We changed our privacy policy. Read more.

Questions tagged [estimator]

For questions about estimators; an estimator is an approximations of a parameter that is using the avaiable data. There could be multiple estimators for a particular parameter.

Filter by
Sorted by
Tagged with
0
votes
1answer
26 views

Showing that $\frac{\bar{X}}{10}$ is an unbiased estimator for p

Let $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}$ be a random sample from a binomial distribution with $n=10$ and $p$ unknown. How do I show that $\frac{\bar{X}}{10}$ is an unbiased estimator for $p$ and then ...
1
vote
1answer
59 views

What is Roberts's method for gradient estimation?

Graphics Gems II, chapter II.9 (Image File Compression Made Easy) says: These constraints are easily met for arithmetic prediction using Roberts’s method L.R., a “poor man’s Lapacian” (gradient ...
0
votes
0answers
33 views

Risk function of an estimator [closed]

We have a random sample of size $n = 2$ from a normal distribution, $X_{i} \sim N(\theta,1)$, where $0 \leq \theta \leq 1$. Our estimator is defined as $ \hat{\theta} = \frac{1}{4} X_{1} + \frac{3}{4}...
0
votes
0answers
13 views

Unbiased asymmetry estimators for non-normal distributions

The sample skewness estimator is not guaranteed to be unbiased for non-normal distributions (see, e.g. this and this). I would like to know alternative measures for asymmetry that have unbiased ...
0
votes
1answer
49 views

variance of score of a game

In a simply setting, we have 2 players, player $A$ and player $B$, playing a game of chess repeatedly. Player 1 wins with probability $p_1$, player 2 with probability $p_2$ and a draw with probability ...
0
votes
1answer
43 views

Modifying a point-estimation to get a $unbiased$ $estimator$ $of$ $variance$

My Exercise Problem: Let x1, x2, ..., x7 be observations of independent random variables X1, X2, ..., X7 with ...
2
votes
1answer
106 views

What is the difference between MVUE and UMVUE

I was going through the minimum variance unbiased estimators and I am confused about the concept of MVUE and UMVUE. Is the unbiased estimator whose variance attaining CRLB a UMVUE or MVUE? I referred ...
1
vote
1answer
60 views

Estimator : Mean of ratio of quantities tends to be equal or equal to ratio of mean of quantities

I have the following estimator : $\hat{O}=\dfrac{b_{sp}^{2}\left(\mathcal{D}_{DM}+B^{C}\right)+B_{sp}}{b_{ph}^{2}\left(\mathcal{D}_{DM}+B^{C}\right)+B_{ph}}$ I would like to demonstrate that the mean ...
5
votes
1answer
78 views

Find Maximum-Likelihood-Estimator (MLE) for $\alpha$

Consider the following PDF: $$w_{\alpha,\beta}(x):=\alpha \beta x^{\beta-1}e^{-\alpha x^{\beta}} \mathbf{1}_{(0,\infty)}(x)$$ This is the Weibull distribution often used in material science. Assume we ...
1
vote
1answer
40 views

Poisson parameter interval estimation vs. the CLT: why not simple?

I've never been much of a mathematician, and I'm now trying to catch up on some things I should have studied properly many years ago. So forgive me if the question is naive! I'm relearning statistics, ...
1
vote
1answer
47 views

Is the estimator $\hat \theta =\max{\{X_1,X_2,\ldots,X_n\}}$ consistent?

This is a follow up question to this: Conceptual question about estimators I am still stuck at showing consistency for the second part. A random number generator produces uniformly distributed random ...
1
vote
1answer
35 views

Show that $E\left[(\hat{\mu}[n+1]-m)^2 \right] \le E\left[(\hat{\mu}[c]-m)^2 \right] $ (minimizing expected MSE)

Give a sample $(X_1,X_2,\ldots, X_n)$ where $X_i$'s are i.i.d exponential random variables. with parameter $\frac{1}{m}$ (mean $m$). Let $$\mathcal T=\{\hat{\mu}[c] \, \vert\, c>0\} \hspace{0.5cm} \...
0
votes
2answers
52 views

Show that the estimator $\hat\lambda_n=\frac{1}{n+a}\sum_{i=1}^n X_i$ is consistent for all $a>0$

Let $X_1,X_2,\ldots$ be independent poisson random variables with $\lambda>0$. Show that for any $a>0$ the sequence $(\hat\lambda_n)_{n \in \mathbb N}$ is consistent given that: $$\hat\lambda_n=\...
2
votes
1answer
39 views

Estimator problem. Can chebyshev's inequality be applied to a sequence?

