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The failure rate properties of a bimodal mixture of normal distributions in an unequal variance case
Statistics and Probability Letters 78 (2008) 2006–2009 www.elsevier.com/locate/stapro
The failure rate properties of a bimodal mixture of normal distributions in an unequal variance case Fuxiang Liu a,∗ , Yanyan Liu b a Science College, China Three Gorges University, Yichang, Hubei, 443002, PR China b Department of Mathematics, Wuhan University, Wuhan, Hubei, 430072, PR China
Received 17 May 2007; received in revised form 23 August 2007; accepted 9 January 2008 Available online 13 February 2008
Abstract The failure rate of the mixture of two normal distributions in an unequal variance case may have four changes in monotonicity. This work analyzes the conditions for which the four changes occur and others for which increasing failure rate and modified bathtub shape can be shown. c 2008 Elsevier B.V. All rights reserved.
1. Introduction A mixture of two normal distributions may be either unimodal or bimodal and the conditions for bimodality are very complicated (Robertson and Fryer, 1969); Glaser notes that for a density f (t) with minimal assumptions, if η(t) = − f 0 (t)/ f (t) is increasing in t ∈ (t0 , +∞), the failure rate is also increasing in this interval (Glaser, 1980). So the monotonicity of the failure rate (r (t) = f (t)/F(t)) has one or more changes. It is proved that the shape of the failure rate of mixture of two normal distributions in an equal variance case may be either IFR (i.e. increasing failure rate) or MBT (modified bathtub shape, i.e. increasing, decreasing, increasing) type; theoretically, in an unequal variance case, the monotonicity of the failure rate can have at most four changes (Block et al., 2005). When the density of the mixture is bimodal and meanwhile the monotonicity of its failure rate has four changes, there exists a super-bimodal (SB) case, which occurs only in an unequal variance case for the mixture. The conditions that determine the shapes of the failure rate in an unequal variance case of the mixture, such as IFR, MBT, and SB ones, are analyzed with typical examples in the following. 2. Main results The density function of a normal distribution with mean µ and variance σ 2 can be written as ϕ(t; µ, σ ); then the general mixture of two normal distributions should be f (t) = pϕ(t; µ1 , σ1 ) + qϕ(t; µ2 , σ2 )
p, q > 0, p + q = 1.
∗ Corresponding author.
E-mail address: [email protected] (F. Liu). c 2008 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2008.01.083
(2.1)
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
2007
Hence, the analysis of the following case can be representative: f (t) = pϕ(t; 0, 1) + qϕ(t; µ, σ )
µ > 0, σ > 0, and σ 6= 1
(2.2)
because all the other cases can be transferred by a linear transformation (µ = (µ2 − µ1 )/σ1 , σ = σ2 /σ1 ) without influence on the shape of the failure rate. Lemma 1. In the mixture model (2.2), r (t) increases if f 0 > 0. The result is supported by the equation r 0 = (F)−2 ( f 0 · F + f 2 ). Lemma 2. In the mixture model (2.2), the failure rate r (t) and η(t) have the same asymptote, either y = t (0 < σ < 1) or y = (t − µ)/σ 2 (σ > 1). Apparently, this result differs slightly from that for a single normal distribution. 2 µ 1+v 2 2 + 1+v − σ12 [2(σ 2 − 1) ln( σqp v) + µ2 ] with v = (q/ pσ ) exp{ σ2σ−1 2 (t − 1−σ 2 ) + µ /2(1 − v p σ2 σ 2 )}, and v0 = σ 2 − 1 + (σ 2 − 1)2 + σ 2 ; there exist two cases: if h(v0 ) < 0, the monotonicity of η(t) has four changes and its monotonic intervals are symmetrical with t0 = µ/(1 − σ 2 ) as the center and η0 (t0 ) > 0; if h(v0 ) ≥ 0,
Lemma 3. Let h(v) =
η(t) increases constantly (Block et al., 2005).
