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Maximum likelihood estimation in uniform distributions
Tiago M. Magalh˜ aes
∗Department of Statistics, University of S˜ao Paulo, Brazil
Abstract
We have some difficulties to obtain the maximum likelihood estimador (MLE) in non regularity cases, for example, when the support of an distribution depends on an unknown parameter. We exemplify possible situations for obtaining the MLE through the uniform distribution.
Keywords: Maximum likelihood estimation, non regularity cases, uniform distri- bution
1 Preliminary considerations
IfX has an uniform distribution in the interval (θ1, θ2) (notation: X ∼U(θ1, θ2), your probability density function is given by
f(x) = 1
θ2−θ1 I(θ1,θ2)(x), (1)
whereI(θ1,θ2)(x) is an indicator function, i.e., I(θ1,θ2)(x) =
1, x∈(θ1, θ2) ; 0, otherwise.
2 Examples
2.1 Uniparametric cases
Example 1. Let X1, . . . , Xn be independent random variables, each X`, ` = 1, . . . , n having a density given by (1), where (θ1, θ2) = (−θ, θ). The likelihood function for θ, denoted byL(θ), is given by
L(θ) = 1
2θ n n
Y
`=1
I(−θ,θ)(x`). (2)
∗Email: tiagomm@ime.usp.br; last modification: July 17, 2014
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2 EXAMPLES 2
Note that, theL(θ), in (2), can be write as L(θ) =
1 2θ
n n
Y
`=1
I(0,θ)(|x`|). (3)
Now, note that, the expression (2) has maximum when Qn
`=1I(0,θ)(|x`|) = 1, i.e., if 0<|x1|< θ, . . ., 0<|xn|< θ, if 0<max{|x1|, . . . ,|xn|}< θ. But
I(0,θ)(max{|x1|, . . . ,|xn|}) = I(max{|x1|,...,|xn|},+∞)(θ). Then, the expression (3) can be write as
L(θ) = 1
2θ n
I(max{|x1|,...,|xn|},+∞)(θ). Finally, the θ that maximizes L(θ) is given by
θˆ= max{|x1|, . . . ,|xn|}.
Example 2. Let X1, . . . , Xn ∼ U(δ − θ, δ +θ), where δ is a known parameter, i.e., eachX`, ` = 1, . . . , n have an uniform distribution centered in δ. In this case, note that X`−δ∼ U(−θ, θ), then
θˆ= max{|x1 −δ|, . . . ,|xn−δ|}
is the MLE ofθ.
Example 3. LetX1, . . . , Xnbe independent random variables, eachX` ∼U(θ−δ, θ+δ), whereδ is a known parameter. The likelihood function forθ is given by
L(θ) = 1
2δ n n
Y
`=1
I(θ−δ,θ+δ)(x`). (4)
The expression (4) has maximum whenQn
`=1I(θ−δ,θ+δ)(x`) = 1, i.e., ifθ−δ < x1 < θ+δ, . . ., θ−δ < xn< θ+δ. Then, the expression (4) has maximum when
θ−δ < x(1)< x(n)< θ+δ, (5)
where x(1) = min{x1, . . . , xn} and x(n) = max{x1, . . . , xn}. In Figure 1, we present the area represented by expression (5) .
By (5), we can write (4) as L(θ) =
1 2δ
n
I(θ−δ,x(n)) x(1)
I(θ−δ,θ+δ) x(n)
(6)
= 1
2δ n
I(x(1),θ+δ) x(n)
I(θ−δ,θ+δ) x(1) .
By (5), we also have thatθ < x(1)+δ and θ > x(n)−δ. Finally, the expression (4) can be write as
L(θ) = 1
2δ n
I(x(n)−δ,x(1)+δ) (θ). (7) The θ that maximizes L(θ), in (7), is given by
θˆ∈ x(n)−δ, x(1)+δ , i.e., in this case the MLE ofθ is not unique.
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2 EXAMPLES 3
x(1) x(n)
θ − δ θ + δ
θ−δθ+δ x(1)=x(n)
Figure 1: Area where Qn
`=1I(θ−δ,θ+δ)(x`) = 1.
2.2 Biparametric case
Example 1. LetX1, . . . , Xn ∼U(θ−ϑ, θ+ϑ), whereθand ϑare unknown parameters.
The likelihood function for (θ, ϑ) is given by L(θ, ϑ) =
1 2ϑ
n n
Y
`=1
I(θ−ϑ,θ+ϑ)(x`). (8)
The expression (8) has maximum when Qn
`=1I(θ−ϑ,θ+ϑ)(x`) = 1, i.e., if θ−ϑ < x1 <
θ+ϑ, . . .,θ−ϑ < xn< θ+ϑ. Then, the expression (8) has maximum when
θ−ϑ < x(1) < x(n) < θ+ϑ. (9)
By inequalities (9), we have the following relations x(n)−ϑ < θ < x(1)+ϑ
ϑ > x(n)−θ and ϑ > θ−x(1) (10)
From second row of (10),
ϑ >max
θ−x(1), x(n)−θ . (11)
If, in (11), max
θ−x(1), x(n)−θ =x(n)−θ, we have that x(n)−θ > θ−x(1), solving this inequation, we have θ ≤ x(1)+x2 (n). If, in (11), max
θ−x(1), x(n)−θ = θ−x(1),
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2 EXAMPLES 4
θ−x(1) > x(n)−θ, in this case, we haveθ ≥ x(1)+x2 (n). Then, the MLE of θ is θˆ= x(1)+x(n)
2 . (12)
Replacing (12) in (11), we have ϑ >max
x(1)+x(n)
2 −x(1), x(n)− x(1)+x(n) 2
= x(n)−x(1)
2 . (13)
The MLE of ϑ is obtained by the inequation (13). But, note that, in (8), ∀θ fixed, L(θ, ϑ) is maximized by the lower ϑ (see the ratio 1/2ϑ), such that L(θ, ϑ)>0. Then,
ϑˆ= x(n)−x(1) 2 is the MLE ofϑ.
