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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Identification Problem in Minimum Distance Estimation

I have the following problem with a system of minimum distance equations I want to solve. The objective is to estimate the parameters of the random variables in the following DGP: $$ x_t= \phi_t(\...
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38 views

Approximating standard error that contains a parameter, by replacing the parameter with its estimate

I am a bit confused about the following step I have seen in the stats literature which seems to me a bit circular. Say you are approximating the standard error of the MoM estimate of an exponential ...
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Check Computation of MME and MLE

Let $X_1$, . . . , $X_n$ be i.i.d random variables having pdf $$f(x\mid\theta) = (\theta+ 1)x^{\theta}I_{(0,1)}(x)$$ where $\theta \gt−1$ (a) Give a MME of $\theta$ based on the first ...
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101 views

Discrete, finite L-moment problem

Suppose that we have a real-valued discrete random variable, whose probability distribution has finite support on some set $S$ of real numbers. Then if $N = |S|$, we know that we can construct the ...
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102 views

Does diagnolizing higher-order cross-moment matrices lead to independent variables?

Diagonalizing the covariance matrix transforms multivariate data into uncorrelated variables, but does not make them independent necessarily. Does it follow from this that if I were to diagonalize ...
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186 views

Method of moment estimator

Consider $U_i \sim^{iid} Bernoulli(\pi)$. Also consider: $$Y_i | U_i = 0 \sim exp(1/\gamma) \text{ and } Y_i | U_i = 1 \sim exp(1/2\gamma) $$ What are the method of moment estimators of $\pi \text{ ...
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1answer
960 views

Estimating parameters in a truncated negative binomial distribution?

I would like to find the estimates of the parameters in a truncated (at zero) negative binomial distribution. Suppose $Z$ has this distribution with parameters ($\alpha,\beta$). (The parametrization ...
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24 views

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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32 views

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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63 views

a question about method of moment estimator

I have a question about method of moment estimator. Say I have a IID sample $X_1, X_2, ..., X_n$ from an exponential distribution $Exp(\theta$), say I want to find the method of moment estimator of $...
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25 views

Estimated Standard Error of a Method of Moment Estimator, Poisson Example

For X randomly sampled from a Poisson(lambda) population, the method of moment estimate of lambda is the sample mean: lambda_hat = X_bar Now, say we are interested in the variance of lambda_hat (...
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277 views

Solving a system of equation by moment condition reports error but minimum distance works

I do not understand the following error message I get using the gmm function in R. The code below creates two moment conditions (...
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77 views

Numerical method to compress empirical probability distribution

I am trying to grapple with the following problem. I have an application that develops empirical distributions. In essence, I end up with a histogram of equally spaced $x$ values, with both a $max$...
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240 views

Covariance of parameter estimates in Method of Moments

I have a $3\times1$ vector function $f(x_i;\theta)$ where $X$ is a rv and $\theta$ is $3 \times 1$ parameter vector, such that \begin{equation} E \, f(X;\theta) = {\bf 0}.\end{equation} If I have a ...
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659 views

When is the MLE equivalent to the MM estimator

I know that in natural exponential families the general method of moments estimator we get from working with the sample and the sufficient statistic gMM is equivalent to the maximum likelihood ...
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491 views

Degrees of freedom of J-Test distribution

This book states, on page page 256 ( the GMM section) that the J-test(for over-identifying restrictions) is of the following form $G_n'J_n^{-1}G_n \approx \chi^2(m-p) $, where $G_n=G_n(\theta)=1/n \...
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2k views

Estimating the parameters of a beta distribution with zeroes and ones in the sample

I have a list of values in [0,1] that I want to fit to a Beta distribution in order to get the corresponding alpha parameter. I can't use a beta fitting function because my values might be 0's and 1'...
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17 views

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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39 views

why is method of moments estimates asymptotically normal

I have noticed that a lot of statistics textbooks contain lengthy discussions and detailed proofs on showing that MLE estimates are asymptotically normal (under regularity conditions). On the other ...
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187 views

Method of Moment Estimator — Uniform Dist

Find the two method of moment estimators for $\theta$ given that $Y_i | \theta$ is distributed i.i.d U(0,$\theta$). We know that E($Y_1$) = $\frac{\theta}{2}$ and Var($Y_1$) = $\frac{\theta^2}{12}$ ....
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64 views

Can I use the method-of-moments estimator to get the adjustment factor for a generic transformation of the regressand?

When the regressand is in logarithmic form, the model looks like this: $$log(y)=X\beta+u$$ When we are ultimately interested in predicting $y$, we cannot simply use $exp(X_0\hat\beta)$ as a ...
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313 views

approximate a probability distribution by moment matching

I have a 60-40 weighted distribution, of uniform(0,7.5) and uniform(7.5,10) respectively, i.e. $$f_X(x)=(0.6/7.5)1_{x∈[0,7.5)}+(0.4/2.5)1_{x∈[7.5,1]}$$ I have worked out that $$E(X) = 0.6(7.5/2) + ...
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265 views

Is it possible to calculate mutual information by moments generating functions?