In the former case,namely, when h(v0 ) < 0, from t0 to +∞, η(t) first increases, then decreases, and finally increases again; and the η(t) monotonicity varies with the left part and the right part of t0 symmetrical to each other. Lemma 4. If η0 (t) > 0 when t ∈ (t ∗ , +∞), r (t) increases in this interval. This conclusion can be drawn from Glaser’s analysis (Glaser, 1980). Theorem 1. If the mixture model (2.2) is bimodal, η(t) monotonicity shows four changes. If f (t) is bimodal, there exists t ∈ (0, µ), s.t. f 0 (t) = 0 and f 00 (t) > 0; then η0 (t) = [( f 0 (t))2 − f (t) · f 00 (t)]/ [ f (t)]2 < 0. Thanks to the continuity and symmetry of η0 (t), η(t) has four changes of monotonicity according to Lemma 3. The equation h(v) = 0 can be simplified to σp σ2 v+ + 2(1 − σ 2 ) ln v = µ2 + 2(σ 2 − 1) ln v − σ 2 − 1 (v defined as above). (2.3) v q If the mixture is bimodal, the Eq. (2.3) has two real roots: v1 and v2 which determine four t values. They can be µ 0 written as ti (i = 1, 2, 3, 4) with t1 < t2 < 1−σ 2 < t3 < t4 . If f (t) is bimodal, there exists η > 0 in these intervals: (−∞, t1 ), (t2 , t3 ), and (t4 , +∞); and η0 < 0 within (t1 , t2 ) and (t3 , t4 ). µ If σ > 1, there exists t1 < t2 < 1−σ 2 < 0 < t3 < t < t4 < µ; according to Lemmas 1 and 4, r (t) increases within (−∞, 0) and (t, +∞). Moreover, r (t) has another property: r 0 (t) = r (r − η)
and r 00 (t) = r [(2r − η)(r − η) − η0 ].
So if r 0 ≥ 0 is constantly true within (0, t), r (t3 ) is not less than η(t3 ) (the maximum of η(t) is in this section). In this case, r (t) increases constantly within (−∞, +∞), i.e. the failure rate of the mixture is in its IFR shape. Meanwhile, if r (t3 ) < η(t3 ), r (t) decreases within a certain section of (0, t); i.e. the failure rate of the mixture is in its MBT shape. Hence, we give the following two corollaries: Corollary 1. If σ > 1,the failure rate has only two shapes: IFR (r (t3 ) ≥ η(t3 )) and MBT (r (t3 ) < η(t3 )). µ If 0 < σ < 1, there exists 0 < t1 < t < t2 < µ < 1−σ 2 < t3 < t4 . According to Lemmas 1 and 4, the failure rate of the mixture increases within these intervals: (−∞, 0), (t, µ), and (t4 , +∞). The interval (0, t) can be analyzed in the same way: if r (t1 ) ≥ η(t1 ), r (t) increases in this interval; and if r (t1 ) < η(t1 ), r (t) decreases in a certain section of (0, t). However, the case is different for the interval (µ, t4 ). According to Lemma 2 and the properties of η(t), it is
2008
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
(a) p = 0.05, q = 0.95, µ = 6, σ = 5.
(b) p = 0.4, q = 0.6, µ = 6, σ = 5.
Fig. 1. (σ > 1) density function f, η function and the failure rate r (t).
(a) p = 0.5, q = 0.5, µ = 1.3, σ = 0.3.
(b) p = 0.885, q = 0.115, µ = 3, σ = 0.3.
Fig. 2. (0 < σ < 1) density function f, η function and the failure rate r (t).
