Questions tagged [estimation]

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Formulae to Convert between z Critical Value and Confidence Level [duplicate]

What are the two formulae to convert between the two? For example, I have a z critical value of 3.00. With what formula can you convert this to ...
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What's the meaning of estimator standardization?

Just to give some context: Two different types of alloy (named here A and B) are used to manufacture experimental specimens of a small tension link to be used in a certain engineering ...
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Posterior of risk ratio as ratio of samples from posteriors of risks

I want to sample from the posterior distribution of a risk ratio. My idea is: define the posterior of risks $R_A$ and $R_B$) for two subgroups (e.g, exposed and non-exposed) in my data, using Beta(1....
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In SPSS, how to obtain a single prediction of the dependent variable for each fixed factor, rather than for each random factor combination?

I have a number of thing-types X and for each type want to obtain a single estimate of its true value Y. When I have measured the same thing-type several times in the same session and several times ...
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1answer
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Parameter Estimation via MCMC

In general, we use MCMC method to sample from a distribution which is hard to compute. In Bayesian setting, we sample from the posterior distribution of the random parameters defining the underlying ...
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39 views

Finding Average Wait Time From Number of People Waiting

Problem: We are trying to estimate the average wait time for a process. Only data we have is how many people are in the system at a given time, how many have entered this period, how many exited this ...
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1answer
17 views

MAP, MLE and parametrised data

It is often said that maximum likelihood is used to obtain estimates of distrubtion's parameters. However, what is unclear is whether it will produce consistent estimate parameters other than those of ...
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constract a 95% confidence interval and deter mine lower and upper limit [closed]

suppose for the entertainment indusrty as a whole it is known that the variance of wages for full time employees are birr 1764 per hour a random sample of 196 employees frm entertainment industry ...
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1answer
58 views

Classification vs. Prediction

Recently there has been much discussion of the distinction between classification and prediction. The main claim as I understand it is that prediction must be a probability - specifically $E[Y|X=x]$, ...
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1answer
24 views

Standard deviation in Direct-Sampling

In a computational physics course, I was asked to do direct-sampling for the numerical value of $\pi$ and then I estimated the standard deviation of $\pi$ , ...
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12 views

Statistical method for estimation of Penetrance

I was thinking on a problem: there is a rare disease, associated with some ultra-rare genomic variant (which means ~ 1 out of millions), and there are estimations of penetrance (how many individuals ...
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GARCH mean parameter estimation in R (rugarch)

I cannot seem to figure out how rugarch calculates the parameter $\mu$ when fitting a specific model For example: ...
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AIC and model selection with multimodel

For a simple example, I am doing a multi-model regression/likelihood estimation. The data is $(y_1,y_2,x_1,x_2,x_3,x_4,x_5)$. The first model (A) consists with two regressions: $y_1=e^{a_1x_1}+e^{...
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why the sigma(volatility) is so small when useing the maximum likelihood estimation for vasicek model

This is my monthly data(in percentile)-2year japan government bond's yield to marturity from year 2001 to year 2019.And my LL function comes from "Maximum likelihood estimation using price data of the ...
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How are SARIMA models (which are non-linear in coefficients) generally estimated?

Very simple question but couldn't find answer to this online. Taking $(1,0,0)(1,0,0)_{12}$ as an example, $$(1-\Phi B^{12})(1-\phi B)Y_t=\epsilon_t$$ $$\implies Y_t=\phi Y_{t-1} + \Phi Y_{t-12} - \...
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1answer
18 views

Multiple regression when the dependant variable is unmeasured or hidden

Say I was measuring the individual performance of each of a group of athletes every week. I measure things like running speed, jumping height, grip strength etc. I want to use these scores with ...
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1answer
62 views

Convergence of kernel density estimate as the sample size grows

Let $X\sim\text{Normal}(0,1)$ and let $f_X$ be its probability density function. I conducted some numerical experiments in the software Mathematica to estimate $f_X$ via a kernel method. Let $\hat{f}...
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1answer
40 views

Constrained MLE

How to write R-code for obtaining MLEs of a ,b and $\theta$ for the density function \begin{equation} f(x)=\theta(a+bx)e^{-(ax+\frac{1}{2} b x^2)}\left(1-e^{-(ax+\frac{1}{2} b x^2)}\right)^{\theta-1} \...
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how can I get the P-value for each parameter in vasicek model then do the simulation in Excel? [duplicate]

I use the solver in Excel to estimate the parameter, the out put is b=0.001153,a=0.095516,sigma=0.0013. I follow the steps at https://www.youtube.com/watch?v=X17cpkkwG_4 The method is the Maximium ...
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Can measuring a linear combination of signals improve SNR?

