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

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what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula (http://math.stackexchange.com/questions/1410492/what-is-the-intuition-behind-slovins-formula), and learned shortly thereafter that it was derived from this ...
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1answer
18 views

MLE of Gamma when only given observations [on hold]

i've been given 10 observations of X, where X is a random variable. the observations are 141 16 46 40 351 259 317 1511 107 567 and now given they are gamma distributed, how do you find the MLE using ...
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39 views

Convergence of Sequence Random Variables

I'm a bit confused by the definition of convergence of a sequence of random variables. I'm reading Wasserman's All of Statistics and at the start of Chapter 5, he writes: "Suppose that $X_1, X_2...$ ...
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14 views

Comparing categorical and numerical results simultaneously

This question is in regards to options for analysing a dataset. I have a dataset where we have 23 video clips that have been treated using three different algorithms to change how each clips looks. ...
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6 views

Fitting a power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
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23 views

Terminology confusion re: sample outcomes

I've just started Wasserman's All of Statistics and he starts by saying: "The sample space $\Omega$, is the set of possible outcomes of an experiment. Points $\omega$ in $\Omega$ are called sample ...
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2answers
11 views

Quantifying the irregularly of a data set

I have two 1D finite domains consisting of $N$ points each. In one domain the points are all regularly spaced, but in the other they are irregularly spaced. I would like to somehow quantify just how ...
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3answers
81 views

Nonrandomized better performance

On the footnote of page 497 in Jaynes' "Probability Theory: The Logic of Science" he writes Of course, Fisher’s randomized planting methods – which we think to be not actually wrong, but ...
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26 views

How to interpret prediction accuracy based on error analysis in test and validation sets?

I would like to calculate accuracy of my prediction model based on polynomial regression and I got some values for the test and validation errors 3.895 and 4.0125. I am wondering how these values for ...
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30 views

Sample size for clinical research

I determined the sample size already (25), but I want to know if it has to be increased because of a minor change. Our study consists of two groups - one will be sedated and the other will not. For ...
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10 views

Normalize data points against an expected value

I have set of data points(y coordinates) representing peak points in an graph. Each data set consist of 4 groups and each group contains list of data points representing y coordinates in graph. I want ...
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25 views

Multiclass classification - Optimizing for recall in subset of classes

As part of a wider analysis, I'm seeking a method to reliably recall the outer margins (~10% each) of a data set. I don't mind if precision is lower as a result. I've separated the data into 3 ...
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15 views

Probability of a team scoring a goal?

I am trying to work out the probability of teams scoring a goal in a football match. Obviously there are many factors to consider, and the probability is impossible to predict correctly as there are ...
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11 views

Confidence intervals: how do we have to interpret them? [duplicate]

I have a very basic question regarding confidence intervals. I am confused between two interpretations of confidence intervals. Interpretation 1: If repeated samples were taken and the 95% ...
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1answer
35 views

Moment-generating function (MGF) of non-central chi-squared distribution

I need to compute the moment-generating function of the non-central chi-squared distribution, but I have no idea where to begin.
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40 views

How many trials are needed to determine the probability of an outcome?

Say I have a coin, how many flips would I need to do and record to determine that results of said flips are accurate, that there is indeed a 50% chance to land on heads or tails? 100, 1000, 10000? ...
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95 views

Calculating $\operatorname{var} \left(\dfrac{X_1-\bar{X}}{S}\right)$

Suppose $X_1,X_2,\ldots, X_n$ are random variables distributed independently as $N(\theta , \sigma^2)$. define $$S^2=\frac{1}{n-1}\sum_{i=1}^{n} (X_i-\bar{X})^2 ,\qquad ...
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14 views

In a random sample, is the range of a variable within the sample a random variable?

I recently saw a line in my textbook with an example that said the range of a variable within the sample is not a random variable. I was confused with this because it appears that while we do know the ...
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1answer
31 views

Can we calculate Z-score for any distribution?

