will be the posterior mean. Absolute error loss function - the estimator ...
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1 answer
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How to verify the convergence rate in Monte Carlo simulation?

Given a iid random samples $X\sim N(\theta,1)$, we have a unknown parameter $\theta$ and its estimator $T_n=T_n(X_1,\dots,X_n)$. If we have strictly proved that the convergence rate is $$ |T_n-\theta|...
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Estimation of the residual covariance matrix

Is the estimation of the residual covariance matrix under multivariate least squares the same as under maximum likelihood in a vector autoregression framework if we assume Gaussian white noise for the ...
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Can I say that $T_n$ is a consistent estimator of $\theta$ by Monte Carlo simulation under this setting?

Given a iid random samples $X\sim N(\theta,1)$, we have a unknown parameter $\theta$ and its estimator $T_n=T_n(X_1,\dots,X_n)$. If we have strictly proved that $T_n$ is a consistent estimator, can ...
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Estimation of mean, variance, proportion for cakes baking in 30 ± 2 minutes, or estimation of parameters of a law for a distribution: is it the same? [closed]

I'm a beginner in statistics and I'm trying to figure things to ensure that I can consider parameters the same manner in two cases : I believe that if we have enlighten a distribution with the ...
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1 vote
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Optimal combination of correlated estimations

Consider two random unbiased estimates $\hat X_1,$, $\hat X_2$ of a parameter (complex number) $x$, with estimation errors $E_1 = \hat X_1-x$, $E_2 = \hat X_2-x$. If the random variables $E_1$, $E_2$ ...
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How to calibrate a multi-dimensional Ornstein-Uhlenbeck process?

There is abundance of literature out there on methods for calibrating a one-dimensional OU process, namely: \begin{equation} dy_t=\kappa(\theta-y_t)dt+\sigma dW_t \end{equation} where $(y_t,\kappa,\...
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Find unbiased estimator for a and b based on X and Y

I need help with the following question(sorry for not formatting, I do not know how): X and Y are random variables, each have standard deviation of 3. The pearson correlation equals to 0.6(in this ...
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7 votes
1 answer
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A Kinder Egg Problem

My friend and I recently saw our old passions for Kinder Surprise toys reignited with a new animal toy line which resembled the old toys we were missing. To our dismay, however, this series did not ...
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1 vote
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Standard errors of Monte Carlo plus linear combination

I'm using Monte Carlo to estimate some quantity $V(x)$. To get an approximation of $V'(x)$ I would use the following $$ V'(x)\approx\frac{V(x+h)-V(x-h)}{2h} $$ so I can simply evaluate it with two ...
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1 vote
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MLE is undefined for "densityless" distribution like Cantor distribution

I thought of a situation where we are given a random variable $X$ that has a Cantor "rescaled" distribution. That means that for a parameter $p>0$, $X$ has CDF $F_X(x)=C(\frac xp)$. This ...
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MLE for distribution having most general form

Consider a random sample {8,4,1/2,1} from a distribution having most general form of the probability mass function f(x,Θ) = (x/Θ)^(Θ A'(Θ)) exp(A(Θ)+C(x)) where A'(Θ) is the derivative of A(Θ) with ...
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4 votes
1 answer
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How to construct a confidence interval for the coefficients of a multivariate regression with dependence between dependent variables?

Suppose we have two linear regression models $y_1=a+bx+\epsilon_1$ where $\mathbb[\epsilon_1]=\sigma_1$ and $y_2=c+dx+\epsilon_2$ where $\mathbb[\epsilon_2]=\sigma_2$. In other words, I am using the ...
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24 views

Estimating the Cramér–Rao bound

Given a random vector $\boldsymbol{X}=(X_1,X_2,...)$, which can be described by the sum of a multivariant Poisson distribution $\alpha P(\boldsymbol{\lambda})$ with a scaling factor $\alpha$ and ...
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Gaussian process regression without a kernel [duplicate]

I am working on a sequential estimation problem that involves a Bayesian update to a multivariate Gaussian prior from a measurement with Gaussian noise. Specifically, I have a mean vector of length n ...
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0 answers
28 views

Proving a result in maximum likelihood theory: accuracy of the quadratic approx. to loglikelihood

I need help proving a result shown in a paper. I am reading Assessing the Quadratic Approximation to the Log Likelihood Function in Nonnormal Linear Models by Salomon Minkin. The paper defines several ...
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We have a sample of random variables constructed from the Euler Maruyama scheme of a stochastic model, is it idependent and identically distributed?

