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Questions tagged [estimation]

Any statistical process which seeks to approximate an unknown value, such as a distribution, a point estimate (e.g. mean), or confidence interval.

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1answer
18 views

How can we estimate (conditional) probabilities from a dataset?

Given three random variables $X$, $Y$ and $Z$, how can we estimate $P(X)$, $P(X\mid Y)$ and determine whether $P(X \mid Y, Z) = P(X \mid Z)$ from a dataset (of e.g. $N$ observations) which contains ...
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0answers
7 views

Estimator of covariance in continuous time

In the discrete time (stationary) case, the covariance $R(\tau) = E\{(X_{t+\tau} - \mu)(X_{t} - \mu)\}$ can be estimated using $\frac1n \sum\limits_{j=1}^{n-\tau} (x_{j+\tau}- \hat{\mu})(x_{j}- \hat{\...
2
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1answer
287 views

Difference between function and distribution?

I know this is dumb question, but i am confused to understand it. I have constant function as $$y = exp(-x^2)$$ This also represent gaussian distribution with mean zero and variance of 0.5 Now if ...
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0answers
14 views

Calibration vs. Estimation in RBC Models [closed]

What are the pro(s) and con(s) of estimation and calibration in RBC models in macro. Can you compare it?
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2answers
38 views

Maximum Likelihood Estimator (MLE) for $2 \theta^2 x^{-3}$

I'm having a bit of trouble solving this. $$ f(x_i; \theta) = 2 \theta^2 x_i^{-3}, 0 \le \theta \le x_i \lt \infty $$ I start by finding $f(\textbf{x}; \theta)$: $$ f(\textbf{x}; \theta) = \prod{f(...
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0answers
18 views

Strange rugarch error: $ operator invalid for atomic vectors? [closed]

So, I am trying to make a huge nested for loop (optimizations be left for later) to fit all of the GARCH models available from rugarch. This is my MLE that ...
0
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0answers
13 views

Determining variance of UMVUE

Let $X_1,...,X_n$ be iid with pdf given by $f(x;\theta)=\frac{log\theta}{\theta^{x-1}}I(x>1)$. My task is to determine if the $\mu=E[X]=1+\frac{1}{log\theta}$ can be estimated efficiently, i.e. if ...
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2answers
52 views

Estimating the min and max of a distribution

I have a measurement problem where I am attempting to measure the minimum and maximum height of a surface by taking point samples of heights. If I then look at the distribution of all height values, ...
1
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0answers
23 views

Adjust two coordinates based on the distance between them [closed]

I have two positions $\overline{X_1} \in \mathbb{R}^{2}$, $\overline{X_2}\in \mathbb{R}^{2}$ with the respectively measurement errors $\Sigma_{X_1}\in \mathbb{R}^{2\times 2}$ and $\Sigma_{X_2} \in \...
1
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1answer
17 views

Is the boundary of an HDR a region of the sample space with equal density value?

After reading this question, I read in the reference provided (Hyndman, 1996, The American Statistician) the following: It follows inmediately from the definition that the boundary of an HDR ...
3
votes
1answer
36 views

Name of mean- or median-like values?

Consider the data points $x_i \in \mathbb R$ for $i=1,\ldots,n$ as well as following definition: $$\hat x_c := \underset{{r\in \mathbb R}}{\operatorname{argmin}}\sum_i \vert r - x_i \vert^c $$ This ...
1
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1answer
109 views

Credibility evaluation - how to model conditional continuous density from multiple variables of various types?

I recently got dataset for 37000 households with declared income and a few dozens of other variables of various types: continuous, discrete, binary. The task is to automatically (unsupervised) ...
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0answers
16 views

Observation Operator - Data Assimilation

In data assimilation, one assumes the existence of a observation operator $\mathcal{H}$ that maps the model-state vector $\bf{x_b}$ to $ \bf{y_b}$ (the model-equivalent of the observations $\bf{y_o}$) ...
3
votes
1answer
59 views

Do GEE and GLM estimate the same coefficients?

In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are $$\sum_i^n (\frac{\partial \mu_i}{\partial \eta_i}) \frac{...
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0answers
26 views

Frequentist vs. Bayesian bias-variance decomposition

Iv'e read the answer to this related question and still have some issues. Suppose that given some data $X$, we want an estimator $\hat{\theta}$ for some parameter $\theta$. A common approach is to ...
1
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1answer
25 views

How to improve a model that is consistently underestimating

I've been trying to predict house prices (real data from my country) and I noticed that initially, errors are centered around zero, but around the $2,500,000 mark, the model starts underestimating ...
3
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1answer
217 views

Which estimation technique minimizes the MAPE?

