Questions tagged [estimation]

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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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Do we need to check normality of the residuals when using 'CSS' method to fit ARIMA model?

I'm using auto.arima to fit my model. When I used the default CSS-ML method, I noticed that the residuals are not normal. So I ...
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OLS estimator question: using a subset versus using a dummy-interacted variables

Suppose that we are interested in the following model: $$y_i=\beta_1+\beta_2x_{i2}+\beta_3x_{i3}+u_i$$ Here, there is a dummy variable $d_i$. I am wondering whether the following estimators are ...
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VAR coefficients

Say I have a VAR(p) model without any noise (a multidimensional AR model without noise). How would I go about calculating the coefficients that are MSE optimal? Is there an extension to the Yule ...
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Nonlinear constrained optimization for a CIR model

I want to calibrate a CIR model which is commonly used to model the evolution of interest rates. Briefly speaking, we know that its dynamics is of the form \begin{equation} r_t = \kappa (\theta - r_t) ...
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Single-sample self-normalized importance weighting

Self-normalizing sampling schemes (https://artowen.su.domains/mc/Ch-var-is.pdf) seem to require at least two samples to give non-trivial weightings under an importance sampling distribution. Is there ...
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Markov Chain Monte Carlo with known normalisation

I would like to compute the expectation value $\langle O \rangle = \sum_x P(x) O(x)$ of some random variable over an extremely large sample space that I cannot simply exhaustively go through. Usually ...
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Infinitesimal Robustness, influence function of $T$ at $F$

This text is taken from Introduction to robust estimation and hypothesis testing. Wilcox R. First I will write down the description that leads to definition of relative influence on $T(F)$ and then I ...
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Translating Parameter Estimation Principles of Quantum Weak Measurement to a Classical Problem

I am not an expert in quantum theory. I want to carry out some parameter estimation on a set of data I have. I have a model for the data with the parameter(s) of interest as variable(s). The data ...
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Renewal counting process with inter-arrival time gamma distribution: Model estimation

Let's start with the Poisson process: If $N_t$ is a Poisson process with parameter $\lambda$, then we know that the inter-arrival time distribution is an exponential distribution with parameter $\...
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How to efficiently estimate number of individuals with n+ successes from a series of bernoulli trials?

I have a situation where I need to estimate the number of persons exposed to a given event n or more times. For each person, I have an array of probabilities ...
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Statistical Inference: Definition of contrast function

Reading a paper recently regarding results on parameter estimation and I came across the terminology "contrast function" which was a function constructed out of a sample. If I compare it to ...
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If $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, how to prove that the MLE of $p = p_1 - p_2$ is $\hat{p}_1 - \hat{p}_2$?

Suppose $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, where $X_1$ and $X_2$ are independent, and let $p = p_1 - p_2$. How can I prove that $\hat{p} = \hat{p}_1 - \hat{p}_2$? (...
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Using IV when regressor is not endogenous

Suppose I have a single regressor model and the regressor itself is uncorrelated with the error term. If I were to use IV estimation to estimate the coefficient, would the estimate be incorrect, and ...
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Finding covariance matrix for bootstrapped errors in OLS

Let's say we have matrix $x \in \mathbb{R}^{n \times k}$, $y \in \mathbb{R}^n$ and $\beta^*$ vector, which $\beta^* = \arg\min_\phi\sum_i (y_i - x_i\phi)$, i.e. we have classic regression problem and $...
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Minimax testing vs estimation for normal means (and other parametric problems)

Suppose $X_1,\ldots,X_n\sim N(\mu,\Sigma)$ with $\Sigma$ known (equivalently, $\Sigma=I$). Consider the problems of (a) Estimating $\mu$ (say, in $\ell_2$) and (b) Testing $$ H_0: \mu=0\, \\ H_1: \mu\...
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Is inference about interior bear population from observation boundary an un"bear"able estimation problem?

