Questions tagged [estimation]

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565 views

How to calculate the expected value of an estimator?

According to my book : An estimator, say, T, of the parameter $ \theta $ is said to be an unbiased estimator of $\theta$ if $ E\left( T\right) = \theta$. It then explains how to calculate $E\left( T\...
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34 views

Maximum Likelihood Estimator for a given density function

I have the following problem: Assume you observe $Y_1,...,Y_N$ independently from the distribution $f_y$: $$ f_{Y}(y)=\frac{12}{12-\theta}\left\{\begin{array}{ll}-\theta(y-0.5)^{2}+1 & \text { if ...
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1answer
45 views

Relation between OLS, MM and ML

What is the relation between OLS, MM (method of moments) and ML (maximum likelihood)? During my studies, the three concepts got taught completely separated from each other. However, they seem to be ...
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2answers
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Regression with flexible functional form

I am assuming a model of the form $$Y_i=\alpha+\beta X_i+g(\mathbf{Z}_i)+\epsilon_i,$$ here $\mathbf{Z}_i$ is an $m$ dimensional vector and $\epsilon_i$ is i.i.d. white noise. I would like to ...
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39 views

Parameter estimation with differential equation?

I have an unknown function in an ODE as follows: dx/dt=Q(t)-a*x(t), a is a function of time and unknown. I plan to consider a as an unknown values at different time points. Then I use MCMC to estimate ...
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30 views

What are some beginner's references on algebraically structured models, and what's their connection with group actions and Fourier transform?

Since I didn't get an answer, I asked it on mathoverflow. I'm looking at a short-term position, whose project is on estimation in algebraically structured models. It also mentions in the required ...
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66 views

Rate of covariance matrix estimation in Frobenius norm for Gaussian distributions

Given i.i.d. observations from an unknown $d$-dimensional Gaussian distribution $\mathcal{N}(\mu,\Sigma)$, it is "well-known" that $n=\Omega(d^2/\varepsilon^2)$ are necessary to produce an ...
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30 views

Combining Estimates with Actual Values

I am exploring stature (living height), calculated from arm and leg bones, from an archaeological sample. Unfortunately, there are very few complete bones that can be measured and entered into the ...
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22 views

Hidden Markov Model

For part b, would the answer be p = 7/10 since the left hand is biased, we would look at every q that has L and check the observation for every L that has H?
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Estimation of Linear Dynamical System with diagonality constraints

I am trying to estimate the parameters of the following linear dynamical system \begin{align} X_t &= \phi X_{t-1}+\varepsilon_t, \quad \varepsilon_t\sim N(0, \Sigma_\varepsilon)\\ Y_t & = h^...
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91 views

Multivariate Normal distribution Cramer-Rao bound

How to compute the Cramer-Rao lower bound for $\mu$ of a multivariate normal distribution $\mathcal{N}_{p}(2\mu,\Sigma_0)$. Also I would like to know the minimum variance unbiased estimators for $\mu$....
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23 views

Applying Pearson's correlation, hypothesis tests etc. to non-iid data

(This is a very loose question. I'm aware that any theoretically sound answer to this will be specific to $(T,X)$ and involve sufficient assumptions on $X$ such as various ergodicity and mixing ...
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21 views

Where can I find an example of calculating a GARCH model without packages?

Every example I've come across uses predefined packages like 'import arch_model'. I can't determine the method given these black boxes. Thank you.
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1answer
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Estimating the number of apples in an apple tree using MCMC

I'm trying to estimate the number of apples in an apple tree by repeatedly kicking the tree and counting how many apples fall down. This process, I believe, is called removal sampling. The only ...
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27 views

When transforming data, why use resampling to estimate transformed values?

I am working through a machine learning book and am using the caret package in R. There is a function, preProcess, that uses ...
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1answer
36 views

Existence of least squares and maximum likelihood estimators?

In statistical parameter estimation where there is a deterministic and stochastic component to the observation-generating model, do least squares and maximum likelihood estimators always exist? ...
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31 views

maximum likelihood estimation of the variance [closed]

In what situations the maximum likelihood estimation of the variance of distribution can severely ruin the estimation?
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15 views

Does it work by generating more sample by GAN to increase the accuracy of estimation?

