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Questions tagged [method-of-moments]

A method of parameter estimation by equating sample and population moments then solving the equations for the unknown parameters.

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Method-of-moment of n IID random variables

The method-of-moment of $\sigma$ for the following pdf is $$ \text{pdf}(x,\sigma) = \frac{x}{\sigma^2}\exp(-\frac{1}{2}\frac{x^2}{\sigma^2}) $$ $$ E[x] = \int_{0}^{\infty}\frac{x^2}{\sigma^2}\exp(-\...
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Is every method of moments estimator (MME) asymptotically normal distributed?

As already written in the question, I am asking myself whether every method of moments estimator (MME) is asymptotically normal distributed? Formulated differently, is every (central or non-central) ...
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$\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

Let $X$ be a non-negative random variable bounded on $[0,1]$. Is it true that $\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$ for some constant $k$? If not, are there any minimal assumptions on $X$ where this ...
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Check if method of moments estimator is unbiased for $X_1...X_n$ being a random sample from $\mathcal{U}_{[-\theta,\theta]}$

I am not sure how to do this. To find the method of moments estimator I did: $$E[X] = \frac{-\theta + \theta}{2} = 0$$ use 2nd moment: $$E[X^2] = \frac{(-\theta)^2 + -(\theta^2) + \theta^2}{3} = \frac{...
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Method of Moments of Uniform Distribution

Let $x_1=2, x_2 = 1, x_3 = \sqrt5, x_4 = \sqrt2$ be the observed values of a random sample of size 4 from a uniform distribution $U(-\theta, \theta)$ where $\theta>0$. Then the method of moments ...
Rhea Agarwal's user avatar
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Why can the method of moments be expressed as a minimization problem?

Generalized method of moments (GMM) estimation seems to be called generalized method of moments because the standard method of moments (MoM) is a special case, following the following logic. MoM is ...
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Why is method of moments (MoM) not unique? What is uniqueness?

I just learned that a MoM estimator is not unique... What does that mean? What does being unique mean in general? For example, if we were to find the mean using a MoM estimator for a uniform ...
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OLS and the Method of Moments estimator

I have been taught that the Method of Moments Approach could be used to obtain estimates of a bivariate regression of the form y = B0 + B1x + u. Specifically, the 2 Moment conditions are E(u) = 0 and ...
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Population and sample moment notation

A population moment is often stated as $E[x]$ the sample moment counterpart is stated as $1/n \sum_{i=1}^{n} x_i$. I do not quite understand what is meant by $E[x]$. Why not $E[x_i]$? What does $E[x]$ ...
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Is a correction needed in the weighting matrix when using the simulated method of moments?

I am following several guides on the simulated method of moments (SMM) and one practical way of getting the weighting matrix, W, is to bootstrap the empirical moments and invert their covariance ...
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Matching Simulated Moments Perfectly in Practice

Several sources suggest that when estimating a model using the simulated method of moments (SMM), one ought to always be able to get the difference between the empirical and simulated moments to be 0 ...
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6 votes
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Why is OLS related to Moment Estimation?

I am trying to understand the relationship between OLS and the Method of Moments. Specifically, why OLS is considered a special case of Method of Moments ... and why Method of Moments is a special ...
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Estimating the population parameters using the sample moments

I have distribution , which is not normal and highly skewed. I take a sample from that distribution and calculate the first four moments of the sample distribution. The first four moments calculated ...
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For Gamma distribution, use MLE or MoM?

For Gamma distribution, is it better to use MLE(maximum likelihood estimation) than MoM(method of moments) to estimate the shape and scale parameters? Also, in python SciPy, does gamma.fit use MLE? I'...
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Method of moments for the Logarithmic/Log-Series distribution

I'm looking for some more insights in applying the method of moments for the Logaritmic (or also called Log-Series) distribution. The Logarithmic distribution only has one distribution parameter $0 &...
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Why do popular ML and statistical packages simply ignore classical estimation and detection algorithms for statistical signal processing? [closed]

For those who had a hard time to study and understand classical estimation and detection algorithms, and unfortunately realized that these algorithms are simply ignored by many packages that have the ...
Rubem Pacelli's user avatar
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How to calculate Gumbel with LMoments and GEV with method of moments

I need to calculate the values for certain return periods of a flood event (up to 5000). It has to be GEV with method of moments and Gumbel with L-Moments. But I am not sure about how to calculate ...
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How to build the covariance matrix with different weighted moments via GMM [closed]