Is z-score only confined to normal distribution or can it be used for any distribution
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18 views

ROC curve analysis

I wanted to ask a question relating to the ROC curves. Suppose I have a drug intervention data set with pre-drug and post-drug values. Can I use a ROC curve for such paired data analysis? I know that ...
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1answer
31 views

Meaning of Dummy Variable

I am working on a case study from Kaggle: https://www.kaggle.com/c/liberty-mutual-fire-peril Here, in the variables, there is a variable: "Dummy" about which the description says: (dummy: Nuisance ...
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1answer
13 views

Confidence Intervals And Probability Mean Is Not Closer To Tails

I'm curious if there's a good way to estimate the probability that the true mean of a population falls closer to the observed sample mean than the tail ends of the margin of error. That is, I believe ...
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25 views

Adding two distributions with proportional means and standard deviations

I need to know the mean $\mu_z$ and standard deviation $\sigma_z$ of a log normal distribution $Z(\mu_z,\sigma_z)$ which is the sum of two other log normal distributions--$X(\mu_x,\sigma_x)$ and ...
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39 views

What is the necessary condition for a unbiased estimator to be UMVUE?

According to the Rao-Blackwell theorem, if statistic $T$ is a sufficient and complete for $\theta$, and $E(T)=\theta$, then $T$ is a uniformly minimum-variance unbiased estimator (UMVUE). I am ...
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7k views

Who Are The Bayesians?

As one becomes interested in statistics, the dichotomy "Frequentist" vs. "Bayesian" soon becomes commonplace (and who hasn't read Nate Silver's The Signal and the Noise, anyway?). In talks and ...
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35 views

receiver operating characteristic (ROC) on a test set

The following image definitely makes sense to me. Say you have a few trained binary classifiers A, B (B not much better than random guessing etc. ...) and a test set composed of n test samples to ...
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21 views

What value of (adjusted) McFadden R square or other pesudo R square means good fitting

I got adjusted MaFadden R square for logistic regression: 0.918772 , 0.6135568 , 0.3407252 respectively, which value is good? I just heard the value between 0.2 and 0.4 is good for McFadden R square. ...
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17 views

Properties of the KL topology [reference request]

I'm trying to understand better what are the implications of a sequence of random variables $X_n$ converging toward some limit $X$ in the KL topology, ie the probability density functions are such ...
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1answer
12 views

Function that relates two models

Suppose I have a response variable modeled by a linear model y1=mt+b and another response variable modeled by y2=d*exp(ct) i.e. an exponential model. Suppose I have fitted these models separately and ...
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19 views

Developing a statistical index

I have some variables like countable (water consumption in litres per day, medicine intake etc.) and some uncountable variables (mood status, happiness etc.) To develop a happiness index, what kind of ...
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40 views

Generalized gamma as a member of the exponential family

I want to show if this generalized gamma (GG) distribution is a member of the exponential family. I don't know how to start since the exponential family has only 2 paramters and the GG has 3 ...
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20 views

Which statistical test to use for comparing frequency of cells in four different sample groups?

I am currently writing up my thesis for my MSc and I wondered if someone could help with the type of statistical test I should be using and why. I am not very good at the statistics part and any help ...
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26 views

Conditional expectation of error based on multivariate normal variables

I have the following situation; I know the "true" model behind my regression but I am intentionally omitting some variables/regressors to simplify the problem. Suppose the true model is: ...
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11 views

How to match two fuzzy membership functions with max-min?