As in the estimation theory, more samples can help to enhance the accuracy of the estimation. If we only get a small amount of samples at first, we then use these samples to train a generative ...
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1answer
21 views

Can I build an equation from just few observations?

I have the following data that has 5 observations and 2 missed observatiosn. I want to build an equation that can help estimate the missed observations. ...
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38 views

Unbiased estimator of the reciprocal of a truncated normal distribution

I have a random variable $y$, which is a function of $x$, i.e. $y = \frac{1}{x} - 1$, where $x$ is a random variable with a truncated normal distribution with mean $mu$ and variance $\sigma^2$. In ...
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1answer
57 views

When is cross validation necessary to estimate a parameter?

In 2013, @Donbeo asked whether there were any theoretical results supporting use of Cross Validation to choose the lasso penalty, and was scolded in the comments for asking "a pretty generic ...
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31 views

Do correlation estimates have increased error for small sample data?

When we calculate the correlation between two random variables, we seem confident that the estimated correlation is 100% accurate since I never heard of estimation error being a problem for ...
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29 views

Moving Average Model over error terms

This is a basic question on Box-Jenkins MA models. As I understand, an MA model is basically a linear regression of time-series values Y against previous error terms et,...,et−n. That is, the ...
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14 views

Why do complete ANOVA tables matter?

I'm trained as a mathematical statistician; my degree is technically "mathematics" but I study statistics. So let's consider classical, one-way ANOVA. Textbooks will teach that to perform ...
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48 views

Estimation of CDF in multiple points

Suppose we have a sample $X_1, \ldots, X_n$ of i.i.d. real-valued random variables with an (unknown cumulative) distribution $F$. The goal is to estimate the value of $F$ in multiple points. That is, ...
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2answers
43 views

Except estimation and testing, is there something else to statistics?

Statistical methods can be used for hypothesis testing in various forms, such as null hypothesis testing. They can also be used to estimate the value of some parameter plus an uncertainty of that ...
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40 views

Least Squares for projection matrix estimation produces unwanted behaviour

I am estimating a non-linear correspondence between 3D and 2D points using scipy.optimize.least_squares. I found that setting the method to be Trust Region ...
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0answers
34 views

Covariance matrix computation VECM, Lutkepohl (2005) p.287

In Lutkepohls book "New introduction to multiple time series analysis" (2005) on p.287 is outlined how to calculate your estimated parameters $[\hat \Pi,\hat \Gamma]$ sample covariance ...
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24 views

UMVUE of $\exp(t \mu)$ for $N(\mu, \sigma^2)$ where $\mu$ is not known but $\sigma^2$ is known [duplicate]

I am new to UMVUE calculations and struggling a bit with this question. I want a hint about how to approach the derivation of the UMVUE for $\exp(t [\mu)$ for $N(\mu, \sigma^2)$ where (\mu is not ...
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52 views

distribution of least squares estimates for random design regression

Suppose we consider the least squares problem where the objective is to find $\beta$ that minimizes $(\boldsymbol{y} - \boldsymbol{X} \beta)^T (\boldsymbol{y} - \boldsymbol{X} \beta)$. We know that an ...
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28 views

Constructing confidence interval for estimator with unknown standard deviation and sample size

Suppose you want to estimate $\pi$ using the estimator $\hat{\pi}_n = 4\times \frac{N_{circle}}{n}$, where $N_{circle}$ is the number of points that lie within the unit circle embedded in the square ...
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87 views

How to correct zero cell counts for odds ratio calculations when variable has >2 categories

I am trying to calculate odds ratios for several univariate models with a binomial outcome. The problem is, for some of my variables, I have at least 1 zero count cell. It seems that the Haldane ...
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Parameter estimation for basis function model in Elements of Statistical Learning (ESL)

In the book Elements of Statistical Learning, section 2.8.3 describes Basis Functions, citing an example of a radial basis function as $f_{\theta}(x) = \sum_{m=1}^M \beta_M \sigma(\alpha_m'x + b_m)$, ...
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52 views

Estimating Joint Probabilities from Marginal Probabilities

Given 2 categorical variables and their marginal probabilities of variable C. What is a good simple way of estimating the joint probabilities? Let C be a Boolean T/F survey variable. For example: ...
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9 views

what metrics to use to estimate accuracy of range prediction?

