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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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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22 views

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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74 views

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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54 views

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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43 views

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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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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26 views

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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98 views

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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18 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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57 views

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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54 views

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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25 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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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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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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88 views

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 ...
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Method of moments when there's no closed form expression

I am trying to code up a method of moments algorithm for parameter estimation. I have a closed form for the moments as a function of the parameters, but these expressions are complicated, so there's ...
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41 views

Method of moments estimator, $P_\theta(X = x) = \frac{1}{\theta}$

I am struggling with finding a method of moments estimator for (seemingly) simple situation: pdf is given by $P_\theta(X = x) = \frac{1}{\theta}$, $x \in$ {1,2,...$\theta$}, where $\theta \in N$. My ...
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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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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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155 views

What is the limiting distribution of $Y_n = \sqrt{n}(\bar{X}_n-1)$ as $n \to \infty$?

Let $X_1,\cdots,X_n$ be independently and identically distributed with pdf $f(x)=e^{-x}, 0 < x < \infty$. Let $Y_n = \sqrt{n}(\bar{X}_n-1)$. What is the limiting distribution of $Y_n$ as $n \to ...
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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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122 views

inverse of an exponential distribution

I have a question regarding this. Say I have $X_1, ..., X_n$ be random sample from an exponential distribution i.e. $Exp(\theta)$, and let $\gamma = \theta^2$. Let denote $\gamma^{mme}$ as the ...
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37 views

Tau2 in random-effect multiple meta-regression

I'm doing random effect multiple meta-regression with two predictors and using method of moments to estimate between-studies variance (tau2). Tau2 formula for one predictor case is: with F being: Q -...
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38 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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281 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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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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301 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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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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241 views

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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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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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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53 views

Moment generating function of binomial distribution

I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomial 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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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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329 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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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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2k 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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257 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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776 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 ...