Let $g: \Theta \to \mathbb R$ be a function of an unknown parameter $\theta$. We want to estimate $g$. Let $\hat g_n$ be a sequence of estimators for $g(\theta)$ such that for all $\theta \in \Theta$:...
1
vote
1answer
36 views

Conceptual question about estimators

A random number generator produces uniformly distributed random numbers on the intervall $[0,a]$ where $a>0$ is unknown. We can draw $n$ independent $\mathcal U_{[0,a]}$ random numbers $X_1,...,X_n$...
0
votes
0answers
8 views

Estimating confidence intervals for the variance of a sampling distribution (from normal), given that the samples are not drawn independently

A random variable $X \sim N(\mu, \sigma^2)$. One day, we draw a sample $X_1, X_2, ..., X_n$ from this population. This gives an estimate of the actual distribution of $X$. What I would like to ...
2
votes
0answers
47 views

PCA for retrieval from absolute values

I have given $(x_i)_{i=1}^n$ $d$-dimensional iid. random variables with $x_i\sim\mathcal N(0,I_d)$ and $y_i=|\langle x_i,\theta\rangle|$ with $\theta\in\mathbb R^d$. First I have to assume that $\|\...
0
votes
0answers
43 views

Finding unbiased estimators for the mean and variance of $X \mid Y = y$ from realizations $x$ with some belief of $Y$

I will start with some background which can be skipped. I am tracking an object in space using a sensor. The variable Y denotes some discrete property that I can track with some amount of uncertainty. ...
2
votes
1answer
53 views

MLE of two variables

Suppose $X$ and $Y$ are random variables and let $x_1, ..., x_n$ be observed values from a random sample of $X$. Assume that $Y_i = 2\alpha x_i + \alpha + \beta_i$ where $\alpha$ is unknown and $\...
0
votes
1answer
58 views

Finding unbiased estimator for $\theta$

Let $f(x)=\frac{2}{(b-\theta )^{2}}(x-\theta )$ be a probability density function of random sample $(X_{1},X_{2},...,X_{n}) $where $\theta < x< b$ ($b$ is known constant) .Find unbiased ...
0
votes
0answers
14 views

Find a unbiased estimate of k [application type question]

A bag contains four balls, of which k are black, where k is either 1 or 2. Balls are drawn from the bag, randomly, repeatedly, one-by-one and with replacement, until a black ball has been drawn. We ...
0
votes
0answers
19 views

Consistency of Order Statistics Estimator

Let us have distribution with density function $f(x) = 5 \theta^{5} y^{-6} I\{y \geq \theta\}, \theta>0 .$ Where $\theta$ is a parameter to estimate. I use maximum likelihood estimator $\hat{\theta}...
3
votes
1answer
44 views

Confidence interval and moment

Suppose a random sample of size $n = 9$ is taken, where $X$ is normally distributed with unknown mean $\mu$ and unknown variance $\sigma^2$. Consider the following cases (a) Before the sample is taken,...
1
vote
1answer
303 views

How do these results show that $T(\mathbf{X})$ is an unbiased estimator of $E_\varphi[T(\mathbf{X})]$ that achieves the Cramer-Rao lower bound?

Let's say that $X_1, \dots, X_n$ has the joint distribution $f_\varphi(\mathbf{x})$ that belongs to the one-parameter exponential family $$f_\varphi(\mathbf{x}) = \exp{\left\{ c(\varphi) T(\mathbf{x}) ...
1
vote
1answer
41 views

Is there an efficient estimator for $\theta$?