The system is described as follows: $y= max(M x + n,0),\quad n|x \sim N(0,\sigma _0^2+\alpha M x)$ With parameters: $ x\in \mathbb{R}_{q\times1}^+ ,\quad \mathbb{E}[x_i]\approx\sigma_0, \quad M\in[...
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Why one of my estimated paramater (sigma) of vasicek model is so large in R?

I use the yield to maturity of 2year, 3year, 5year, 7year japan government bond from 1989-2019 as my data (i.e.,the name of my data is vasicdata), they are all daily data, and each year contains 261 ...
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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 $$ \...
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Variance estimator that is optimal under absolute loss

Given a random i.i.d. sample from a population with a finite variance $\sigma^2<\infty$, what estimator of $\sigma^2$ is optimal under absolute loss? $$ \arg\min_{\hat\sigma^{2}\in F}\mathbb{E}(|\...
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Maximum Likelihood Estimator with exponential noise

So I need a little help with this please. I'm given N measurements of a signal $Y_{i} = A + v_{i}, i = 1,...,N$, where $v_{i}$ is measurement noise with the exponential pdf $f_{v}(v) = e^{-v}, v \geq ...
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How are $n$ and $Var(\varepsilon)$ affecting to Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression

Once I have derived the variance of $\hat{\beta_1}$ as: $\text{Var}(\hat{\beta_1})= \frac{\sigma^2}{\sum(x_i-\overline{x})^2}$ I would like to know how are affecting to this formula: the size of ...
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26 views

More efficient estimator in this situation?

I am thinking about the following situation (it is an example): There are N families in a city and we have a simple random sample of size n. In that sample we know that there are $x_{0}$ families with ...
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27 views

Why do some people say that an asymptotically unbiased estimator “satisfies a strong law of large numbers”?

If $x\in\mathbb R$, an estimator for $x$ is an integrable random variable $X$. We say that $X$ is unbiased if $\operatorname{Bias}(x,X):=x-\operatorname E[X]=0$. Now, in the context of Markov chain ...
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Who performed the first Maximum Likelihood Estimation?

I am very interested in the historical development of statistical theories. Here is the research I've done: I've tried to read two old papers of Fisher. I think the first theory paper on MLE should be ...
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What regression/estimation is not a MLE?

I just rigorously learned that OLS is a special case of MLE. It surprises me because the popular and "reliable" sources such as researchgate and this do not mention this most important connection ...
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Simple worked-out example to compute mutual information between two random variables that are vectors

Here is a simple worked out example to compute MI between two random variables $X,Y\in \mathbb{R}$. This also does a good job. Suppose now i am dealing with two random vectors $P\in \mathbb{R}^3,\ Q\...
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1answer
41 views

Optimality of Bayesian filtering

In Kalman filter, we can show it's a minimum variance filter, which I believe is due to the linearity of system and the Gaussianity of noise. It comes to me that what is the optimality criterion used ...
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2answers
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Estimation of multiple regression model with cross-product terms [closed]

I am studying statistics and I stumbled on the following question. If we have the following model: Y = b0 + b1X1 + b2X2 + b3X3 + b4X1X2 + e with X3=X1-X2, which of the parameters can be estimated and ...
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1answer
30 views

Estimating a distribution from sums of samples

I'm trying to figure out the parameters of a distribution from real data, but I only get their sums and counts. For either exponential or normal distributions. So, I'll get the sum of 27 samples, ...
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Estimation of population quantile based on mutually exclusive sub-populations?

