Expected Value Of Estimator

The Expected Value of an Estimator: A Deep Dive

Introduction:

In statistics, we often use sample data to estimate unknown population parameters. For example, we might use the sample mean to estimate the population mean, or the sample variance to estimate the population variance. These sample statistics are called estimators. A crucial aspect of evaluating the quality of an estimator is its expected value. The expected value of an estimator tells us, on average, how close the estimator's values are to the true population parameter. Understanding the expected value of an estimator is fundamental to assessing the bias and overall reliability of our statistical inferences. This article will delve into the concept, exploring its implications and providing illustrative examples.

1. What is an Estimator?

An estimator is a statistic calculated from sample data that is used to infer the value of an unknown population parameter. A population parameter is a numerical characteristic of a population (e.g., the population mean μ, the population variance σ², the population proportion p). Since we rarely have access to the entire population, we rely on samples to estimate these parameters. Common estimators include:

Sample Mean (x̄): An estimator for the population mean (μ).
Sample Variance (s²): An estimator for the population variance (σ²).
Sample Proportion (p̂): An estimator for the population proportion (p).

These estimators are functions of the sample data and provide our best guess of the corresponding population parameters.

2. Defining Expected Value in the Context of Estimators

The expected value (or expectation) of a random variable is its average value over an infinite number of trials. In the context of estimators, the expected value E(θ̂) of an estimator θ̂ (theta-hat) represents the average value of the estimator across all possible samples of the same size drawn from the population. This average is taken using the probability distribution of the estimator.

Mathematically, the expected value of an estimator is calculated as:

E(θ̂) = Σ [θ̂ P(θ̂)] (for discrete estimators)

E(θ̂) = ∫ θ̂ f(θ̂) dθ̂ (for continuous estimators)

where P(θ̂) represents the probability of the estimator taking a specific value θ̂ (for discrete cases) and f(θ̂) is the probability density function of the estimator (for continuous cases).

3. Unbiased and Biased Estimators

An estimator is considered unbiased if its expected value is equal to the true population parameter it estimates. Formally:

E(θ̂) = θ

where θ represents the true population parameter. If E(θ̂) ≠ θ, the estimator is biased. The bias of an estimator is defined as:

Bias(θ̂) = E(θ̂) - θ

An unbiased estimator, on average, hits the target. A biased estimator, on average, misses the target; it systematically overestimates or underestimates the true value.

4. Examples of Expected Value Calculations

Let's consider a simple example. Suppose we want to estimate the population mean (μ) of a normally distributed population with known variance. We use the sample mean (x̄) as our estimator. It can be shown that for a random sample from a normal population, E(x̄) = μ. Therefore, the sample mean is an unbiased estimator of the population mean.

Now, consider the sample variance calculated as s² = Σ(xi - x̄)² / (n-1). While this seems intuitive, it's actually an unbiased estimator of the population variance (σ²). If we used the formula Σ(xi - x̄)² / n, it would be a biased estimator.

5. Importance of Expected Value in Estimator Selection

The expected value of an estimator is a key criterion in choosing among different estimators for the same population parameter. While unbiasedness is desirable, it's not the only factor. We also consider other properties like:

Variance: A lower variance indicates that the estimator's values are clustered more tightly around its expected value, suggesting greater precision.
Mean Squared Error (MSE): MSE combines bias and variance, offering a comprehensive measure of estimator accuracy. MSE = Variance(θ̂) + [Bias(θ̂)]²

Ideally, we seek estimators that are unbiased or have minimal bias, low variance, and consequently, low MSE.

Conclusion:

The expected value of an estimator provides a crucial measure of its accuracy and reliability. Understanding its calculation and interpretation allows us to assess the quality of our statistical estimates. An unbiased estimator is desirable but not always attainable. By considering the expected value alongside other properties like variance and MSE, statisticians can select estimators that provide the most accurate and precise inferences about population parameters.

