# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### Derivation of M-step for pLSA

I was looking at section 6 of these notes and trying to understand the derivation of the M-step at the top of page 10. I understood the derivation for the model without background, but I do not ...
215 views

### Proper similarity measure for clustering

I have problems in finding a proper similarity measure for clustering. I have around 3000 arrays of sets, where each set contains features of certain domain (e.g., number, color, days, alphabets, etc)....
479 views

### Variance explained for binomial proportions

In Rosenthal and Rubin (1979) A Note on Percent Variance Explained as A Measure of the Importance of Effects'', they give an example of where $r^2$ is deceptively low: Suppose half the patients ...
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### For linear regression, what's the distribution of error term from Classical and Bayesian point of views?

I know that linear regression is based on the assumption that the errors are normally distributed (from both bayesian and classical views). I'm just trying to verify this assumption based on the final ...
342 views

### Integral of survival function

If X is a nonnegative random variable representing the life of a component having distribution function F, and S is the survival ...
11k views

### Mean residual life

If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life ...
15k views

### A robust (non-parametric) measure like Coefficient of Variation -- IQR/median, or alternative?

For a given set of data, spread is often calculated either as the standard deviation or as the IQR (inter-quartile range). Whereas a standard deviation is ...
239 views

### Finding expectation of log

Does somebody have an idea to find the following expectation: $$E(X\log(a+bX)),$$ How can I proceed, can we go like this: $$=E(X)E[\log(a+bX)],$$if yes then what?
3k views

### Eigenvectors of a covariance matrix with only positive elements

If all the elements of a positive-definite covariance matrix are positive, how can I prove that the coefficients [elements] of the first principal component [first eigenvector] are all of the same ...
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### Sufficiency in Lehmann Scheffe

We are wondering what sufficiency in the Lehmann Scheffe Theorem is needed for. Our reasoning was: If an unbiased estimator is uncorrelated with all unbiased estimators of 0, it is UMVUE If the ...
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### Comparing two opposite ranked sets of data

Looking for the best formula to compare two sets of numbers Set #1 - Rank: Smaller numbers indicate higher "rank" (Ex: 1 is the best, 1,000,000 is the worst) Set #2 - Number of Items: The more items ...
592 views

### Why is the amount of variance explained by my 1st PC so close to the average pairwise correlation?

What is the relationship between the first principal component(s) and the average correlation in the correlation matrix? For example, in an empirical application I observe that the average ...
517 views

### Simple regression proof using general formula

I want to derive the least square estimation of the coefficients for $y=\beta_0+\beta_1*x_1+\varepsilon$ using the general formula. Can someone walk me through how to get from B to D in the image ...
4k views

### Showing that the order statistic $X_{(n)}$ is sufficient

I have some trouble showing sufficiency for largest order statistic ${x}_{n}$. This is from Casella's text, problem 1.6.3. Let ${p}_{\theta}$ be a density function. ${p}_{\theta}(x)=c({\theta})f(x)$ ...
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### Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
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### Estimating misclassification rate from summary classification values

I'm trying to work out the misclassification rate in a published study, where the results do not include this value. However, I have the sensitivity, ...
255 views

### Representation within a RKHS framework

Given a p.s.d kernel $Q$, can minimization/maximization of $Tr(X^TQX)$ over X be represented within a reproducing kernel Hilbert space (RKHS) framework? If there is a primary concern with the trace ...
2k views

### What test will tell me a normalized percentage of the data?

I'm working with baseball statistics and I have a list of the players' batting percentages. I want a formula that will tell me what percentage of the data is represented from a player's score ...
849k views

### What's the difference between variance and standard deviation?

I was wondering what the difference between the variance and the standard deviation is. If you calculate the two values, it is clear that you get the standard deviation out of the variance, but what ...
I am dealing right now with a lot of distributions, e.g., $F$, $t$, $\chi^2$. I was wondering why do these degrees of freedom signify for distributions such as the $F(m,n)$ distribution?