# Does “Inference” include estimation or only testing?

Does the term "statistical inference" include only hypothesis testing or does it also include point estimation, interval estimation etc.

Authoritative references will be greatly appreciated.

• @ttnphns Testing and estimation are different concepts: parameters can be estimated without any test being performed (e.g., many opinion polls provide only parameter estimates and no tests) and not all hypothesis tests require anything to be estimated (e.g., what is the Kolmogorov-Smirnov test estimating?). – whuber Jul 30 '13 at 20:12

In a typical problem of statistics it is ... a class of laws [which is specified], any of which may possibly be the one which actually governs the chance device or experiment whose outcome we shall observe. We know that the underlying probability law is a member of this class, but we do not know which one it is. The object might then be to determine a "good" way of guessing, on the basis of the outcome of the experiment, which of the possible underlying probability laws is the one which actually governs the experiment whose outcome we are to observe. ...

... statistical inference [is] the subject of obtaining good guessing methods. ...

It is possible to discuss all of the important ideas of modern statistical inference ... and we shall try to do this.

-- Jack Karl Kiefer, Introduction to Statistical Inference, pp 1-3. Springer Verlag, New York (1987).

Kiefer's discussion of "all of the important ideas" fills the rest of this text. Thus the main chapter headings (following preliminary general material) may serve to document what he felt comprised statistical inference:

• Linear Unbiased Estimation (the general linear model)

• Sufficiency (concepts of the maximum likelihood theory)

• Point Estimation

• Hypothesis Testing

• Confidence Intervals

Notably, statistical prediction is not included in any of this.

• Prediction might be a method by which one tests the adequacy of the inference of course. – Peter Ellis Jul 31 '13 at 7:17
• @PeterEllis: seems to me that prediction falls in the definition suggested by A. H. Welsh in the sense it is indeed a question about the underlying $F_0$. – JohnRos Jul 31 '13 at 7:51
• JohnRos The difference between prediction and estimation lies in their targets: estimation is about $F$ while prediction is about a sample from $F$. Whether the latter, which is only indirectly about $F$, is considered "inference" is likely to differ from author to author. – whuber Jul 31 '13 at 13:17
• @whuber: I admit I would call the quantiles, or the mode of $F$ "properties of $F$". I agree however that different authors will differ-- which is precisely why I posted the question :-) – JohnRos Jul 31 '13 at 13:30

It includes any procedure in which you try to draw conclusions about an underlying population or data generating process from data using statistics. Yes, this definitely includes point estimation and interval estimation, etc.

References? - I would start with any book with "statistical inference" in the title, but Wikipedia would also do.

Edit/addition: here are some specific references.

First and very directly to your point is from page 1 of Paul H. Garthwaite, Ian T. Jolliffe and Bryon Jones (1995), Statistical Inference, Prentice Hall.

"In statistical inference we use a sample of data to draw inferences about some aspect of the population (real or hypothetical) from which the data were taken. Often the inference concerns the value of one or more unknown parameters, which describe some attribute of the population such as its location or spread.

There are three main types of inference, namely point estimation, interval estimation and hypothesis testing..."

And here is my favourite, A. H. Welsh (1996), Aspects of Statistical Inference, John Wiley & Sons

"Statistical inference is concerned with using data to answer substantive questions. In the kind of problems to which statistical inference can usefully be applied, the data are variable in the sense that, if the data could be collected more than once, we would not obtain identical numerical results each time." (p.1)

"The components of the inference problem are:

• a substantive question
• data z which we interpret as a realization of a random variable Z with a distribution $F_0$
• a model for $F_0$

The objective of inference is to answer the substantive question by reformulating it as a question about the underlying distribution $F_0$ and then using the data z, the model and any other information we have to answer the question about $F_0$. The kinds of questions we ask about $F_0$ are typically of one or both of two types:

• Can the model be viewed as a reasonably close approximation to the data generating process?

or

• Can we determine a set of plausible values for a parameter $\theta(F_0)$, or can we determine whether a particular value of a given parameter $\theta(F_0)$ is plausible?

The answers to these questions are derived from the data through the calculation and interpretation of the realized values $t(z)$ of statistics $t(Z)$, which are functions of the data which do not depend on any unknown parameters." (pp. 31-32)

• My understanding is the same as yours. But as I got into a dispute with a referee over the definitions, Wikipedia will not do and not all books are in agreements. I am thus looking for some authoritative reference. – JohnRos Jul 30 '13 at 20:36
• Literally, any book with Statistical Inference in the title will do - when I get back to where my books are I'll expand out my answer with some specific references and quotes. – Peter Ellis Jul 30 '13 at 21:28
• I've added some references. Thanks for the question which gave me a chance to include an excerpt of my favorite description (Welsh's) of the inference problem! – Peter Ellis Jul 31 '13 at 7:11