I have two sets of moment conditions, one is IV moment with N observations but the second moment only has N_1 observations, N_1<N. How to build the covariance matrix? Appreciate for any replies!
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Confidence interval for continuous uniform distribution [duplicate]

If X is a random variable from the uniform distribution on the interval [0,θ], how would one construct a 95% confidence interval for the method of moments estimator of θ? I have found that and also ...
Mari Mtchedlishvili's user avatar
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Motivation for using moment about mean versus moment about the origin

When calculating moments of probability distributions, what would motivate you to take the moment about the mean (called "central moment") versus the standard moment about the origin?
Alexander Mills's user avatar
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Find nth central moment for exponential distribution

I am trying to figure out what the nth central moment is for the exponential distribution. Here is the formula for the nth moment: $$ \mathop{\mathbb{E}}{[x^n]} =\dfrac{n!}{\lambda^n} $$ My question: ...
Alexander Mills's user avatar
5 votes
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Maximum likelihood vs generalized method of moments

I am trying to understand how maximum likelihood (MLE) and generalized method of moments (GMM) are related to each other. In particular, I often see people saying that MLE can be written in terms of ...
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How to take 1000 samples from distribution X and then use MLE to prove they came from distribution X?

I am trying to do: find 1000 points that represent samples from distribution X with parameters $(a,b,c,\ldots, d)$ be guaranteed that the MLEs for those 1000 points are $(a,b,c,\ldots, d)$ with $99\%$...
Alexander Mills's user avatar
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Is there any intuitive explanation for MoM in estimating parameters?

I found from some literature that when we use the method of moments to fit the Gumbel distribution, the estimated (On page 24) A comparison of the variance formulas in (1.66) with the CramBr-Rao ...
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Confidence Interval for a MOM Estimator of a uniform distribution

If $X$ is a random variable from the uniform distribution on the interval $(\theta, 2\theta)$, how would one construct a 90% confidence interval for the method of moments estimator of $\theta$? The ...
Mike K's user avatar
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Are these moments estimators asymptotic unbiased?

In this paper, authors consider method of moments of fitting Gumbel distribution: We know that maximal likelihood estimators are asymptotic unbiased. But are these moments estimators asymptotic ...
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fit gamma distribution based on median interquartile range

I would like to create a gamma probability distribution in R with a median of 3 and interquartile range of [2,5]. I am familiar with the methods of moments to estimate the scale and shape of a gamma ...
Elliot's user avatar
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Is there a formula for estimating confidence intervals for indirect inference estimates?

Indirect inference is usually deployed to estimate parameters $\theta$ of simulation models, i.e. models for which likelihood is unknown or intractable but that can be "run forward" ...
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Using the Yule-Walker equation to calibrate an autoregressive model with the method of moments

Consider the following discrete autoregressive $\epsilon_t$, where $\epsilon_t \in (\pm 1 ) \forall \ t \geq 1$. We think of $\epsilon_t$ as the child of a previous sign at time $t-l$, where $l$ is ...
J_derry's user avatar
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Using the method of moments or GMM to estimate the parameters of a specific problem

Given $(X_t)_{t \in \mathbb{Z}}$ an AR(1) process: $$X_t = c+ \phi X_{t-1} + \epsilon_t, \quad \epsilon_t\sim WN(0,\sigma^2)$$ We can show that $E(X_t) = \frac{c}{1- \phi}$ and $E(X_t^2) = \frac{\...
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Method of Moments for a mixing proportion

Suppose we have densities $f_1, f_2$ from the random variables $W_1$ and $W_2$ where $W_i$ has known mean $\mu_i$ and variance $\sigma_i$. Consider the mixture of the two densities $$ f(x;\theta)=\...
Geigercounter's user avatar
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In a regression model, can we use $E(u)=0$ as the moment condition?

Consider the linear model with stochastic regressors: $$y_t = \beta_0^\prime x_t + u_t, \quad E(u_t | x_t)= 0$$ So that $E(u_t | x_t)= \beta_0^\prime x_t $. Using the e Law of Iterated Expectations (...
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Two supposedly equivalent approaches to the method of moments

I am reading the book "An Introduction to Econometric Theory" by A. Ronald Gallant. In the section of the book on the Method of Moments, I get a little confused about the method as I know it....
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A description of the mean of the Geometric Distribution - is it unorthodox or just incorrect?

