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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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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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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1answer
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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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44 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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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 $$\theta x^{\theta-1}, \ \ \ \text{ for 0 < x < 1, and } 0 \ \text{ ...
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Moments method and plug-in method
We have the following discrete distribution on $\{1,2,3\}$:
$$\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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1answer
24 views
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, Ā· Ā· Ā· , XT of an AR(1) process Xt = Āµ + ĻXtā1 + et.
Derive the Yule-Walker/Method of Moment estimate ĻĖMM for Ļ.
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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1answer
38 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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30 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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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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1answer
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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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1answer
104 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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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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1answer
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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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1answer
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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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1answer
239 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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1answer
201 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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1answer
124 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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1answer
47 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)$$
where $\theta \gtā1$
(a) Give a MME of $\theta$ based on the first ...
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Estimated Standard Error of a Method of Moment Estimator, Poisson Example
For X randomly sampled from a Poisson(lambda) population, the method of moment estimate of lambda is the sample mean:
lambda_hat = X_bar
Now, say we are interested in the variance of lambda_hat (...
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2answers
263 views
Reasons for different parameters via MoM and MLE
I got a study of 210 samples and I tried fitting gamma distribution to them. I used method of moments and maximum likelihood estimation to calculate the parameters, but parameters came out quite ...
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Method of moments giving super sensitive estimates
I'm trying to study a process that produces, in theory, an equilibrium distribution where the $i$th raw moment is given by:
$$ \mu_i = \exp(-\theta_1 \sum_{j=0}^{i-1}(1 + j\theta_2)^{-\theta_3}) $$
...
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Method of Moment Estimator — Uniform Dist
Find the two method of moment estimators for $\theta$ given that $Y_i | \theta$ is distributed i.i.d U(0,$\theta$).
We know that E($Y_1$) = $\frac{\theta}{2}$ and Var($Y_1$) = $\frac{\theta^2}{12}$ ....
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1answer
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Confidence Interval for a Uniform Distribution based on Method of Moments
Let $X_1,..,X_n$ be a random sample of $X$~$U[\theta,\theta+1]$. Given a sample $n=100$ from that distribution, the following statistic was calculated:
$\sum\limits_{i=1}^n X_i = 350.492$
I need to ...
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Solving a system of equation by moment condition reports error but minimum distance works
I do not understand the following error message I get using the gmm function in R. The code below creates two moment conditions (...
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1answer
581 views
By conditioning on $N$, show that the moment generating function of $Y$ is given by $m_Y(t)=m_N(\ln(m_X(t)))$
I am having a difficult time using moment generating function properties to prove this: (any direction or key properties will be very helpful)
Let $X_1$, $X_2$, . . . be independent and identically ...
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1answer
598 views
Two Parameter Method of Moments Estimation
I know this is a very basic question, but I am getting a bit confused due to some variation between resources I'm using for a statistics course.
Say you have some $iid$ random samples $X_1$,$X_2$,......
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1answer
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MME for exponential family
Let $X_1, X_2,...,X_n$ be iid random variables having pdf
$$f(x|\theta) = \frac{1}{x \sqrt{2\pi\theta}}e^{(-[\log x]^2/[2\theta])} I_{(0,\inf)}(x)$$ where $\theta > 0$. Determine the MME of $\theta$...
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1answer
47 views
Find the Method of moments estimate
So here i am getting E(X)=1/2,E(X^2)=0 and E(X^3)=Theta squared/4
How do i proceed now?How do i use the given x values ?
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1answer
588 views
Deriving method of moments estimator for AR(1) process
The method of moments estimator for AR processes can be had with the Yule-Walker equations. But how is it derived?
The equation for AR(1):
$$Y_t =aY_{t-1}+\epsilon_t$$
Where $\epsilon $ ~ $N(0,\...
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1answer
329 views
How can we determine the parameters of a Beta-Binomial Distribution for given mean and variance?
Given a beta-binomial random variate $X$ with $N$ known, how can I choose $\alpha$, $\beta$ such that the distribution's mean matches a chosen quantity $\mu$ and its variance matches a chosen quantity ...
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1answer
187 views
It is required to obtain the method of moment estimator and maximum likelihood estimator of a exponential distribution with two parameters
I have the following density:
$f(x)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}$ where $x>\theta$. It is required to obtain the maximum likelihood estimator and method of moment estimator for $\...
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49 views
How does a method of moments estimate compare with a true value of the parameter?
To cut the long story short, I've been given this information:
pdf:$$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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1answer
445 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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1answer
100 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 ...