easy to observe that η(t) becomes very close to its asymptote y = t beginning from t4 . Because of the properties of r (t), namely, r 0 (t) = r (r − η) and r 00 (t) = r [(2r − η)(r − η) − η0 ], it can be assumed that r 0 (t) is constantly nonnegative in this interval (µ, t4 ); and the comparison of the failure rate monotonicity with the monotonicity changes of η(t) in the interval (µ, +∞) proves that if the assumption is true, the failure rate can no longer vary close to its asymptote. This is obviously impossible. In other words, r (t) absolutely decreases in a certain section of (µ, t4 ). When 0 < σ < 1, the following can be concluded in the same way: Corollary 2. If 0 < σ < 1, the failure rate has only two shapes: MBT (r (t1 ) ≥ η(t1 )) and SB (r (t1 ) < η(t1 )). 3. Examples Case I (IFR, σ > 1) p = 0.05, µ = 6 and σ = 5; the mixture is bimodal and the failure rate is in the shape of the IFR (Fig. 1(a)); Case II (MBT, σ > 1) p = 0.05, µ = 12 and σ = 5; the mixture is bimodal and the failure rate is in the shape of the MBT (Fig. 1(b)); Case III (MBT, 0 < σ < 1) p = 0.5, µ = 1.3 and σ = 0.3; the mixture is bimodal and the failure rate is in the shape of the MBT (Fig. 2(a)); Case IV (SB, 0 < σ < 1) p = 0.885, µ = 3 and σ = 0.3; the mixture is bimodal and the failure rate is in the SB shape (Fig. 2(b)).
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
2009
Table 1 Failure rate shape
σ condition
Example
IFR MBT
σ >1 σ >1 0<σ <1 0<σ <1
r (t3 ) ≥ η(t3 ) r (t3 ) < η(t3 ) r (t1 ) ≥ η(t1 ) r (t1 ) < η(t1 )
SB
4. Conclusion The failure rate properties of the mixture model (2.2) can be summarized as in Table 1. Acknowledgements The authors are very grateful to the referees for their valuable remarks and suggestions. References Block, H.W., Li, Y., Savits, T.H., 2005. Mixtures of normal distributions: Modality and failure rate. Statist. Probab. Lett. 74, 253–264. Glaser, R.E., 1980. Bathtub and related failure rate characterizations. JASA 75, 667–672. Robertson, C.A., Fryer, J.G., 1969. Some descriptive properties of normal mixtures. Scand. Actuar. Tidskr. 52, 137–146.
Fig. 1(a) Fig. 1(b) Fig. 2(a) Fig. 1(b)
The failure rate properties of a bimodal mixture of normal distributions in an unequal variance case Fuxiang Liu a,∗ , Yanyan Liu b a Science College, China Three Gorges University, Yichang, Hubei, 443002, PR China b Department of Mathematics, Wuhan University, Wuhan, Hubei, 430072, PR China
Received 17 May 2007; received in revised form 23 August 2007; accepted 9 January 2008 Available online 13 February 2008
Abstract The failure rate of the mixture of two normal distributions in an unequal variance case may have four changes in monotonicity. This work analyzes the conditions for which the four changes occur and others for which increasing failure rate and modified bathtub shape can be shown. c 2008 Elsevier B.V. All rights reserved.
1. Introduction A mixture of two normal distributions may be either unimodal or bimodal and the conditions for bimodality are very complicated (Robertson and Fryer, 1969); Glaser notes that for a density f (t) with minimal assumptions, if η(t) = − f 0 (t)/ f (t) is increasing in t ∈ (t0 , +∞), the failure rate is also increasing in this interval (Glaser, 1980). So the monotonicity of the failure rate (r (t) = f (t)/F(t)) has one or more changes. It is proved that the shape of the failure rate of mixture of two normal distributions in an equal variance case may be either IFR (i.e. increasing failure rate) or MBT (modified bathtub shape, i.e. increasing, decreasing, increasing) type; theoretically, in an unequal variance case, the monotonicity of the failure rate can have at most four changes (Block et al., 2005). When the density of the mixture is bimodal and meanwhile the monotonicity of its failure rate has four changes, there exists a super-bimodal (SB) case, which occurs only in an unequal variance case for the mixture. The conditions that determine the shapes of the failure rate in an unequal variance case of the mixture, such as IFR, MBT, and SB ones, are analyzed with typical examples in the following. 2. Main results The density function of a normal distribution with mean µ and variance σ 2 can be written as ϕ(t; µ, σ ); then the general mixture of two normal distributions should be f (t) = pϕ(t; µ1 , σ1 ) + qϕ(t; µ2 , σ2 )
p, q > 0, p + q = 1.
∗ Corresponding author.