The following map shows a density-coloured grid of observations of American black bear reported on iNaturalist. In the top half of the province you can see a loose boundary of observations around an ...
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Estimating experimental resolution of an instrument from data

Say I have measured some quantity $x$ with an instrument producing a set of observations $x_i$. I don't have the specifications of my instrument, and I have to estimate the experimental resolution of ...
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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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Confining the function approximation of NN

I was recently studying about LMSE estimates and came across an idea. Is there a technique to reduce the hypothesis space of a Neural Network by introducing regularisation techniques of some sort ...
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2 answers
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Markov Chain Monte Carlo doesn't converge

I have a synthetic measurement model that looks like this, $$ x(t) = e^{j u t \frac{4}{\lambda}}, $$ $\lambda$ is a constant. $$ z(t) = x(t) + n(t) $$ The quantity $j = \sqrt{-1}$, the imaginary unit. ...
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Bayesian estimation under transformation on the paramater

Consider the classical model Normal-Normal-Inserse-Gamma model: $$ x=(x_1,...,x_n)|\mu,\sigma^2\sim N(\mu,\sigma^2)\,\,(iid),\,\,\mu\sim N(m_0,\tau),\sigma^2\sim IG(a,b), $$ where $m_0,\tau,a,b$ are ...
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How would we define a "model" in terms of its relation estimators and statistics?

I found this to be an interesting post but I want to hone in on the definition of a model. It defines a model as: the function (or pooled set of functions) that you may accept or reject as being ...
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Estimating parameters GARCH model

Suppose I have the following model \begin{equation} R_{i t}-r_{t}=\alpha_{i}+\beta_{i}\left(R_{m t}-r_{t}\right)+s_{i} S M B_{t}+h_{i} H M L_{t}+\varepsilon_{i t} \label{eqn:egarch} \end{...
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3 votes
1 answer
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Sample proportion vs mean proportion for sample size estimation

Context I would like to estimate the sample size needed for an experiment. I’m testing a feature on a website and would like to detect a significant change between different variants . One control and ...
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Comparison between true simulated value and parameter estimation

I would appreciate any help regarding the following problem: Assume I am simulating time series data, for example of a sinusoidal wave with the equation: y(t)=A*sin(2*pi*f*t) Now, I use this model for ...
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1 answer
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What is the variance of $s^2$?

I am trying to calculate the variance of $s^2=\frac{1}{n-1}\sum (x_i-\bar x)^2$. So what I want to find is $ Var(s^2)$. I have seen different posts, but many of them seem to make the assumption that ...
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3 votes
2 answers
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Is the parameter of interest uniformly distributed over the confidence interval?

Let's assume that we are estimating a parameter of the population (e.g., mean height or weight). We've gathered our sample and calculated confidence intervals around the parameter of interest. Are we, ...
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Finding the correlation between comments within the same thread

I have a collection of data of numerical ratings of comments from threads on a website. I want to determine how strongly these ratings are correlated with other comments within the same thread. More ...
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Is the within estimator from plm() estimated by Ordinary Least Squares (OLS)?

I have a panel data set with `N = 17 Spanish regions and T = 32 years. I performed some models like this one with the plm() function and the within estimator for individual fixed effects: ...
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3 votes
1 answer
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Proof that $g(p)$ unbiasedly estimable only if it is a polynomial (Binomial Distribution)

In Lehmann-Casella (Theory of Point Estimation) they state without proof that if $T \sim Bin(n,p)$, then $g(p)$ is estimable only if it is a polynomial in $p$ of degree $\leq n$. How does one go about ...
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Parameter Estimation on S1 model

I have the following model of creating a random graph on the circle: First, N nodes are uniformly distributed on the circle of radius $N/(2\pi)$ to give a node density of $1$. We sample the expected ...
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How to estimate the parameters of a hypergeomtric distribution?