I went to listen to a workshop and some audience asked the presenter how the moments can improve the mutual information. I am learning the MI(Mutual Information) and moments so don't have enough ...
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187 views

How do I apply the method of moments for estimating parameters in a sum-relationship?

We have a model relationship between three random variables like this: $$ U = C + S $$ I have a ton of measurements of realizations of $U$, as well as a ton of realizations of $C$. But the ...
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952 views

Why doesn't the method of moments work when calculating the variance of the inverse gamma distribution?

I'm trying to calculate the variance of the inverse gamma distribution using the method of movements. According to wikipedia the variance should be: $$\sigma^2 =\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}$...
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23 views

Finding the Method of Moments in two Parameters

Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n$ from the following distribution $$f(x;\mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{\frac{-\lambda(x - \mu)^2}{2\mu^2 x}}$$ where $x, \...
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20 views

Method of Moments Estimators

Let $X_1, \ldots, X_n$ be a random sample of size $n$ from the following distribution: $$f(x) = \frac{1}{\theta_2}e^{-\frac{(x - \theta_1)}{\theta_2}}, \;\; x\geq \theta_1, \;\; (\theta_1,\theta_2)\in\...
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16 views

Asymptotic distribution of mme of coefficient of variation

For random sample from unif(0,1) distribution, method of moments estimator for coefficient of variation is sample mean divided by sample standard deviation. Here, coefficient of variation theta is mu/...
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19 views

Derivation of the Satterthwaite appproximation for nonnegative estimators

I am reading Statistical Inference, Berger & Casella (Page 314-315) This is for deriving Satterthwaite approximation, where we want to approximate the distribution of $\sum_{i=1}^k a_i Y_i$ where $...
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27 views

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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21 views

How to generate numbers from Beta distribution in R if one of the parameters is $\theta$?

I have to estimate ( using method of moments) $\theta$ with an estimator $\hat{\theta}$ for a function with pdf given as $$\theta x^{\theta-1}, \ \ \ \text{ for 0 < x < 1, and } 0 \ \text{ ...
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45 views

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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41 views

Approximating distribution by moment matching?

I am going to compare distributions by moment-matching (expected value, standard deviation, skewness, kurtosis etc). The question is simple: As the moment-matching relates to Taylor expansion, would ...
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27 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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66 views

Method of moment through covariance derivation

Given a Bivariate INAR(1) Poisson Process: $Y_t^1 = \rho_1 * Y_{t-1}^1+R_t^1$ $Y_t^2 = \rho_2 * Y_{t-1}^1+R_t^2$ Where $R_t^1$ and $R_t^2$ are the innovation terms and follow the bivariate Poisson ...
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23 views

How to derive the distribution of OLS starting from the sample moments?

I know I am supposed to start from $N^{1/2}[N^{-1}\sum x_{i}u_{i}]$ Then by central limit theorem that that it is asymptotically $ N(E(x_{i}u_{i}),var(x_{i}u_{i})) $ and $E(x_{i}u_{i})=0$ so $ ...
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33 views

Suggesting a method of moments estimator for the chance that some event happens

Let $X_i$~ $\text{Pois}(\lambda)$ be the number of breakdowns a certain ATM machine experiences in the $i^{th}$ week. $\implies$ Let $ \{X_i\}_{i=1}^n$ be iid of the number of breakdowns the machine ...
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21 views

Method of moments giving super sensitive estimates

I'm trying to study a process that produces, in theory, an equilibrium distribution where the $i$th raw moment is given by: $$ \mu_i = \exp(-\theta_1 \sum_{j=0}^{i-1}(1 + j\theta_2)^{-\theta_3}) $$ ...
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51 views

How does a method of moments estimate compare with a true value of the parameter?

To cut the long story short, I've been given this information: pdf:$$f_X=\tau xexp\left({\frac{-\tau x^2}{2}}\right)$$ $x,\tau>0$ I went ahead and found the cdf:$$F(X)=1-exp\left(\frac{-\tau x^2}{2}...
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224 views

Understanding the Method of Moments

This is an understanding question regarding the method moments: It's basic idea is that any $k^{th}$ moment $m_{k}$ converges in probability to its empiric estimator as follows: $$m_{k} = \frac{1}{n} ...
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143 views

Distribution that has shape of normal and uniform distribution

I have a pdf of a random variable that looks like image below (the support is $[-1,1]$). It is actually function of another random variable, let's say $T = aX + bY + cZ$. When I changed parameters $(a,...
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490 views

Confusion about using moment condition in a multiple regression model

The very simple case assumes that we have a model like $y = a + bx + e$ where the condition $cov(x,e)=0$ is true. Hence one can use the relationship of the moment conditions to estimate the parameter $...
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47 views

Find the Method of moments estimate

So here i am getting E(X)=1/2,E(X^2)=0 and E(X^3)=Theta squared/4 How do i proceed now?How do i use the given x values ?
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496 views

Matching moments in Gaussian Mixtures

Consider $$q^{\backslash n}(\theta) = \mathcal{N}(\theta | m^{\backslash n},v^{\backslash n}I)$$ and $$f_n(\theta) = (1-w)\mathcal{N}(x_n | \theta,I) + w\mathcal{N}(x_n|0,aI)$$ Then let $$\hat{P}(\...