Given a simple random sampling with density function: $$f_{\theta}(x) = \frac{\ln(\theta)\theta^x}{\theta - 1} I_{(0,1)}(x)\,, \quad\theta \gt1$$ Is there an efficient estimator for $\theta$? I know ...
2
votes
0answers
18 views

Find $a_1$ and $a_2$ such that estimator is unbiased

Let us consider the sample of independent random variables $X_1,\ldots,X_n$ with $E[X_1]=E[X_2]\ldots=E[X_n]=\mu$. We examine the following estimator of $\mu$: $$a_1(X_1+X_2+\ldots+X_n)+a_2.$$ Find ...
-1
votes
1answer
32 views

Sample mean estimator [closed]

The sample mean of $N$ independent random variables with the same distribution is an estimator that is unbiased and consistent? And if one random variable and calculare the sample mean is that ...
0
votes
1answer
21 views

Can I use another parameter in an unbiased estimator?

Given a multinomial distribution, the task is to make an unbiased estimator for one parameter. However, the parameters all add up to 1, so how can I make an unbiased estimator without using other ...
0
votes
0answers
6 views

Are these estimators linear?

How can you prove that the following estimators are linear? $$ \tilde{b}(X,Y) = \begin{pmatrix} \tilde{\alpha}\\ \tilde{\beta}\end{pmatrix} = \begin{pmatrix} \bar{Y}_3 - \frac{Y_3 - Y_1}{X_3 - X_1}\...
1
vote
1answer
79 views

Finding the variance of MLE and comparing to MOM from a gamma distribution.

This is from Casella Berger chapter example 10 (11) part d. we have the pdf $f(x|\mu,\beta)= \frac{x^{\mu/\beta -1 e^{-x/\beta}}}{\gamma(\mu/\beta) \beta^{\mu/\beta}}$. This pdf was reparameterized ...
2
votes
1answer
145 views

MLE of power $2$

Suppose $X$ and $Y$ are random variables and let $x_1, ..., x_n$ be observed values from a random sample of $X$. Assume that $Y_i = \alpha x_i^2 + \beta_i$ where $\alpha$ is unknown and $\beta_1, ..., ...
0
votes
0answers
47 views

Sufficient statistic as an estimator

Suppose that $X$, with a random sample $X_1, ..., X_n$, is a random variable with the pdf $f(x;t) = 2xt^{-2}$ and support $x\in [0, t]$, where $0 < t < \infty$ is unknown. (i) Show that $W = max\...
1
vote
0answers
61 views

Likelihood estimators and order stats

Suppose that $X$, with a random sample $X_1, ..., X_n$, is a random variable with the following pdf: $f(x; t) = xte^{\frac{-x^2t}{2}}$ with the following support: $x \in [0, \infty)$. Note that $0 <...
2
votes
1answer
32 views

Trouble Understanding a Part of the Method of Moment Estimator Solution for Normal Random Variables

Good day. I've been trying to understand the solution for the $\sigma^2$ of the method of moment point estimator for the normal random variables $X_1, X_2,...,X_n$ of mean $\mu$ and variance $\sigma^2$...
0
votes
1answer
25 views

Normal distributions

X and Y are positive continuous random variables that are approximately normally distributed with E(X) = 50, sd(X) = 6 and E(Y) =30, sd(Y) = 4. Pr( X/Y > 2) is equal to I'm not sure how to do this. ...
1
vote
1answer
48 views

Concluding that MLEs for exponential distribution parameters are biased and then unbiasing them

I have the i.i.d. exponential random variables $X_1, \dots, X_n$ with the density functions $$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)/\sigma} &\text{if}\, x\geq \tau\\ ...
0
votes
1answer
49 views

How to find the smallest variance that can be achieved by an unbiased estimator?