Suppose we have two estimation techniques: Linear Least Squares, which aims to minimize squared residuals Least Absolute Deviation, which aims to minimize absolute residuals We have a model, which ...
2
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0answers
26 views

Is there any case in which unbiased but larger MSE estimator preferred to biased and smaller MSE one?

Let saying we are interested in a population mean $\mu$ and we have two estimators $\hat{\mu}_{n}^b$ and $\hat{\mu}_{n}^{u}$ defined on $n$ samples such that $\hat{\mu}_{n}^b$ : biased (i.e, $\mathbb{...
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0answers
14 views

How to estimate average user review ratings?

For example, in Amazon Books book A : 1 rating of 5. Average rating of 5 book B : 50 ratings. Average rating of 4.5 If use naive average, then book A is higher than book B. However, we all know that ...
3
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0answers
34 views

regression analysis with both slope and value data

I have measured data $\{x_i, y_i, y'_i\}$, to which I would like to fit a polynomial $y=a x^2 + bx + c$ and $y' = 2ax + b$. It occurs to me that the regression problem of fitting $y=a x^2 + bx +c$ to ...
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1answer
27 views

Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
3
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2answers
103 views

Independence of events in real-life data

Most of statistical methods (if not all) rely on independence of events. How do we know that this assumption is valid in real-life problems like clinical trials or web crawling? What might be the ...
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0answers
22 views

Skewness and Kurtosis estimation methods [closed]

In finance time series literature ARCH and GARCH models are used to get an estimate of volatility. Are there similar models to estimate skewness and kurtosis?
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1answer
21 views

VAR estimation-How to interpret the results?

I have these results when I estimate a VAR with two variables:Growth and Debt and p=2.How to interpret the result for each equation? Thank you. VAR Estimation Results: ...
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3answers
140 views

Deriving likelihood function of binomial distribution, confusion over exponents

This question focuses on a specific aspect of this one: How to derive the likelihood function for binomial distribution for parameter estimation? In my own derivation, I start with: $$f(x\mid p) = ...
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0answers
9 views

Definition of a 'design adaptive' fit?

When studying non-parametric regression, I've been told that local linear fitting is often better than local constant fitting at the job of estimating regression functions because local linear fitting ...
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0answers
15 views

Reference request: Least Squares Concentration Bounds

I've recently become interested in parameter estimation for regression and time-series type models and I often encounter, and indeed need to understand, results using concentration of measure-type ...
2
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0answers
15 views

Estimating the number of tokens in a pile

We have a pile of tokens numbered from $1$ to $n$, where $n$ is unknown. $k$ tokens are drawn from the pile at random without replacement(i.e. the numbers that we get from tokens are unique). Say the ...
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0answers
12 views

Simulating size distrotion with a VAR(1) model

Firstly, I am not asking for code, I would like the intution of how I would do this. I am testing for size distortion. I have estimated and VAR(1) model and I have the parameters. I want to ...
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0answers
66 views

Simultaneous estimation of a group of linear model (regression) parameters

Suppose $y=ax+z$ where $x, y, z$ are random variables with range in $\mathbf R$, $\mathbf E[x]=\mathbf E[z|x]=0$ and $a$ is a constant. Note the distribution of $z$ conditioned on $x$ depends on $x$. ...
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0answers
9 views

Parzen density estimates convergence

I am trying to understand why $lim_{||u|| \rightarrow+\infty}{\varphi(u)}\prod_{i=1}^{d}u_{i} = 0$ is necessary for convergence of Parzen density estimates. Similar question has been asked here ...
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0answers
21 views

Error variance estimation with weights

I am using data which include a (sample-)weight for each observation, i.e. the data is from a survey that has weights to make the sample representative for the US-population. I perform OLS to get some ...
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1answer
37 views

The UMVUE of ratio of parameters for two uniform distributions,

Let $X_1,\ldots,X_m$ be i.i.d. having the uniform distribution $U(0, \theta_x)$ and $Y_1,\ldots, Y_n$ be i.i.d. having the uniform distribution $U(0, \theta_y)$. Suppose that $X_i$’s and $Y_j$’s are ...
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0answers
16 views

Efrons enhanced bootstrap - estimate calibration equation parameters

"The bootstrap method is as follows. From the original X and Y in the sample of size n, draw a sample with replacement also of size n. Derive a model in the bootstrap sample and apply it without ...
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0answers
33 views

