# Questions tagged [method-of-moments]

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

122 questions
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### Estimation of variance in GMM

In GMM, the efficient weight matrix minimizes the asymptotic variance of the GMM estimator by setting: $$W_T^{opt} = S_T^{-1}$$ where $S_T$ is an estimator of the asymptotic variance of the moments,...
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### “Appropriate conditions” for method of moments estimator to exist, be consistent, and asymptotically normal?

My statistics text has the following theorem, and alludes to "appropriate conditions on the model", but never specifies what those conditions are. What conditions are necessary? Let $\hat{\theta}_n$...
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### Deviations of Method of moments estimators for linear regression with constant

I am new to method of moments and want to figure out how to derive the method of moment estimator for $\beta$ in the linear equation with a constant term and three corresponding moments, namely, I ...
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### 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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### 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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### 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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### Method of moments for linear regression?

I have been reading about the method of moments, and now I understand how to obtain the method of moments estimator for a random sample $x_1,...,x_n$ from a distribution $f(x;\theta)$, in the ...
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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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### 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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### Reasons for different parameters via MoM and MLE

I got a study of 210 samples and I tried fitting gamma distribution to them. I used method of moments and maximum likelihood estimation to calculate the parameters, but parameters came out quite ...
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### Interpreting weighted MLE, MoM estimates with stochastic weights

I deal with the weighted probability density $h(x)=w*f(x, \theta)$, where x is the stochastic variable, f is the unweighted probability density, w is stochastic weight depending on x and also measured ...
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### 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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### 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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### Confidence Interval for a Uniform Distribution based on Method of Moments

Let $X_1,..,X_n$ be a random sample of $X$~$U[\theta,\theta+1]$. Given a sample $n=100$ from that distribution, the following statistic was calculated: $\sum\limits_{i=1}^n X_i = 350.492$ I need to ...
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### 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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### By conditioning on $N$, show that the moment generating function of $Y$ is given by $m_Y(t)=m_N(\ln(m_X(t)))$

I am having a difficult time using moment generating function properties to prove this: (any direction or key properties will be very helpful) Let $X_1$, $X_2$, . . . be independent and identically ...
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### Two Parameter Method of Moments Estimation

I know this is a very basic question, but I am getting a bit confused due to some variation between resources I'm using for a statistics course. Say you have some $iid$ random samples $X_1$,$X_2$,......
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### MME for exponential family

Let $X_1, X_2,...,X_n$ be iid random variables having pdf $$f(x|\theta) = \frac{1}{x \sqrt{2\pi\theta}}e^{(-[\log x]^2/[2\theta])} I_{(0,\inf)}(x)$$ where $\theta > 0$. Determine the MME of $\theta$...
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### 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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### Method of Moments for $\nu$ of standard t-distribution: what if true $\nu=2$?

Note I am considering the standard $t$ distribution $(\mu=0,\sigma=1)$ The method of moments for $\nu>2$ is derived in this question My question is, if the true (population) value of $\nu$ is $2$,...
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