Let $X_1, \dots, X_n$ denote a random sample from the PDF $$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{...
0
votes
1answer
21 views

Estimator in statistics: Correlating random variable with real life example

An estimator is a process to construct estimates for a quantity q based on a random sample $X_1$,$X_{2}$, ..., $X_{n}$. So suppose we want to check how many hours ...
1
vote
1answer
42 views

Estimating a bivariate moment generating function [closed]

Assume there are many identical and independent sample pairs e.g. $(X_1, Y_1), (X_2, Y_2), (X_3, Y_3), \dots, (X_n, Y_n)$. How do you consistently estimate the following function $M(t_1, t_2)$, such ...
2
votes
0answers
53 views

Which tosses should one choose to better estimate $p$?

Puzzle $1$: A coin $C_1$ has probability $p$ of turning up head, while a coin $C_2$ has probability $2p$ of turning up head. All we know is that $0 < p < \frac12$. Now, $20$ tosses are given. ...
0
votes
0answers
20 views

Why do we use sufficient statistics in Rao-Blackwell theorem

Suppose we have an estimator $\theta_0$ for a random variable $X$. The Rao-Blackwell theorem states that the variance of the new estimator $\theta(t)=E[\theta_0|T=t]$ is smaller than the variance of $\...
0
votes
0answers
16 views

Find the MRE for b of E[0,b]

$X_1,\dots , X_n$ i.i.d. from the E(0,b) distribution. Find the MRE (Minimum Risk Equivalent estimator) for b under the scale transformation group with the standardized square loss $𝐿(𝑏,\delta)=(\...
2
votes
1answer
48 views

MLE and biasedness

Question: Let $X_1, ..., X_n \sim \text{Exp}(\lambda)$ where $E(X) = \lambda$. Find MLE for $\log(\lambda)$, and show if it is biased or not. From the description, $f_X(x) = \lambda^{-1}\exp(-\lambda^{...
0
votes
0answers
25 views

understanding the proof that the average of sum of i.i.d cauchy is not a consistent estimator of location parameter

Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$ To prove that it is inconsistant, consider the characteristic function of $X_i$ and $\bar{X}$, which ...
0
votes
0answers
40 views

Compute a Monte Carlo estimate. Which of the variances (of $\hat{\theta}$ and $\hat{\theta}^{*}$) is smaller, and why?

Compute a Monte Carlo estimate $\hat{\theta}$ of $$ \theta = \int_{0}^{0.5} e^{-x} dx $$ by sampling from Uniform$(0, 0.5)$, and estimate the variance of $\hat{\theta}$. Find another Monte Carlo ...
0
votes
1answer
47 views

How to estimate population by using capture-recapture method

Recently I am studying using statistical methods to estimate animal abundance. It seems that the mark recapture method is a widely used statistical method to do it. However, there are some concepts I ...
2
votes
1answer
87 views

understand the solution to the unbiased estimator of area of circle when given n independent radius $R$ measurement with error $\sim N(0,\sigma^2)$

$S = \pi R^2$ $E(\bar{X}^2) = Var(\bar{X})+(E(\bar{X}))^2=\sigma^2/n + R^2$ then it states thats an unbiased estimator of $\sigma^2/n = \frac{1}{n(n-1)} \sum_{i=1}^n(X_i-\bar{X})^2$. However, when i ...
2
votes
0answers
8 views

Independence between $2$ Pearson Correlation Coefficients

Suppose that $X_1, X_2, Y$ are three scalar Gaussian distributed variables. Suppose we know that $X_1$ independent from $X_2$. However, $X_1, Y$ could be correlated. Similarly, $X_2, Y$ could be ...
0
votes
0answers
24 views

Can the moment estimator be not sufficient?

As the title, can the moment estimator sometimes be NOT sufficient? Consider the following example where Y ~ Uniform(1-b, 5+b): From method of moment estimator: E[Y] = 1/2((1-b) + (5+b)) = 3 => ...
1
vote
0answers
25 views

Non-existence of unbiased estimators for normal distibution

Let $X = (X_{1},...,X_{n})$ be a random sample from the normal distribution with mean $\theta \in \cal{R}$ (real numbers) and variance 1. Whether the estimand $g(\theta) = |\theta|$ has an unbiased ...

1
2 3 4 5