I am looking for some guidance on how to estimate the parameters of a hyper-geometric distribution, based on a random sample. For example, if I generate a distribution as follows: ...
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Best estimate of a variable from two observations with different efficiencies

Problem Suppose to measure the frequency of a certain rare event (e.g. particle count) with two instruments $I_1$ and $I_2$ for a time $\bar{t}$, the same for both instruments. We expect the same ...
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Why does $E(\epsilon_ix_i)$ have the best linear predictor and $E(\epsilon_i|x_i)$ have the best predictor

Why does $E(\epsilon_ix_i)$ have the best linear predictor and $E(\epsilon_i|x_i)$ have the best predictor? Each $x_i$ is a $1\times K$ vector. All of information I could understand is $E(\epsilon_i|...
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2 votes
1 answer
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Student's t-distribution and ML in R [duplicate]

I am a newbie to R and am trying to learn it. I have some financial data which is normal distributed. Now, the Student's t-distribution is given by $$g_v(z):= \...
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Friendly reference for point estimation that covers the following

I took a mathematical statistics class in college which covered the following: Cramér–Rao bound Lehmann–Scheffé theorem Rao–Blackwell theorem I found these theorems very beautiful and want to ...
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1 answer
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Regression specification: what if one regressor is function of another

Consider the following regression model $$ Y_i=D_{i}(\alpha_1+\beta_1 X_i)+(1-D_i)(\alpha_2+\beta_2 X_i)+\epsilon_i $$ where $D_i$ is a binary variable. Suppose the researcher has an i.i.d. sample $\{...
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Bayesian estimator under transformation of the parameters

Suppose we have $x=(x_1,...,x_n)|\mu,\sigma^2\sim f(x_i|\mu,\sigma^2)$ $iid$, also let $\mu\sim p(\mu)$ and $\sigma^2\sim \pi(\sigma^2)$ be prior distributions. Here $f,p,\pi$ are generic ...
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Using s to estimate $\sigma$ when finding the sample size in confidence interval questions

I am trying to learn sample confidence interval for $\mu$ , in this topic , there is a subtopic which is finding the sample size. I know that if $\sigma$ is given (standard deviation of population) , ...
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Empirical variance of simulation estimate

Consider the following quantity of interest: $$I[a,b]=\int_{a}^{b}g(\theta)h(\theta), \ldots (1)$$ that is, the expected value of some function $h(\theta)$, of $\theta$ distributed $g(\theta)$. ...
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3 votes
1 answer
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How can population variance be estimated from a bivariate sample?

Let's assume a bivariate population with a correlation $\rho$ and a common $\sigma$ so that $\Sigma = \sigma^2 \begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}$. I would like to know the ...
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2 answers
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How to calculate a sense of error for estimated number of events with known probabilities?

I have a long series of entities: x1, x2, x3, ... xn for each of which, there is a probability of an event occurring. The probability for each x may be different, ...
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How to show Neyman orthogonality of the score

In section 4.1 of this paper, the authors talk about the PLR model: \begin{aligned} &Y=D \theta_{0}+g_{0}(X)+U, \quad E_{P}[U \mid X, D]=0 \\ &D=m_{0}(X)+V, \quad E_{P}[V \mid X]=0 \end{...
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Follow-up to "Standard deviation of standard deviation"

This question asks about a way to calculate the standard deviation of a standard deviation. The answer with most votes derives a formula for an unbiased estimator of $\text{SD}[s]$, which confuses me ...
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2 votes
3 answers
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We're estimating mean, variance, proportion or compare samples with them. I understand for mean and variance. But why is it required for proportions?

I have learned how to estimate a mean, variance or proportion from a sample. and also, how to compare those for samples. I'm understanding well why we might need to estimate or compare means or ...
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Finding Monte Carlo estimate $\hat{K}$ using Monte Carlo integration

Let $$f\left(x\right)=K\left[sin^2\left(6x\right)+3cos^2\left(x\right)sin^2\left(4x\right)+1\right]e^{-\frac{x^2}{2}},\:-\infty <x<\infty $$ be the probability density function of a random ...
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Pivotal quantity of exponential model

I'm attempting to solve this problem but it's been driving me crazy. The goal it's to find a pivotal quantity of the model below and a symmetric confidence interval for $\theta$ with an $(1-\alpha)$ ...
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Kernelized Decision Trees

I came across a simple example that shows where decision trees may have difficulty solving a classification problem efficiently: "[...] For example, if we have a two-class problem and the ...
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does assumptions effect the bias or variance?

in machine learning text it is often said that assumptions affect bias like the following text from Kevin Murphy: "Given the large variety of models in the literature, it is natural to wonder ...
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