Page 338 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 315

and then write F = {f(x; µ) = φ (x – µ) : µ ∈ ℜ, x ∈ ℜ}. Next, the collection
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of N(0, σ ) distributions, with σ ∈ ℜ , forms a scale family. To see this, let us
simply write F = {f(x; σ) = σ φ (x/σ) : σ ∈ ℜ , x ∈ ℜ}. The collection of
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N(µ, σ ) distributions, with µ ∈ ℜ, σ ∈ ℜ , forms a location-scale family. We
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simply write F = {f(x; µ, σ) = σ φ ((x – µ)/σ) : µ ∈ ℜ, σ ∈ ℜ , x ∈ ℜ}. We
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have left other examples as exercises.
In a location family, the role of the location parameter θ is felt in the “move-
ment” of the pdf along the x-axis as different values of θ are contemplated.
The N(0, 1) distribution has its center of symmetry at the point x = 0, but the
N(µ, 1) distribution’s center of symmetry moves along the x-axis, to the right
or left of x = 0, depending upon whether µ is positive or negative, without
changing anything with regard to the shape of the probability density curve.
In this sense, the mean µ of the normal distribution serves as a location
parameter. For example, in a large factory, we may look at the monthly wage
2
of each employee and postulate that the distribution of wage as N(µ, σ )
where σ = $100. After the negotiation of a new contract, suppose that each
employee receives $50 monthly raise. Then the distribution moves to the right
with its new center of symmetry at µ + 50. The intrinsic shape of the distri-
bution can not change in a situation like this.
In a scale family, the role of the scale parameter δ is felt in “squeezing or
expanding” the pdf along the x-axis as different values of δ are contemplated.
The N(0, 1) distribution has its center of symmetry at the point x = 0. The
N(0, δ ) distribution’s center of symmetry stays put at the point x = 0, but
2
depending on whether δ is larger or smaller than one, the shape of the density
curve will become more flat or more peaked, compared with the standard
normal, around the center. In this sense, the variance δ of the normal distri-
2
bution serves as a scale parameter. Suppose that we record the heights (in
inches) of individuals and we postulate the distribution of these heights as
N(70, σ ). If heights are measured in centimeters instead, then the distribution
2
would appear more spread out around the new center. One needs to keep in
mind that recording the heights in centimeters would amount to multiplying
each original observation X measured in inches by 2.54.
In a location-scale family, one would notice movement of the distribu-
tion along the x-axis as well as the squeezing or expansion effect in the
shape. We may be looking at a data on the weekly maximum temperature
in a city recorded over a period, in Fahrenheit (F) or Celsius (C). If one
postulates a normal distribution for the temperatures, changing the unit of
measurements from Fahrenheit to Celsius would amount to shifts in both
the origin and scale. One merely needs to recall the relationship 1/5C =