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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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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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16 views

“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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48 views

Finding method of moments estimator of $\theta$ in $\Gamma(\theta,\theta)$ distribution

Please refer to the question in image I have tried to find $ E(x) $ but i ended up with $\overline x $ = $\frac{\theta + 1}{\theta} $ which statisfies no option , i also tried to find $ E(x-1)^2 ...
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27 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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58 views

Method of Moments Bernoulli

We have this pdf for $x_1, x_2,\dotsc, x_n$ : $$\theta x^{\theta -1 }$$ with indicator variable 1 for $ 0 \le x \le 1$. We decide not to observe the $x_1,x_2,\dotsc,x_n$ but $y_1,y_2,\dotsc,y_n$ ...
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141 views

95% Confidence interval of $\lambda$ for $X_1,…,X_n$ IID exponential with rate $\lambda$

I know how how to find the estimation of $\hat{\lambda}$ using the method of moments. I can take the first moment and equate it to the empirical to get, $E(X) = \frac{1}{\lambda} = \frac{\sum_{i=1}^{...
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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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22 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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94 views

Estimate Information Entropy from Moments

I hope I'm using the right terminology below. I have access to the moments statistics of a large sample. That is, I have $\sum(x)$, $\sum(x^2)$, ..., $\sum(x^k)$. I also have access to max and min, ...
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30 views

Moment generating function of binomial distribution

I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomail distribution iwth $p=\frac{1}{2}$. With the standardized test statitics is $S=\frac{S(\theta_0)-(\frac{n}{2})}{...
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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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460 views

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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107 views

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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803 views

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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110 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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1answer
163 views

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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408 views

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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81 views

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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39 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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323 views

Deriving method of moments estimator for AR(1) process

The method of moments estimator for AR processes can be had with the Yule-Walker equations. But how is it derived? The equation for AR(1): $$Y_t =aY_{t-1}+\epsilon_t$$ Where $\epsilon $ ~ $N(0,\...
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171 views

How can we determine the parameters of a Beta-Binomial Distribution for given mean and variance?

Given a beta-binomial random variate $X$ with $N$ known, how can I choose $\alpha$, $\beta$ such that the distribution's mean matches a chosen quantity $\mu$ and its variance matches a chosen quantity ...
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125 views

It is required to obtain the method of moment estimator and maximum likelihood estimator of a exponential distribution with two parameters

I have the following density: $f(x)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}$ where $x>\theta$. It is required to obtain the maximum likelihood estimator and method of moment estimator for $\...
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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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246 views

Truncated Beta parameters - method of moments

Context Let me first introduce some context. The probability density function and cumulative distribution function of a Beta random variable with parameters $\alpha>0$, $\beta>0$ and support ...
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60 views

Some questions about the Generalized Method of Moments

Here are three (somewhat related) questions about the (Generalized) Method of Moments. I have only just today started studying this method. Concerning the following statement by Greene: Why is $...
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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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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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Is independence assumption needed for Method of Moments estimator

I read about Method of Moments estimator (MOM) in Statistical Inference by Casella and Berger. In MOM description, I do not see the requirements that the sample should be iid. However, in the examples,...
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185 views

Empirical Bayes: method of moments

The model for the data is $X_{i}$ ~ $Bin(n_{i},\theta_{i})$ (iid, $i=1,...,k$). The prior distribution is $\theta_{i}$ ~ $Beta(\alpha,\beta)$. How do we choose (and deduct) moment estimators for $\...
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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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500 views

Hyper-parameter estimation for Beta-Binomial Empirical Bayes

I am reading a paper Illustrating empirical Bayes methods and in the paper the author uses method of moments to derive the value of an estimate. In equation 17 the author gives the following marginal ...
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345 views

Solving log-logistic distribution parameters from moments

Let's say we have a log-logistic random variable $X$ with probability density function: $$f(x)=\frac{(\beta/\alpha)(x/\alpha)^{\beta-1}}{(1+(x/\alpha)^{\beta})^{2}}$$ where $\alpha>0$, $\beta>...
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751 views

Beta distribution parameter estimation: method of moments

In a paper: Topics over time, method of moments was applied to estimate $\alpha$ and $\beta$ for a Beta distribution. My question is that how $\alpha$ or $\beta$ should be calculated if there are no ...
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1answer
93 views

Hawkes processes: interpretation of maximum likelihood estimates disagreeing with moment estimator estimates?

So, I have some data.. and a parameterized Hawkes process which I estimate parameters for via maximum likelihood... the residuals ( the compensator aka the dual-predictable projection) are good in ...
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1answer
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How to use method of moment to find Pareto distribution estimator?

I have $f_{\alpha, \beta}(y)=\frac{\alpha}{\beta}(\frac{\beta}{y})^{\alpha +1}, y\ge\beta,\ \ \alpha,\beta\gt 0$. Both $\alpha, \beta$ unknown. To find estimators using the method of moment, we ...
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Real life uses of Moment generating functions

In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in ...
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Estimating variance of dgp that measured as Bernoulli Parameter

I have some parameters $p_1,\ldots,P_n$ distributed iid. with mean and variance $\mu, \sigma^2$. Then, for each $t\in\{1,\ldots,n\}$ I have $\{X_{1, t}, \dots X_{N, t}\}$ realizations of the ...
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55 views

Method of moments to estimate a parameter

Suppose that $Y_1, Y_2, ..., Y_n$ is a random sample from a distribution with density function $$ f(y) = \begin{cases} \theta y^{\theta - 1}\ \ \ \ 0 < y < 1, \\ 0\ \ \ \ \ \ \ \ \ \ \ ...
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1answer
123 views

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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200 views

method of moments for skew normal distribution

I have a random Variable $X$ is $ SN(\lambda)$ and is pdf is given by: $f(x)=2\phi(x)\Phi(\lambda x)$. The model of the variable X is given by:$X=\frac{1}{\sqrt{1+\lambda^2}}Z_1+\frac{\lambda}{\sqrt{...
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288 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}(\...
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1k views

Principle of Analogy and Method of Moments

I am studying method of moments and GMM in the context of econometrics. Can someone explain on intuitive level, what does it mean to match moments? And how does this differ from the classical linear ...
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2k views

Method of moment estimates for n Bernoulli trials

Let $X_1, X_2 \dots X_N$ be the indicators of $n$ Bernoulli trials with probability of success $p$. What is the method of moments estimate of $p$? Exhibit method of moments estimates for $p \cdot (1 ...