Frequently Asked Questions (FAQs):

1. Q: What does it mean if an estimator has a negative bias?
A: A negative bias means the estimator, on average, underestimates the true population parameter.

2. Q: Is unbiasedness always the most important property of an estimator?
A: No, while unbiasedness is desirable, a slightly biased estimator with significantly lower variance might be preferred in practice, especially if the bias is small.

3. Q: How does sample size affect the expected value of an estimator?
A: Increasing the sample size generally reduces the variance of the estimator but doesn't directly change its expected value if the estimator is unbiased. However, larger samples lead to more precise estimates.

4. Q: Can we calculate the expected value of an estimator without knowing the population distribution?
A: It's difficult, if not impossible, to calculate the expected value without knowing (or assuming) something about the population distribution. The calculation requires the probability distribution of the estimator, which is derived from the population distribution.

5. Q: What is the difference between expected value and mean?
A: In this context, they are essentially the same. The expected value of an estimator is its mean value across all possible samples. The term "expected value" is more commonly used in a theoretical statistical sense, while "mean" might refer to the average of a specific set of sample data.

Search Results:

Expected Value Calculator 7 Feb 2024 · Unlock the power of statistics with our expected value formula calculator. Learn how to calculate the expected value swiftly. Try it today!

Expected Value of an Estimator - Sites Expected Value of an Estimator The statistical expectation of an estimator is useful in many instances. Expectations are an “average" taken over all possible samples of size n. The process is fairly simple when working with discrete random variables. As an example, we examine a population of 4 rats (rat A, B, C,

Expected Value of estimator - Expected Value of an Estimator Expected Value of an Estimator. The statistical expectation of an estimator is useful in many instances. Expectations are an “average" taken over all possible samples of size n. The process is fairly simple when working with discrete random variables.

Expected Value: Definition, Formula & Finding - Statistics by Jim The expected value in statistics is the long-run average outcome of a random variable based on its possible outcomes and their respective probabilities. Essentially, if an experiment (like a game of chance) were repeated, the expected value tells us the average result we’d see in the long run.

3 Ways to Calculate an Expected Value - wikiHow 19 Jan 2024 · To calculate an expected value, start by writing out all of the different possible outcomes. Then, determine the probability of each possible outcome and write them as a fraction. Next, multiply each possible outcome by its probability. Finally, add up all of the products and convert your answer to a decimal to find the expected value.

Expectation value of an estimator - Mathematics Stack Exchange 24 Nov 2022 · So the expected value of the estimator is the population mean. This is good because in expectation, as we sample over and over again, the sample mean will give us the correct answer. The variance is. Var(x¯) =Var(1 n ∑i=1n xi) = 1 n2 ∑i=1n Var(xi) = 1 n2nσ2 = σ2 n.

Expected Value Calculator With our intuitive Expected Value Calculator, you'll be able to compute expected values with ease, saving you time and effort. In statistics, the Expected Value (EV) is the average outcome of a random variable over a large number of experiments or trials.

How to Calculate Expected Value in Excel using AI 6 Feb 2025 · If you roll a six, you win $5. If not, you lose your dollar. You might want to know if it’s worth playing. The expected value helps you understand the average gain or loss per game in the long run. To calculate the expected value, you multiply each outcome by …

Chapter 7. Statistical Estimation - Stanford University Bias measured whether or not, in expectation, our estimator was equal to the true value of . MSE measured the expected squared di erence between our estimator and the true value of . If our estimator was unbiased, then the MSE of our estimator was precisely the variance.

Statistical Methods: Exploring the Uncertain - 4.3: Properties of ... In this section, we continue our study of estimators of population parameters. An estimator, denoted θ ^, is a formula or rule that we use to estimate the value of an unknown population parameter θ. For a single parameter θ, there are many (possibly infinite) different estimators θ ^ from which to choose from.

