In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean.
There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators.
Estimation theory is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. However, in robust statistics, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having less good properties that hold under wider conditions.
Hello,
I understand that we have a population of values. We don't know the parameters of this population. The parameters are numbers, each one describing the population in a collective sense. Examples of parameters are the mean, the median, the mode, the variance, skewness, kurtosis, etc.
We...
Hello everyone,
I am trying to close the loop on this important topic of estimators.
An estimator is really just a function to calculate point statistics that are close estimates (with low variance) of population parameters. For example, given a set of data, we can compute the mean and the...
The proportion of individuals that carry a certain gene mutation in the population is unknown. A research assistant at a medical laboratory wants to estimate this proportion. The research assistant is thinking of two approaches:
A. Take blood samples from all individuals that come to the...
Hi,
IIRC, Maximum Likelihood Estimators ( MLEs) satisfy an " Inheritance" property , so that if ##m_1,m_2,..,m_n## are MLEs for ##M_1,M_2,...,M_n## respectfully and f is a Random Variable of the ##M_i##, then the MLE for f is given by ##f(m_1,m_2,...,m_n)##. Is this correct? If so, is there a "...
I make confusions in the using of Maximum Likelihood to find (approximately) the original signal knowing the data observed and the using of Maximum Likelihood to find estimations of parameters of the PSF
1) Find (up to some point) the original signal :
I start from this general definition (in...
Hi, I am doing the following question:
https://i.gyazo.com/f2e651334bcbd5f1dcb6d661e4160956.png
I have estimated both n and theta. But the part that is throwing me off is what problem could you encounter in estimating n here? My only idea is that it might be something to do with the sample...
Suppose that a given population is endowed with a pair of characteristics T and K. Let's think of these characteristics as random variables
(T,K)∼BiNormal((μT,μS),(σT,σS),ρ)
I observe the realisations of T for a sample consisting of those individuals with K<a, where the selection threshold a...
Find maximum likelihood estimators of an sample of size $n$ if $X\sim U(0,\theta].$
Hello MHB :)! Can any user help me please :)! I don't how follow...
I need help on this problem, anyone know how to do it?
Suppose you have n independent observations from a uniform distribution over the interval [𝜃1, 𝜃2].
a. Find the maximum likelihood estimator for each of the endpoints θ1 and θ2.
b. Based on your result in part (a), what would you expect...
So I'm trying to show that the estimators for the normal distribution by the method of moments are consistent. So to show consistency, I have to :
1) Show E(θ(estimator) = θ (parameter)
2) lim n-->∞ Var(θ(estimator) = 0
So Since there are two estimators in the normal distribution (...
Homework Equations
L(x,p) = \prod_{i=1}^npdf
l= \sum_{i=1}^nlog(pdf)
Then solve \frac{dl}{dp}=0 for p (parameter we are seeking to estimate)
The Attempt at a Solution
I know how to do this when we are given a pdf, but I'm confused how to do this when we have a sample.
Why is/was "consistency" of estimators desired?
In an article, I found while researching another thread ("Revisiting a 90-year-old debate: the advantages of the mean deviation", http://www.leeds.ac.uk/educol/documents/00003759.htm ), the author states this bit of statistics history:
I...
Theorem: If "θ hat" is an unbiased estimator for θ AND Var(θ hat)->0 as n->∞, then it is a consistent estimator of θ.
The textbook proved this theorem using Chebyshev's Inequality and Squeeze Theorem and I understand the proof.
BUT then there is a remark that we can replace "unbiased" by...
May I ask what is needed to be computed in order to compare the estimators of the population mean \bar{y}_{N}, in terms of accuracy? That is to compare the multivariate ratio estimator \bar{y}_{RM} with the Y-only estimator \bar{y}_{n} in terms of accuracy.
Hey All,
I am comfortable with the idea of biased and unbiased estimators, but what I don't understand is why you would ever want to use a biased estimator ? at the end of the day doesn't it mean that the sample statistic is different from the population statistic you are trying to estimate ?
Homework Statement
suppose that 14, 10, 18, 21 constitute a random sample of size 4 drawn from a uniform pdf defined over the interval [0, theta], where theta is unknown. Find an unbiased estimator for theta and y'3, the third order statistic. What numerical value does the estimator have for...
Homework Statement
Show that the product of the sample observations is a sufficient statistic for theta if the random sample is taken from a gamma distribution with parameters alpha = theta and beta = 6.
Homework Equations
The Attempt at a Solution
So I need to make sure that Y...
Homework Statement
See attached.
Homework Equations
The Attempt at a Solution
I have no issues with part A. I simply took the expected value of T1 and T2 and everything turned out fine. What I'm having issues with is part B.
I have:
T1 = (4 / n) * 1997 - 2 = 0.08075
T2 = (4 / n) * 32 =...
If Y~N(mu,sigma) and y=logX, with X~LN(mu,sigma),
with a*=exp{ybar+1/2*theta*sample variance of y}, where ybar=sample mean of y and a=E[X]=exp{mu+1/2*sigma^2}, theta is constant.
If theta=1, a* is consistent but biased and we can reduce the bias by choosing a different value of theta. Use a...
Hey guys, long time lurker, first time poster!
Just having some trouble with something..Im probably just looking at it the wrong way, but I was wondering if anyone could help me with this..
Im trying to prove that by choosing b0 and b1 to minimize...
Homework Statement
Q1) Theorem:
An asymptotically unbiased estimator 'theta hat' for 'theta' is a consistent estimator of 'theta' IF
lim Var(theta hat) = 0
n->inf
Now my question is, if the limit is NOT zero, can we conclude that the estimator is NOT consistent? (i.e. is the theorem...
1) Theorem:
An asymptotically unbiased estimator 'theta hat' for 'theta' is a consistent estimator of 'theta' IF
lim Var(theta hat) = 0
n->inf
Now my question is, if the limit is NOT zero, can we conclude that the estimator is NOT consistent? (i.e. is the theorem actually "if and only...
Homework Statement
A library has been given 3 books. these books carry volume numbers X1, X2 and X3 where
X1<X2<X3. But it is not known how many volumes there are altogether in the set. Suppose there are n volumes, numbered 1,2,3,...n in the set, and the 3 volumes in the library are...
I would very much appreciate if someone could explain the following:
- What is the use of the MSE (Mean Square Error) i.e. why do we use it?
I understand that MSE(t) = Var(t) + {E(t-&)}^2, but what does this tell us?
- Why/ How does E{A*Sx^2 +b*Sy^2} = a*Var + b*Var
(I am using ^ to...
1. Let X1,X2, ... ,Xn be independent identically distributed random variables with ex-
pected value \mu and variance \sigma^2: Consider the class of linear estimators of the form
\mu\widehat{} = a1X1 + a2X2 + ... + anXn (1)
for the parameter \mu, where a1, a2, ... an are arbitrary constants.
a)...
have a final exam on monday, and cannot figure out the stuff on estimators:
1) a random sample of size 2, y1, y2 is drawn from the pdf f(y, theta) = 2y(theta^2), 1 < y < 1/theta.
what must c equal if the statistic c(y1 + 2y2) is to be an unbiased estimator for 1/theta.
I really don't...
I dotn know, I'm still lost on this whole MLE thing...but here is my attempt at some problems...please critique(just the concept is still bugging me).
find the MLE:
p_x(k;\theta) = \theta^k (1-\theta)^{1-k}, . k = 0, 1, 0 < \theta < 1 .
so here is what I did.
L(\theta) =...