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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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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Method of moments - what determines the number of moment conditions?

Given the following definition: What is the function $f$ here? For me, the method that is presented in the wikipedia article is very clear: https://en.wikipedia.org/wiki/Method_of_moments_(statistics)...
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How many components of a gaussian mixtures do I need to match moments up to the $r$-th order?

Suppose I have a ($k$-dimensional) random variable $X \sim D$ and I want to find a Gaussian Mixture $GM \sim \sum_{i=1}^C \pi_i \mathcal{N}(\mu_i, \Sigma_i)$ such that the moments of order $r'$, for $...
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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 ...
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Method of moments estimator for a probability of an event

I need to find the method of moments estimator for $P(pois(\lambda)=0)$. I already worked out the MME $\hat{\lambda}=\bar{X}$ but I'm not sure how to proceed here because I can only see how this ...
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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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Compare quality of fit, MLE vs method of moments

I have many different datasets that presumably should follow the same distribution type (but with distinct parameters). I've identified one distribution type that seems to describe best the data (...
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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)=\...
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Different method of moments technique for Lognormal Distribution

We consider a random sample $X_1,...,X_n$ from a lognormal distributed random variable $X$ with density function $$f_X(x;\mu,\sigma)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(\ln(x)-\mu)^2},\...
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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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Why Horvitz-Thompson Estimator is a type of Method of Moments?

I notice Donald Rubin once remarked the IPW estimator (resembling the Horvitz-Thompson estimator in sampling theory) in a conference: "Horvitz-Thompson is just glorified Method of Moments. We've ...
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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,\...
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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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Find method of moments of estimate

Let X be a discrete random variable such that P(X = 2) = (1−θ)/2 , P(X = 3) = (1+θ)/3,P(X = 4) = (1+θ)/6 , P(X = x) = 0 for all x $\notin$ {2, 3, 4}. Here θ is an unknown parameter such that θ ∈ {0, 1/...
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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 ...
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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 \ ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 = \...
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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 ...
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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 ...
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Method of Moment, MLE and Information matrix

We have $$\mathbb{E}[Y_i| X_i] = β_0 + β_1X_i$$ What would be the Method of Moments estimator and MLE for this model?
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Why methods of moments is not widely used in machine learning?

If I recall correctly, in most machine learning context I have encountered, only maximum likelihood estimation and maximum a posteriori method are used. Why not method of moments?
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MLE and Methods of Moments of Negative Binomial in R

Question Let's say we define the Negative Binomial as follows: $$f(x) = {x+r-1 \choose x} p^x (1-p)^r$$ With mean and variance: $$E(x) = \frac{rp}{1-p} \quad \quad V(x) = \frac{rp}{(1-p)^2}$$ We are ...
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Variance of the method of moments estimator for $\mu$ of log-normal distribution

Suppose the data has originated from a log-normal distribution with parameters $\mu$ and $\sigma$ (i.e. the mean and standard deviation of the underlying normal distribution). I have only the sample ...
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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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Poisson Process Method of Moments

Disclaimer: This is a homework problem A School of Ornithology researcher wants to estimate the number of red-tailed hawks in Ithaca. She radio tags 10 birds, and then sets up a feeding station with ...
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Method of Moments for Mixture distribution

The restaurant owner also wants to reconfigure her seating layout, and has asked you for help in modeling her clients. She gives you a dataset of past reservations, and tells you that she gets a mix ...
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Is this estimator biased or unbiased?

A random variable X constructed as follows: $$X = \sum_{i=1}^{N} Z_i \ $$ where $N$~Poisson$(\lambda)$ with $\lambda > 0,\space$ and$\space$ {${{Z_i}}$}$^N_{i=1}$ is an independent and identically ...
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Weighted sum of negative binomial distributions - approximate fast parameter calculation

Let's suppose we have a convolution (weighted sum) of three negative binomials (parameterised as mean and overdispersion). ...
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Method of moments as non-parametric model

In Wikipedia article (https://en.wikipedia.org/wiki/Nonparametric_statistics) about nonparametric statistics there is "Method of moments (statistics) with polynomial probability distributions&...
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Can we use pooled ols on dynamic panel without individual fixed effect (but with group fixed effects) and no serial correlation in error

Is it correct that dynamic panel without individual fixed effect (but with group fixed effects), no serial correlation in error can be estimated consistently through pooled ols? For example, suppose ...
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Confusion about method of moments for linear regression

It is known that linear regression estimator can also be viewed as a method of moment estimator derived using the moment condition $E[X\epsilon]=0$. This moment condition follows from exogeneity ...
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An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]

Can someone construct an example of this? i.e., $E[X^2] < \infty$ but $E[X^3] = \infty$. Results could be in terms of pdf, or cdf, or survival function. Justification would be appreciated
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Exhaustive list of techniques used to estimate population mean and variance?

In beginning stats, we were told that: $\bar{x}$ is an unbiased estimate of $\mu$ $\frac{1}{n - 1}\sum(x - \bar{x})^2$ is an unbiased estimate of $\sigma^2$ As I am reading more, I have learned that ...
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Method of moments and MLE estimates for Lomax (Pareto Type 2)

I have this dataset, on which I am supposed to fit Lomax distribution with MM and MLE. Lomax pdf is: $$f(x|\alpha, \lambda) = \frac{\alpha\lambda^\alpha}{\left(\lambda+x\right)^{\alpha+1}}$$ For MM, ...
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Higher moments of Lognormal Distribution

I am experiencing a weird problem modeling lognormal distributions and I am quite stuck on this one. For a normal distributed variable X following a $N(\mu,\sigma^2)$ distribution, we have that $Z=e^...
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Moments method and plug-in method

We have the following discrete distribution on $\{1,2,3\}$: $$\mathbb{P}(X=1)= \theta^2, \mathbb{P}(X=2) = 2\theta(1-\theta), \mathbb{P}(X=3)=(1-\theta)^2$$ with $\theta \in (0,1)$ and want to use ...
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Which estimator is preferred for a random sample from $P_\theta(X=x)=\theta^x(1-\theta)^{1-x}, x=0,1; 0 \le \theta \le \frac{1}{2}$?

Let $X_1,\cdots,X_n$ be an i.i.d sample from $P_\theta(X=x)=\theta^x(1-\theta)^{1-x}, x=0,1; 0 \le \theta \le \frac{1}{2}$. Its the method of moments estimator of the MLE better? Why? My work: I ...
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Intuition behind Method of Moments estimators of Binomial distribution

The method of moments estimators of the binomial distributions ($x \sim Binom(n, p)$) are a bit weird... I got $\hat p = \bar x + 1 - \frac{\sum x_i^2}{\sum x_i}$ and $\hat n = \frac{\bar x}{\hat p}$. ...
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How to estimate parameter using yule-walker/method of moments?

Suppose you observe the first T periods. X1, X2, · · · , XT of an AR(1) process Xt = µ + φXt−1 + et. Derive the Yule-Walker/Method of Moment estimate φˆMM for φ. I thought YW was used to solve for ...