I trained a model that predicts customer's income given the features: age, declared income , overdue total amount active credit limit, total credit limit Metrics used: NIRDM - not in range distance ...
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27 views

VAR($p$) estimation

I am bit stumped by this result. Source: Remark on page 46 of Multivariate Time Series Analysis by Rsay. ... one can obtain the GLS estimate of a VAR($p$) model equation by equation. That is, one can ...
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1answer
57 views

Monte Carlo standard error for a sum

Suppose that I want to compute $E[X+Y]$ using Monte Carlo simulation and compute the standard error. (Note: $X,Y$ are not necessarily independent) The standard way to do this is to Consider the ...
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5 views

relationship between the confusion matrix of a detector and the variance of an estimator

Say I have a detector with a confusion matrix. Also, I am interested in the estimation of the number of the detected cases $(\hat{N})$ rather than in the parameters given by the confusion matrix. For ...
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1answer
45 views

Question about estimating the standard error of the regression- notation and intuition

in a standard linear regression frame work: $y_{i}=\beta x_i + \epsilon_i$ when calculating standard errors, we find an unbiased and consistent estimator of $var(\hat{\beta})$. Assume spherical errors....
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1answer
36 views

While simulating the value of a double integral , why do we need to draw different samples everytime?

Suppose I want to simulate the value of the integral $\int_{0}^{1} \int_{2}^{3} 2xy \ dx dy$ using Monte Carlo methods. So, now, I draw a random sample from $U_1,U_2,...,U_n$ from $U(0,1)$ and for ...
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1answer
42 views

Asymptotic distribution after replacing quantities by consisent estimators

Suppose that we wish to estimate $T(\theta_1,\theta_2)$, a continuous function of several parameters. Suppose that we know the asymptotic distribution when $\theta_1$ is replaced by an estimator $\hat{...
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1answer
34 views

SEM: can GLS estimation be used under severe nonnormality?

In Randall E. Schumacher's and Richard E. Lomax's book "Beginer guide to SEM", the writers keep saying that if non-normality is sensed, where you can't use ML, you can go for GLS/ADF/WLS. ...
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1answer
80 views

In SEM, what is the difference between ADF and WLS estimation methods?

In some books both WLS and ADF are considered different methods. In other books, they acknowledge that ADF = WLS, so they are used interchangeably throughout the book. How solve this confusion?
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1answer
28 views

How to estimate population statistics for each category?

I have data from a simple random sample having n elements which I’m using as an unbiased estimate for my population statistics (mean). My population can be divided into k categories, and I know the ...
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1answer
47 views

How can i find bias of estimator for specific value?

I have $X_1,...,X_n~ Ber(p)$ with MLE estimator $\hat p$ which is equal to sample mean. I need to find bias of estimator $\hat p(1 - \hat p)$ for $p(1-p)$. I presume $p(1-p)$ is variance of my RVs, so ...
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41 views

What test to use for threshold

I've got a problem where, with my basic knowlegde of statistical tests, I'm not sure how to solve it. I have a dataset with data on the execution of a process. This includes: Time the process lasted ...
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48 views

How to calculate confirmatory factor analysis by hand?

I am trying to learn how confirmatory factor analysis works and the way I learn is best by understanding how the calculations work by hand using a pen, paper, and a calculator, and then replicate this ...
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36 views

Empirical estimation of conditional distribution $Y|X$ at the boundaries of X

I want to estimate conditional distributions of Y | X. Where X contains several continuous covariates. I'm coding in R. I tried several methods so far, but none gives me entirely satisfactory results ...
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28 views

How to solve a negative binomial regression?

From this https://content.wolfram.com/uploads/sites/19/2013/04/Zwilling.pdf : But I don't know how to solve it . I know let the derivative of the function L with respect to $\alpha$ and $\beta$ equal ...
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9 views

Runtime Estimation for Multi-threaded processes with varying loads

This is much more of a computer-related question but I believe this requires statistical knowledge (which I'm not well versed and currently reading on the basics) so i hope you can bear with me. I ...

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