We have the mean reversion stochastic model $dX_t=(\alpha-\beta X_t)dt+\sigma X_t dB_t$, the Euler-Maruyama scheme for this model is $X_{t_i}=X_{t_{i-1}}+(\alpha-\beta X_{t_i})\Delta+\sigma \sqrt{\...
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Condition for the asymptotic non-zero point estimation of the variance

we know that a condition for a non-zero point estimate of the variance for a finite sample is that there exist at least two integers $i,j$ such that $X_i\neq X_j$. In other words $\frac{1}{n}\sum\...
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2 votes
2 answers
82 views

Using the Median to Estimate a Parameter

I am trying to use the Median to estimate the value for the parameter $a$ in the following PMF. \begin{equation} \label{eq1:givenpdf} \mathbb{P}\left[X=\frac{a}{n}\right] = \frac{36}{5}\frac{n^2}{\...
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Confusion about consistency of time series model parameter

Can someone clear this confusion. Lets say I have a time series model: $$X_t \text{ follows Poisson}(\lambda_t)$$ $$\lambda_t=a*X_{t-1}+b*\lambda_{t-1}$$ Then I find estimators for $a$ and $b$ called: ...
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What is the Best Neural Network architecture to estimate a convex function?

I am currently working on a Q learning algorithm for multi-agent systems and sub-classes of Dec-POMDPs .. It has been shown before that the Q value at any time step can be reduced to a piecewise ...
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1 vote
0 answers
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Tradeoff when estimating the variance of the autocorrelation estimator

Autocorrelation estimator Given a wide-sense stationary process $\{X_t\}_{t\in\mathbb{N}}$ one can estimate its autocorrelation defined as $R[k]=\mathbb{E}[X_tX_{t+k}]$ from $N$ observations $\{X_1,...
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Deviation estimated by humans

I'd like to know which type of deviation humans will naturally estimate if asked to. Like, if you give them some points on a line and ask them to estimate the "plus or minus error" around a ...
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1 vote
1 answer
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Iterative proportional fitting with constraints

I am trying to determine if it is possible to conduct iterative proportional fitting with some constraints. To give a dummy example of my goal: Say I had data for two towns, A and B, on the ice cream ...
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Obtaining a national total from total of some cities

We are interested in estimating a total of a variable $x$ on the national range. If the country is divided in cities $C_1,\cdots,C_5$ and the population frame has units in all the five cities, and we ...
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Estimation of total and breakdowns from SRS

Suppose I want to estimate a total and the breakdowns of some variable $x$ using a simple random sample (SRS). From the $N$ units of the population we extract $n$ units by SRS. Let $\displaystyle\bar ...
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How is the objective function of the different flavors of GARCH different?

How does the objective function/likelihood function of these different GARCH variations differ? Is it convex in all cases? Knowing convexity tells me whether some are not possible to find a globally ...
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1 vote
1 answer
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Estimating a population total from a stratified simple random sample

I have a stratified population. I want to estimate a population total $T$ from a stratified simple random sample. I have two strategies: I compute $\displaystyle T=\sum_h N_h\bar x_h$ where $x_h$ is ...
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0 votes
1 answer
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Estimate a parameter that is not in a PDF, makes sense?

I have been reading some posts about the proof of the invariance of MLE because I did not fully understood the proof given in Statistical inference by Casella Berger in the page 320. My doubt is the ...
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2 votes
1 answer
79 views

Double dipping/p-hacking and interim sample size reestimation

When one applies interim sample size reestimation based on nuisance parameter estimates (observe fraction of initial sample size, estimate nuisance parameter(s) based on the obtained data, reestimate ...
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2 votes
1 answer
53 views

Why do we need to Define "Valid" State Transitions in a Multi-State Model?

I was watching this video (https://www.youtube.com/watch?v=Wy-WmY6x4tg) and the presenter mentions (@ 8:10) that in a Multi-State Model, the user is required to specify number of "States" ...
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2 votes
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Probability that interval arithmetic estimate of mean covers the true mean of a normal variable?

Background Suppose there exists a normal variable $X \sim \mathcal{N}(\mu, \sigma)$ where $\mu, \sigma$ are known. Also suppose that you only observe rounded instances $Y = \operatorname{round}(X, d)$ ...
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What is the difference between MVB UMVUE and MVUE.?

Cramer Rao inequality gives MVB and if MVB exist it is MLE. Rao Blackwell gives UMVUE, but isn’t when we have MVB estimator for unbiased it is UMVUE? Then what is MVUE? MVB minimum variance bound ...
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1 vote
1 answer
45 views

Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t \, dt + \sigma X_t \, d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\...
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1 vote
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Applications of Mortality Tables and Life Tables?

Recently, I showed my friend what Life Tables and Mortality Tables are (e.g. https://en.wikipedia.org/wiki/Life_table) - I tried to explain possible uses for these kinds of tables but my friend argued ...
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1 vote
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Applications of "Dose Response" Outside of Biostatistics?

I was reading about a class of models called "Dose Response Models" (https://cran.r-project.org/web/packages/drda/vignettes/drda.pdf) - in the traditional sense, these are typically used to ...
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3 votes
1 answer
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Ideal Settings for Longitudinal Models?

The way I see it, logically speaking - Longitudinal Data (e.g. medical patients being measured repeatedly over a period of time) can have one of two forms: Case 1: All patients are measured exactly &...
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2 votes
1 answer
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Testing and conidence interval in a clinical trial

In a clinical trial, let's say I want to test $$H_0: \mu_1 \leq \mu_2$$ $$H_1: \mu_1 > \mu_2$$ $\mu_1$ belongs to the placebo group and $\mu_2$ belongs to the trt group. I used an independent two-...
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