I have a homework assignment where I'm asked to propose an estimator for the mean of a geometric random variable. This seemed simple enough, given that I've always understood the mean of the geometric ...
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Deriving unbiasedness of estimators (involving method of moments idea) of normal distribution with heterogeneous variance

I am currently reading this paper and in pp.127, 128, there are unbiased estimators that I cannot derive its unbiasedness. The setting is simple. Let $$X_i\sim N(\mu,\tau^2+\sigma_i^2),\quad i\in\{1,\...
sepiabrown's user avatar
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Estimating $1/a$ for following pdf using method of moments estimation

A random sample of size $n$ is being drawn from a population with pdf as: $$f(x) = \begin{cases} (a + 1)x^a & \text{for }0<x<1, \\ 0 & \text{otherwise.} \end{cases}$$ Can we express the ...
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Using the methods of moments in R for the dirichlet distribution

I'm trying to build a distribution of transition probabilities to randomly sample from in a Markov model where individuals can transition from one health state to another (assume that in the image ...
James Moore's user avatar
2 votes
1 answer
831 views

Method of moments estimate of Pareto Distribution

The Pareto distribution has the following $cumulative \ distribution \ function$ : $$F(x;\alpha ,\Theta ) = \left\{\begin{matrix} 1 - (\frac{\alpha}{x})^{\theta}\ \ if \ \alpha \leq x\ & \\ 0 \ ...
Kalvin's user avatar
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2 answers
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Finding method of moments estimate for density function $f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1}$

Suppose that $X_1, X_2, ..., X_n$ are i.i.d random variables on the interval $[0,1]$ with the density function $$ f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1} $$ where ...
iobtl's user avatar
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Efficiency of IV vs GMM

I am trying to understand how IV/just identified GMM and overidentified GMM compare when it comes to efficiency. The way I understand it, we are able to identify the vector of coefficients in IV and ...
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Method of moments for symmetric mean zero distributions

When using the method of moments to fit a symmetric mean zero distribution, does it make more sense to fit higher order moments or lower order absolute moments? I could not find any resources which ...
Trevor Hansen's user avatar
4 votes
3 answers
285 views

What's special about moments that allows "method of moments" to work?

The idea behind Method of Moments (MOM) is quite intuitive: find the parameter values so that the population moments (which are functions of those parameters of interest) matches the sample moments. ...
Heisenberg's user avatar
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2 votes
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Consistent but inefficient GMM

Consider the following linear model $$y_t = x_t' \beta +u_t$$ where $t =1,...,T$ and $x_t = (x_{1t} x_{2t} ... x_{kt})'$ , $ \beta$ is $k \times 1$ vector of unknown coefficients, $u_t$ is an iid ...
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Discuss that method of moment estimation is inefficient

Discuss that method of moment estimation is inefficient. Then model is Consider the following linear model $$y_t = x_t' \beta +u_t$$ where $t =1,...,T$ and $x_t = (x_{1t} x_{2t} ... x_{kt})'$ , $ \...
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When is the Optimal weighting matrix in GMM singular?

currently I am trying to estimate a simple linear regression: \begin{equation} y_t = X \beta + \varepsilon_t, \end{equation} where I try to find 4 coefficients and one specific predictor is an ...
user326465's user avatar
2 votes
1 answer
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How does the information in the problem statement and this solution align with the provided description of the method of moments?

I have the following problem: Let $Y_1, Y_2, \dots, Y_n$ be i.i.d. $\text{Uniform}(\theta, 1)$ random variables, and let an estimator be $\hat{\theta} = \min\{ Y_1, Y_2, \dots, Y_n \}$. You may find ...
The Pointer's user avatar
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3 votes
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Negative-Binomial Method of moments with an offset

Given the method-of-moments approach to estimate the parameters of the NB-2 distribution $\mu$ and $\phi$: $$ \mu = \bar{y} $$ $$ \phi = \frac{\bar{y}^2}{s^2 -\bar{y}} $$ How can this be extended to ...
AndrewrJ's user avatar
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127 views

Method of Moment for Gamma Distribution

I know that the empirical $r$-th moment is defined as: $$\hat E(X^r) = \frac{1}{n} \sum_{i=1}^n x_i^r $$ So for the first moment I did: $$E_{\lambda,\alpha}(X) = \hat E(X) = \bar X $$ $$\bar X = \...
René Martínez's user avatar
2 votes
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204 views

Confidence Interval for Estimator using Delta method

The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
Suriya's user avatar
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Method of moments and MLE

Suppose a random variable follows the logistic distribution, $X ∼ Logistic (\mu, \sigma)$ and we restrict our attention to random samples drawn from this random variable $X$. What would be the MoM and ...
Maybeline Lee's user avatar