Bolet´ın de Estad´ıstica e Investigaci´on Operativa

Vol. 31, No. 2, Julio 2015, pp. 108-117

Estad´ıstica

A Note on Optimal Intervals in Normal Populations

Jose M. Gavilan and Francisco J. Ortega

Department of Applied Economics I

University of Seville

!

gavi@us.es,

!

fjortega@us.esAbstract

In the setting of one and two normal populations, the shortest con-

fidence interval (SCI) involving location parameters coincides with the

classic equal-tails confidence interval (ETCI). However, for confidence in-

tervals involving scale parameters, the ETCI fails to provide the SCI and

results can differ notably. In order to obtain such SCIs, either constrained

optimization problems or nonlinear systems of equations have to be solved.

In this setting, two tables are provided to find the SCIs at 95% confidence,

which can be then used in classrooms to compare the results with the

ETCIs usually obtained by the students and provided by the statistical

software.

Keywords:

Confidence intervals, Normal populations, Shortest confi-

dence interval.

AMS Subject classifications:

62F99

1. Introduction

Given a real-valued random variable X of probability density function

(pdf)

f

(

x

;

θ

)

, x

∈

R

,

where

θ

∈

Θ is an unknown parameter (a fixed real

quantity) to be estimated, a confidence interval (CI) at confidence level 1

−

α

is

a random interval [

q

∗

1

, q

∗

2

] built from a random sample

X

1

, X

2

, . . . , X

n

such that

P

(

q

∗

1

< θ < q

∗

2

) = 1

−

α.

(1.1)

The most common way of obtaining a CI for

θ

is the well-known

method of

the pivotal quantity

(see for example Casella and Berger, 2002) which is based

on a random variable

Q

=

Q

(

X

1

, X

2

, . . . , X

n

;

θ

)

,

called the pivot or pivotal

quantity, whose distribution is independent of

θ

. Therefore, it is possible to find

two quantiles

q

1

and

q

2

of

Q

such that

P

(

q

1

< Q < q

2

) = 1

−

α,

(1.2)

c

⃝

2015 SEIO