I have read somewhere that "it's obvious" that the estimator $\hat{q}\%$ where $q\%$ is the q-quantile% of a $X_i iid \tilde{ } Dn$ for some distribution $Dn$ for $...
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12 views

Count distinct from a bootstrapped sample

I have a dataset of N items, with attribute V which can be repetitive, for example: ...
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13 views

Checking unbiasedness of an estimator

After finding the method of moments estimator, how do we check for unbiasedness? For an exponential distribution, I found the method of moments estimator as (theta + 1), the book concluded that ...
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21 views

Combining estimated model distribution

I estimate separately a univariate model distribution $f_d(x;\widehat{\mu_{d}},\widehat{\sigma_{d}^{2}})$ with data of $d$ day. I have 5 models according with the 5 days of week. The behavior of ...
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12 views

Using expectation maximization for robust regression

What are the advantages/disadvantages of using EM for robust estimation vs. the robust estimation with Huber or Tukey loss functions?
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1answer
29 views

How to identify one-one correspondance in Sufficient Statistics?

The correct answer to the given question is (1),(3) and (4). I understood how 3 and 4 are correct but I could not understand how (1) is also a correct answer. I know that here $\sum_i X_i$ is a ...
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51 views

Different loss functions for estimation and cross-validation

Assume that the goal is to estimate some parameter $\theta$ (finite or infinite dimensional) based on some data available. Also, assume that there are other nuisance parameters are present in the ...
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31 views

Efficient online (rolling window) estimation of a GARCH model

I have a time series $x_t$ of length $n$. I would like to model it using rolling window approach with window length (width) $w$: window $1$: $x_1,\dots,x_w$, window $2$: $x_2,\dots,x_{w+1}$, $\dots$, ...
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16 views

Fisher Information with respect to the Standard deviation of Normal distribution

Let $X\sim\mathcal{N}(0,\sigma^2)$ be given. I computed the Fisher Information to be $I(\sigma)=\frac{2}{\sigma^2}$. Note that the Fisher Information for the variance is given by $I(\sigma^2)=\frac{1}{...
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Confusion on when you can apply CRLB

I am somewhat confused on when you can apply CRLB when estimating parameters which are not of densities directly. For example a classic problem is estimating the magnitude, phase and frequency of a ...
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1answer
20 views

Non Gaussian QMLE for GARCH(1,1)

What is the difference between QMLE and MLE method to estimate GARCH parameter? Because both maximizes the same log likelihood (?) I tried to estimate the GARCH(1,1) parameter by using quasi maximum ...
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1answer
32 views

MLE $\hat{h(\mu)} = h(\hat{\mu})$ of $h(\mu) = var(Y_1) = \mu^2$

Question: Suppose Y1, · · · , Yn follows an Exponential distribution with $\lambda = \frac{1}{\mu}$. Derive the MLE $\hat{h(\mu)} = h(\hat{µ})$ of $h(µ) = var(Y_1) = µ^2$, and show that $h(\mu)$ is ...
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1answer
43 views

Standard error of the $n^n$ bootstrap means

I need to show that the standard error of the $n^n$ bootstrap means is $SE^*(\bar{Y^*}) = \frac{S\sqrt{n-1}}{n}$, where $\bar{Y^*}$ is the sample mean of a randomly drawn bootstrap sample, and $S^2 = \...
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Data scale for regression analysis

I am working on my PhD thesis.i have dependent variable as daily stock return for many firms and independent variable are inflation, exchange rate etc. My problem is 1-my dependent variable stock ...
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32 views

Expectation of the k-th order statistic of a standard Gaussian sample

Let $(X_1,\dots,X_n)$ be independent random variables with common distribution $\mathcal{N}(0,1)$. The order statistics satisfy $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$. I am interested in the ...
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10 views

How to estimate population size from repetition in random sampling? [duplicate]

I'm periodically sampling values from what's likely a large population. Millions, let's say. And I'm assuming initially that it's random sampling. However, it's only workable to collect thousands of ...
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33 views

Dynamic panel data model with AR(2) process in the errors

I set up the following dynamic panel data model: $$y_{it}=\alpha y_{it-1}+x_{it}^T\beta+v_{it}$$ Additionally, I have the process in the errors: $$v_{it}=\rho_1u_{it-1}+\rho_2u_{it-2}+\epsilon_{it}$$ ...