Questions tagged [definition]
This tag indicates questions about definitions of statistical terms. Use a more general tag [terminology] for questions on statistical parlance that are not specifically about definitions.
561 questions
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A problem with the definition of ergodicity in "Evaluating gambles using dynamics, O.Peters"
In the paper we are presented with a additive time stochastic process:
$$ W_{t+ \delta t \times T} - W_t = \sum^T_{i=1} D_i $$
Where $W_t$ represents the wealth at time $t$ , $\delta t$ is the ...
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definition of regular estimators
In the book "Semiparametric Theory and Missing Data" by Tsiatis, superefficient estimators are defined as "they are unnatural and have undesirable local properties associated with them&...
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3
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Is "categorical data" a synonym of "nominal data"?
So far, it's always been my understanding that nominal data was a type of categorical data, not a synonym of it. For me, categorical data included ordinal data, not just nominal data.
As of November ...
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What are the mathematical structures of nominal, ordinal, interval and ratio data? [duplicate]
In Michael Eysenck's fundamentals of psychology, it says
Another factor to consider when deciding which statistical test to use
is the type of data you have obtained. There are four types of data of
...
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2
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Where should the $\leq$ go in the definition of the hazard function?
I have come across two definitions of the hazard function in several textbooks and online resource.
Definition 1. Here for example.
$$h(t) = \lim_{\epsilon \to 0+}\dfrac{P(t < T \leq t + \epsilon \...
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What does it mean to say that a statistical model is a location family?
I would like to know what it means to say that a statistical model is a location family. Looking at some definitions, I came up with three possible interpretations of the definition of location family....
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Can manual feature extraction be considered a part of a learning algorithm?
We can view a learning algorithm as a tuple $(\mathcal{H}, \mathcal{O}, \mathcal{L})$ where $\mathcal{H}$, $\mathcal{O}$ and $\mathcal{L}$ are the hypothesis class, optimizer and loss function ...
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Can you arbitrarily define your sample space?
Let's say you flip a fair coin. A sample space is a set of possible outcomes of an experiment. So, intuitively, the sample space will be {Head, Tail}.
However, I have seen that it is also possible to ...
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1
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Formal definition of sufficient statistic
Let $(\Omega_X,\mathcal{F}_X)$ and $(\Omega _T,\mathcal{F}_T)$ be measurable spaces. Let $\mathfrak{M}$ be a family of probability measures on $(\Omega_X,\mathcal{F}_X)$. Let $X:\Omega\to \Omega _X$ ...
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When is it important for a practitioner to understand CIs?
I have a physician friend who asks me questions about stats. He gets confused about stuff, e.g., the definition of a confidence interval (CI) and its intricacies. For example, he finds the following ...
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Fixed vs random: how can a fixed effect be constant across individuals?
For years, I have been trying to find a definition for fixed and random effects. I often see statements like this:
Fixed effects are constant across individuals. - almost universal
We define effects (...
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Motivation behind definition of PMF of function of $2$ variables
I am really curious to understand what motivates the definition $$p_{g(X,Y)} (g(X,Y)=z) = \sum_{(x,y)\in g^{-1}(\{z\})} p_{X,Y} (x,y)$$
where $g$ is a two variable function, and $X,Y$ are random ...
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Frequentist probability: Can we prove mathematically what we are setting probabilities equal to, or are they just assumptions/definitions?
Just as an example, let's say I am modeling the rolling of a die.
We can use the frequentist definition of probability to define a probability of an event, say rolling a 6, as the $\lim_{n\to\infty}$$\...
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Unbiased estimator for mean
The question: Given a random sample $X_1,...,X_n$ show that $\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator for $E(X_1)$.
My confusion: Given a statistical model $(\Omega,\Sigma,p_{\theta})$, ...
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Kendall's Notation for Kelly Networks?
Background
Kendall's notation (also see Ciw docs') is a convenience when you work with queues a lot. It not only provides an abbreviation, but its wide use helps make literature searches more specific....
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Finding a source for the definition of "clustering accuracy"
In papers about unsupervised clustering I see a lot of references to a metric "clustering accuracy" or "unsupervised clustering accuracy" (ACC) which is usually defined as ...
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Example of statistical experiment in which the statistical unit is time
Could you give me some examples of statistical experiment/analysis in which the statistical unit is
the time? If I conduct a study on the number of people who have attended a bachelor degree over time,...
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Definition of $\text{“}R^2\text{”}$
I have always taken $\text{“}R^2\text{”}$ to mean the proportion of a sum of squares explained by a model.
The context in which this idea is first encountered is when one explains part of the total ...
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Definition of Uncertainity
I have some confusion regarding Measurement Uncertainty. In some books/articles it is defined wrt true value as "Uncertainty in the average of measurements is the range in which true value is ...
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Do testability and falsifiability have statistical definitions?
Psychology: the Core Concepts says
Psychology differs from the pseudosciences in that it employs
the scientific method to test its ideas empirically. The scientific
method relies on ...
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0
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Definition of exponent measure (extreme value theory)
Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$.
In my textbook, there is the ...
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For a confusion matrix, is there a name for FP / (FP + FN)?
For a confusion matrix, there are a variety of useful rates, ratios and indices. But I cannot find the one I care about:
FP / (FP + FN)
Of course this measure is ...
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What is a formal and authoritative definition of an 'assumption' in a statistical model?
The description of the tag in this website states that it
Refers to the conditions under which a statistics procedure yields valid estimates and/or inference. E.g., many statistical techniques ...
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Definition of "unpredictable"
How do we rigorously define the term "unpredictable" in cases of point and density prediction?