Properties of Estimators - GeeksforGeeks 13 Nov 2024 · Estimators use information from a sample (a small part of a larger group) to guess about the entire group. A Good Estimator has these Five Main Properties: 1. Unbiasedness. An estimator is unbiased if its expected value equals the true parameter value.

Money blog: Blow for holidaymakers as Ryanair scrapping several ... 13 Feb 2025 · One of Britain's oldest department stores is shutting its last shop after more than 140 years due to tax and wage increases. Beales, which first opened in Bournemouth in 1881, said trading at its ...

7.1: Estimators - Statistics LibreTexts 24 Apr 2022 · Thus, for an unbiased estimator, the expected value of the estimator is the parameter being estimated, clearly a desirable property. On the other hand, a positively biased estimator overestimates the parameter, on average, while a negatively biased estimator underestimates the parameter on average.

The Expected Value – Explanation & Examples - The Story of … “The expected value is the average value from a large number of random processes.” In this topic, we will discuss the expected value from the following aspects: What is the expected value? How to calculate the expected value? Properties of expected value. Practice questions. Answer key. What is the expected value?

statistics - What is the expected value of the estimator? 24 Feb 2021 · By definition, an estimator $\hat{\theta}$ is a function that 'estimates' the value of the parameter $\theta$, itself being a random variable. An estimator is unbiased if it's expected value is $\theta$ .

Estimator - Wikipedia 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. [1]

Expectations and estimators - University of Oxford It is desirable that following a long run of experiments the statistic should home in on the correct value of the required parameter. Another way to say this is that the expectation of the statistic should equal the correct value. Such a statistic is called an unbiased estimator. Example of an unbiased estimator - the sample mean

Estimating Parameters from Simple Random Samples 10 Feb 2025 · One common summary of the error of estimators is the mean squared error (MSE), E( (X−t) 2), the expected value of the square of the difference between an estimator and the parameter it is intended to estimate. However, there are …

Expected Value Calculator - Good Calculators This expected value calculator helps you to quickly and easily calculate the expected value (or mean) of a discrete random variable X. Enter all known values of X and P (X) into the form below and click the "Calculate" button to calculate the expected value of X. Click on the "Reset" to clear the results and enter new values.

Moshi Alam – Econ 265: Introduction to Econometrics If yes, how precisely can we obtain those estimates; Will require assumptions; For OLS, we have 5 of them, which will constitute the Gauss-Markov assumptions; Expected Value of the OLS Estimator Unbiasedness of an estimator. An estimator is unbiased if, on average, it produces estimates that are equal to the true value of the parameter being ...

How to calculate the expected value of an estimator? 22 Oct 2020 · But your text says that the expected value of an estimator may be obtained by taking the average value of all possible samples of a given size (here 25) drawn from the population.

distributions - Expectation of an estimator? - Cross Validated 13 Apr 2012 · Basically, your estimate depends on the sample which is random, which makes your estimate a realisation of a random variable called estimator. Hope this helps. Just to complement what @ocram said, this r.v that you formulate is also called a statistic.

How to Calculate Expected Value in Sports Betting 4 Feb 2025 · Since the expected value is +5, this means you are expected to profit $5 per $100 bet in the long run. This is a +EV bet, and placing these types of bets consistently should yield a profit over time. Calculating EV for an NBA Point Spread Bet. Imagine an NBA game where the Lakers are -110 to cover a -3.5 spread against the Celtics. At -110 odds ...

Savers Value Village (SVV) Expected to Beat Earnings Estimates… 13 Feb 2025 · For the last reported quarter, it was expected that Savers Value would post earnings of $0.17 per share when it actually produced earnings of $0.15, delivering a surprise of -11.76%.

5 Examples of Calculating Expected Value in Real Life - Statology 12 Nov 2021 · Expected value is a value that tells us the expected average that some random variable will take on in an infinite number of trials. We use the following formula to calculate the expected value of some event: Expected Value = Σx * P (x) where:

Expected Value Of Estimator

The Expected Value of an Estimator: A Deep Dive

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