Mean and variance of the median estimator [duplicate]

We are given an i.i.d sample of size n, $X_{1},...,X_{2}$. We also know that for all i, $EX_{i}=\mu,Var{X_{i}}=\sigma^{2}$. Let us define $\hat{X}_{median}$ as the median of these samples: $\hat{X}...
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0answers
39 views

Estimating a MS-ARMA(p,q)-GARCH(r,s) parameters via MCMC

I am currently working on a MS-ARMA-GARCH model proposed by Dhiman das on this paper, and trying to fit it on simulated data. So far I understand the model and its construction, but I'm having a hard ...
4
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1answer
108 views

$X_1,X_2,\dots,X_n \overset{\text{iid}}{\sim} N(\mu,\sigma^2)$. Derive a confidence interval for $\mu+\sigma$ and $\mu/\sigma$

I know how to find confidence interval for each of the parameters $\mu$ and $\sigma$ resectively but stuck in finding for the parametric functions above.
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2answers
68 views

Where does the white noise come from in MA(q) model?

I'm having trouble understanding the intuition of the moving average model. How does summing up a bunch of white noises related to predicting your particular time series data? Suppose I have a MA(q) ...
3
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0answers
62 views

Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...
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0answers
19 views

How to create the initial ensemble samples for EnKF

As we know, for the ensemble Kalman filter (EnKF), we need to create a set of samples in the beginning and then to run the predict and analysis step. But for now I have a question of how to create the ...
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0answers
24 views

number of samples needed to determine quantile

I am trying to estimate 16th, 50th, and 84th percentiles of some quantity $b$, and it would be helpful to know if I have enough samples to do so reliably (say, at the 1% level). Sound like a beginner ...
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1answer
50 views

arbitrariness in bootstrap bias estimation

The bootstrap estimates bias by applying the "plug-in" principle to $$E(\hat{\theta}_n) - \theta$$ I got this knowledge from p.124 of Efron, Tibshirani, 1994. equation(10.1) $\text{bias}_F=E_F[s(\...
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0answers
27 views

How does one estimate parameters in a GARCH-M(1,1) model?

Say you have a GARCH-M(1,1) model as follows: $y_t = \beta y_{t-1} + \delta h_t + \epsilon_t, \quad \epsilon_t \sim N(0, h_t) $ $h_t = a_0 + a_1 \epsilon^2_{t-1} + b_1 h_{t-1}.$ How exactly does ...
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0answers
33 views

Consistency of estimators

For $1\leq i\leq K$, I have an estimator of $\mu_{i}$ given by $\hat{\mu}_{i}=\frac{1}{K}\sum_{j\neq i=1}^{K}\frac{Y_{ij}}{n_{ij}}$, where $Y_{ij}\sim N(n_{ij}(\mu_{i}-\mu_{j}),\sigma^{2}n_{ij})$. ...
1
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0answers
19 views

Understanding Rao-Blackwell [duplicate]

From Casella and Berger: Let $W$ be an unbiased estimator of $\tau(\theta)$ and let $T$ be a sufficient statistics for $\theta$. Define $\phi(T) = E[W|T]$. Then $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
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0answers
38 views

Estimating sample variance

Suppose that we have 2 independent samples $X_{11}, X_{12},.., X_{1n_1}$ and $X_{21}, X_{22},.., X_{2n_2}$ from a normally distributed population with $n_1<n_2$. Does that mean that the sample ...
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0answers
13 views

Effect of estimate of one variable on estimate of other variable

I have a linear equation: as Y = -4 + 5X1 + 0.9X2 +0.5X3; Suppose correlation between X1 and X2 is 0.7. Since X1 has more predictive power than X2, regression picked X1 with more weightage. X2 was ...
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0answers
17 views

How can I not show the initialization of the estimation in the Extended Kalman Filter?

I'm making estimates through the Extended Kalman Filter and I have a problem related to the vertical axis of my figure, it's too big, so I can not see population dynamics. However, I wish it did not ...
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0answers
18 views

Identification and estimation of my structural model with a latent variable

I am having some trouble trying to identify the parameters in the following structural model that I am trying to estimate. $$ y = a'x_1 + \beta\eta + \epsilon_1 $$ $$ \eta = b'x_2 + \delta T+\...
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1answer
71 views

Iterated estimation of Taylor series

Say your data generating process is given by the function $y=f(x|\theta)$, where $y$ and $x$ represent variables (data) and $\theta$ represent parameter(s). For convergence reasons (e.g. $f(\cdot)$ is ...