# Tagged Questions

"A rule, method, or criterion for arriving at an estimate of the value of a parameter."

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### Variance computed using Taylor series does not agree with numerical experiment

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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### Variance of Estimator (uniform distribution)

In my script for statistical signals, I have some troubles to get the same result for the variance of an estimator $T$. Here is the example: Given the observations $X_1, \dots , X_N$ of a uniquely ...
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### Minimising MSE of $\sigma^2$ estimator of specific form

I have found a past exam question for a statistics course and can't seem to find the required result. Part A is fine but my working for part B must be incorrect [see below]. Can anyone figure out ...
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### Non-linear least squares and the distribution of an estimator

I have been trying to find the asymptotic normality of the non-linear least squares estimator. If I start with $0=X_t(\beta)'(y_t-x_t(\beta))$. I know that I have to perform Taylor expansion around ...
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### A specific way to minimise the variance when importance sampling

Consider the following problem. We are interested in approximating from samples the expectation of $h$: $$\int_t p(t) h(t) d(t)$$ We seek to obtain a lower-variance estimate by using importance ...
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### What's the difference between asymptotic unbiasedness and consistency?

Does each imply the other? If not, does one imply the other? Why/why not? This issue came up in response to a comment on an answer I posted here. Although google searching the relevant terms didn't ...
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### Example of a consistent estimator that doesn't grow less variable with increased sample size?

I've had it asserted to me that any consistent estimator must necessarily also grow less variable with increased sample size. I felt that this couldn't be correct, since there was nothing in the ...
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### Estimating R90 (radius of a circle expected to include 90% of impacts)

I want to determine how big a target I can hit with a bow at a certain distance with 90% probability. I place some paper targets at that distance and shoot 20 arrows at them. I have a ruler and a ...
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### Estimating a variable from its cosine corrupted by additive Gaussian noise

I observe $y_i=\cos(\theta)+z_i$, $i=1,\ldots,n$, where each $z_i\sim\mathcal{N}(0,\sigma^2)$ is an i.i.d. zero-mean Gaussian random variable. I am interested in estimating $\theta\in[0,\pi]$ with ...
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### Rule of Succession for Unfair Coin

Given the first n flip results from an unfair coin, we wish to estimate the probability that the next flip is a heads. I can take 2 approaches to this: Frequentist:...
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### Comparing large and small populations

Here is a hypothetical that mimics my actual data (sensitive so cannot post actual details). I have 149 students in a class. In preparation for the final exam (which is pass/fail, NOT ranked), I held ...
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### Why is the intercept of linear regression biased?

Out of curiosity, I conducted the following simulation (code below). Why is it that when the variance of the error term is large coefficient associated with the intercept is biased? Can you recommend ...
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### Confidence interval difference for the difference of two estimators

Let us assume that we havo two parameters $\alpha$ and $\beta$ and the two maximum likelihood estimators $\hat{\alpha}$ and $\hat{\beta}$. We have also the confidence interval of these estimators, ...
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### How well does sample average range estimates sigma?

Suppose there are m subgroups of n items each. The following practice is described by Montgomery in the construction of a $\bar{X}-R$ chart: Let $\bar{R}$ be the average range of the m subgroups. ...
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### Is M-estimation valid only for regression models?

Is M-estimation valid only for regression models or does it's working hold good for robust estimation of parameters in other statistical models? I understand that M-estimators are asymptotically ...
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### Square of the Sample Mean as estimator of the variance

Suppose we have the following random variables $X_1$, $X_2$,....$X_n$,.., that are $iid$ but we dont know what distribution they follow. I know that the sample mean $\bar{X}$ is an unbiased ...
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### Reporting Data with Estimates and SE

What do you think? When reporting data in an article, a usual way is to show the mean values +/- the standard errors in a table. For Example, when you measure Abundance of pigeons in parks of five ...
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### Variance of estimator(exponential distribution)

I have exponential distributed data $Exp(\lambda)$ with sample n = 50. Also, The sample mean = 2.17. I need to find the estimator of parameter $\lambda$ by the method of moments and to build 95% ...
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### Are point estimators and set estimators the same thing?

My professor defined a point estimator as A mathematical rule that maps a random sample $\lbrace x_i \rbrace_{i=1}^n$ into a 'best guess' at the parameter $\theta$ I am confused about what ...
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### What are the bias and variance of a model returning the observed mean for a training set?

It seems to me that bias = variance = 0 but MSE > 0, possibly very high, so clearly my intuition, and math, are wrong. For a training set $T$ and a regression problem let $M(T) = \text{Ave}(y(T))$. ...
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### When is a biased estimator preferable to unbiased one?

It's obvious many times why one prefers an unbiased estimator. But, are there any circumstances under which we might actually prefer a biased estimator over an unbiased one?
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### Linear regression: Estimator for OLS and Minimum Absolute Deviations

I've received conflicting information in 2 different statistics classes, and I want to better understand the problem before I ask either of them for clarification Prof 1: We use OLS for linear ...
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### Procedure for calculating a sampling distribution

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
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### Paired comparison of instruments using different measurement samples

I have instruments A, B, C and D - I'm in search of the best one. The problem: For illustrative purposes, let's use an example of evaluating the best among the instruments measuring difference in ...