# What is Estimators: Definition and 28 Discussions

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.

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1. ### I What to do with biased estimators if we don't know the bias term?

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...
2. ### I What are the commonly used estimators in regression models?

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...
3. ### I Estimating Gene Mutation Proportion: A & B Approaches

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...
4. ### I Distributivity/Inheritance of Max Likelihood Estimators

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 "...
5. ### I Maximum Likelihood to find the original data or estimation directly

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...
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### I Problems that could occur in estimating n from a Binomial distribution

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...
7. ### A Information contained in minimum value of truncated distribution

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...
8. ### MHB Maximum Likelihood Estimators for Uniform Distribution

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...
9. ### MHB What is the Maximum Likelihood Estimator for Uniform Distribution Endpoints?

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...
10. ### How to show that the Method of Moment Estimators for the Normal

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 (...
11. ### How to Find Maximum Likelihood Estimators for Sample Data?

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.
12. ### Why is/was consistency of estimators desired?

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...
13. ### Asymptotically unbiased & consistent estimators

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...
14. ### Comparing accuracy of estimators

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.
15. ### Biased and unbiased estimators

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 ?
16. ### Properties Of Estimators Question

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...
17. ### Proving Sufficient Estimators for Gamma Distribution | Homework Solution

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...
18. ### Why are my T1 and T2 values so different if they are both unbiased estimators?

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 =...
19. ### How Can We Reduce Bias in LogNormal Mean Estimators?

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...
20. ### Proof of least Squares estimators

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...
21. ### Statistics: Consistent Estimators

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...
22. ### Statistics: Consistent Estimators

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...
23. ### Estimating the Number of Volumes in a Library Using Two Proposed Estimators

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...
24. ### MSE and Estimators for Random Samples

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...
25. ### Statistics Unbiased Estimators

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)...
26. ### Final exam questions: estimators.

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...
27. ### How can I improve my understanding of Maximum Likelihood estimators?

Can anyoe help with likelihood estimtor problems? :cry:
28. ### Maximum liklihood estimators again help

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) =...