I have two triangular membership functions, let's say A=[0.2,0.5,07] and B=[0.4,0.6,0.8] with the same universe X=[0:0.01:1]. Can someone show me numerically how to calculate the matching function ...
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3answers
89 views

Proof of link between the OLS slope estimate and two sample t test statistic (categorical Xvar)

Regarding a univariate OLS regression with a single categorical predictor (coded 0,1). I am wrestling with the proof that $$t =\frac{b_1}{s(b_1)} $$ starting from the basic OLS estimator for the ...
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0answers
14 views

The winning probability of an unfair game

Say $N+1$ independent random variables $x_0...x_N$ are compared against each other. The expectancy of the first, $E[x_0]$, is higher than the expediencies of the rest, such that the probability ...
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1answer
26 views

Understanding categorical variables in ZINB and other models

This might sound very basic question but not getting the logic from the outcome or unless I need to code my categorical variables in a different way. I am trying to model risk factors for a disease ...
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1answer
46 views

Puzzling Sufficient Statistic

If $X_1\sim U(0,\theta)$ then $X_1$ is a sufficient statistic for $\theta$. Also when $X_2\sim U(0,\theta + 1)$ then $X_2$ is a sufficient statistic for $\theta$. Is that right? Now if $X_1, X_2$ ...
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2answers
61 views

How can I compute $\int F(x \mid a,b)f(x \mid w,z) {}dx$ in closed form?

Suppose $F$ is the cumulative distribution function of the normal distribution with mean $a$ and standard deviation $b$, and suppose $f$ is the probability density function of the normal distribution ...
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0answers
11 views

A Fisher information metric which doesn't refer to any exponential family

If $\mathcal E$ is some exponential family of distributions, then we can view it as a Riemannian manifold with local metric the Fisher information matrix. We can then define the Fisher metric: the ...
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18 views

How many test cases do I need to be (95%) confident in control group testing when the desired outcome is an increase in revenue?

I am implementing a new pricing structure and am going to be conducting control group testing but am unsure how many test cases I need to be statistically confident in the results. The desired ...
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1answer
27 views

Generalized linear models and noncanonical link function

When using the canonical link function for GLMs, it is true that $g(\mu) = \theta = \mathbf{X}\boldsymbol\beta$. Is this not the case when using a link that isn't canonical? For example, say we're ...
2
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1answer
42 views

Mean and variance for unequal samples

I have a sampling of variable sized plots. Each plot contains the number of trees present on the plot. Given: $n=$ the number of plots $s_i=$ the size of the $i^{th}$ plot $y_i=$ the number of trees ...
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32 views

Extreme Value Theory Data Scaling

I have a data set available of almost thirty years of data, with for each month the number of occurrences of a certain event and the total number in the set available. What I would like to compute is ...
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1answer
22 views

Relationship: Regression Coeffecient and Line Plot Trendline Slope

Why does the slope of my line fit plot not match the coefficient of the same variable in my regression? For one of the variables, the coefficient is positive while the line fit plots trendline is ...
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1answer
16 views

Coding factors with non-numeric settings

For my role as a TA I am assisting the instructor in developing curriculum based on DOE and statistics. We have a set of failure strengths (the output/dependent variable) for several glass rods. The ...
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52 views

What is the formal definition for $\sim$ in the context of statistics?

If $X\sim N(\mu,\sigma^{2})$, we say that $X$ follows a normal distribution with mean $\mu$ and variance $\sigma$. How is the operator $\sim$ formally defined?
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64 views

Cramer Von Mises - How to use this test correctly?

I had a problem when I tried to test the fitting of my data with the generalized Pareto distribution. I used the MLE to estimate the two parameters 'shape' and 'scale' and I generated a vector of ...
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54 views

Are this two definitions of survival equivalent

Let $\lambda_i(t)$ denote hazard of the $i$-th individual at time $t$ and let $\lambda(t)=\frac{\sum_{i=1}^n S_i(t) \lambda_i(t)}{\sum_{i=1}^n S_i(t)}$ be the weighted average of individual hazards. ...
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1answer
52 views

What is the formal definition of an “atomless” distribution?

The use of atomless (continuous) distributions is ubiquitous in applied works. While the general idea is somewhat clear to me, I was looking for a formal definition or some useful references on the ...