The term "unpredictable" is employed in various contexts, e.g.
"the outcome of ...
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Clarification: confidence interval of the (slope of the) regression line
When confidence intervals are referenced in the regression context, I often see them mentioned (generally speaking) as "of the slope of the regression line," or as "of the regression ...
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Data vs. Information. What’s the difference between these two terms?
Following are the definitions "representative" of the countless articles I have read about data vs information and yet have nothing to say about them. Can anyone just put some accent upon ...
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Definition of Statistics by Sheldon M. Ross [closed]
Sheldon M. Ross in his book, "Introductory Statistics" states the definition of statistics to be:
"Statistics is the art of learning from data. It is concerned with the collection
of ...
- 11
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282
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meaning of independent samples
I am getting confused about the concept of independent samples, which is related to the concept of pseudoreplications. This question is related to this topic. But reading it does not make it clear for ...
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The link between AQL, RQL and the reliability of a process
The problem is to distinguish between the definitions of AQL, RQL and reliability in the field of quality control. Let us just pose the context. We wish to calculate the reliability of a manufacturing ...
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Questions on what it means when we talk about "overfit"
"Overfit" is a commonly discussed concept in ML community. However, I tend to feel that there might be abuse of using this terminology. I wonder what it means when we talk about overfit, ...
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What's the relationship between these two definitions of martingales?
On wikipedia, the definition of a martingale is given as follows:
A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) $X_1, ...
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How to demonstrate amount of virus required to lyse all cells using Poisson distribution?
In a seminal article in virology ('Sur l’unité lytique du bactériophage'
Comptes rendus des séances de la Societé de biologie et de ses filiales, 1939, 130, pp. 904-907) the Nobel prize winner ...
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Definition p-value and find p-value in practice
I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $...
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marginal cdf and cdf of marginals
given a joint distribution of 2 variables $P(X,Y)$, is the cdf of the Y-marginal distribution equals to the Y-marginal of the cdf of $P(X,Y)$?
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Why are Square Integrable Functions important in Statistics?
I'm reading a paper by Giles Hooker on Functional Decomposition through the use of Functional ANOVA.
In the paper he defines a function:
$$ F(x) : \mathbb{R}^k \rightarrow \mathbb{R} $$
and explicitly ...
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Is there a connection between the Kernels in Statistics and Linear Algebra?
According to this question, the etymology of the terms is related, and Kernel is used to mean the "core" of something. In general it seems to refer to an unchanging transformation at the ...
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What are balanced and orthogonal designed experiments?
I have recently started studying Designed Experiments and have have come across non-rigorous definitions of "balanced" and "orthogonal" experiments and would be interested in ...
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Relationship between blocks, factors and treatments
I have recently began studying a course on Designed Experiments and am having some trouble understanding some of the terminology. I've looked at some other answers on the site and I think that I am ...
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554
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Is there a difference between out-of-domain and out-of-distribution?
When I research papers on the generalisability of ML models, both terms "Out of Distribution" and "Out of Domain" pop up. Are these essentially the same? In my understanding, yes. ...
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Can we consider the loadings as a proxy for correlation, in a Principal Component Analysis (PCA)?
In a PCA, the loadings can be understood as the weights for each original variable when calculating the principal component, or "how much each variable influence a principal component".
Thus,...
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Questions regarding power of test and type II error
I´m preparing for a lecture in decision theory and I´m a little bit confused by the notation used by my prof.
On the first slide under remark 3.2 point v) its written, that $\beta(\varphi)$ is equal ...
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What is independent censoring and what are its assumptions?
I'm reading the Introduction to Statistical Learning book. In chapter 11 (Survival Analysis, page 463), the authors state:
In general, we need to assume that the censoring mechanism is
independent: ...
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Bounded density function: definition?
What is the correct definition of bounded probability density function:
$\sup_{x} f(x)<\infty$. If this is the correct definition of bounded probability density function, can you give the example ...
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Understanding likelihood vs conditional joint pdf
My course notes (3rd-year module in Bayesian Statistics, unpublished) have a paragraph,
Gaussian data with known variance
Suppose we have $\textbf{x}=\{x_1,\dots,x_n\}$ iid given $\theta$ and $\...
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Rationale behind defining distribution function with strict inequality
A friend from Italy pointed me to a nice Probability Theory book written in Italian by G. Dall'Aglio, a well-known Italian probabilist; Original title: Calcolo delle Probabilità, terza edizione, ...
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Are random variables mathematical functions when dealing with continuous sample spaces?
I'm struggling with some basic concepts regarding the definitions of random variables.
If I'm not mistaken, random variables are functions which aim to "translate" outcomes of a sample space ...
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Definition of Hypothesis Testing in All of Statistics
In the book All of Statistics by Larry Wesserman, the author says "Let $X$ be a random variable and let $\mathcal{X}$ be the range of $X$. We test a hypothesis by finding an appropriate subset of ...
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Regression vs. Classification: Is there a clear, generally accepted definition?
As a mathematician/economist, I am not trained to think in classification and regression tasks. This is why I wonder: is there a clear, widely accepted definition of regression and classification ...
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Provide an example of a dataset where maximum likelihood is inapplicable as third moments and fourth moments "assumptions" do not apply
An additional complication arises with estimation, since maximum likelihood estimation may not be feasible without making unrealistically strong ?????"assumptions"????? about third‐ and ...
user318514
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What is the difference between something being "true" and 'true with probability 1"?
In the beginning of chapter 2 of Information Theory, Inference and Algorithms, the author says that he will refrain from being unnecessarily rigorous and provides the example of saying that something ...
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