E-mail address: [email protected] (F. Liu). c 2008 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2008.01.083
(2.1)
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
2007
Hence, the analysis of the following case can be representative: f (t) = pϕ(t; 0, 1) + qϕ(t; µ, σ )
µ > 0, σ > 0, and σ 6= 1
(2.2)
because all the other cases can be transferred by a linear transformation (µ = (µ2 − µ1 )/σ1 , σ = σ2 /σ1 ) without influence on the shape of the failure rate. Lemma 1. In the mixture model (2.2), r (t) increases if f 0 > 0. The result is supported by the equation r 0 = (F)−2 ( f 0 · F + f 2 ). Lemma 2. In the mixture model (2.2), the failure rate r (t) and η(t) have the same asymptote, either y = t (0 < σ < 1) or y = (t − µ)/σ 2 (σ > 1). Apparently, this result differs slightly from that for a single normal distribution. 2 µ 1+v 2 2 + 1+v − σ12 [2(σ 2 − 1) ln( σqp v) + µ2 ] with v = (q/ pσ ) exp{ σ2σ−1 2 (t − 1−σ 2 ) + µ /2(1 − v p σ2 σ 2 )}, and v0 = σ 2 − 1 + (σ 2 − 1)2 + σ 2 ; there exist two cases: if h(v0 ) < 0, the monotonicity of η(t) has four changes and its monotonic intervals are symmetrical with t0 = µ/(1 − σ 2 ) as the center and η0 (t0 ) > 0; if h(v0 ) ≥ 0,
Lemma 3. Let h(v) =
η(t) increases constantly (Block et al., 2005).
In the former case,namely, when h(v0 ) < 0, from t0 to +∞, η(t) first increases, then decreases, and finally increases again; and the η(t) monotonicity varies with the left part and the right part of t0 symmetrical to each other. Lemma 4. If η0 (t) > 0 when t ∈ (t ∗ , +∞), r (t) increases in this interval. This conclusion can be drawn from Glaser’s analysis (Glaser, 1980). Theorem 1. If the mixture model (2.2) is bimodal, η(t) monotonicity shows four changes. If f (t) is bimodal, there exists t ∈ (0, µ), s.t. f 0 (t) = 0 and f 00 (t) > 0; then η0 (t) = [( f 0 (t))2 − f (t) · f 00 (t)]/ [ f (t)]2 < 0. Thanks to the continuity and symmetry of η0 (t), η(t) has four changes of monotonicity according to Lemma 3. The equation h(v) = 0 can be simplified to σp σ2 v+ + 2(1 − σ 2 ) ln v = µ2 + 2(σ 2 − 1) ln v − σ 2 − 1 (v defined as above). (2.3) v q If the mixture is bimodal, the Eq. (2.3) has two real roots: v1 and v2 which determine four t values. They can be µ 0 written as ti (i = 1, 2, 3, 4) with t1 < t2 < 1−σ 2 < t3 < t4 . If f (t) is bimodal, there exists η > 0 in these intervals: (−∞, t1 ), (t2 , t3 ), and (t4 , +∞); and η0 < 0 within (t1 , t2 ) and (t3 , t4 ). µ If σ > 1, there exists t1 < t2 < 1−σ 2 < 0 < t3 < t < t4 < µ; according to Lemmas 1 and 4, r (t) increases within (−∞, 0) and (t, +∞). Moreover, r (t) has another property: r 0 (t) = r (r − η)
and r 00 (t) = r [(2r − η)(r − η) − η0 ].
So if r 0 ≥ 0 is constantly true within (0, t), r (t3 ) is not less than η(t3 ) (the maximum of η(t) is in this section). In this case, r (t) increases constantly within (−∞, +∞), i.e. the failure rate of the mixture is in its IFR shape. Meanwhile, if r (t3 ) < η(t3 ), r (t) decreases within a certain section of (0, t); i.e. the failure rate of the mixture is in its MBT shape. Hence, we give the following two corollaries: Corollary 1. If σ > 1,the failure rate has only two shapes: IFR (r (t3 ) ≥ η(t3 )) and MBT (r (t3 ) < η(t3 )). µ If 0 < σ < 1, there exists 0 < t1 < t < t2 < µ < 1−σ 2 < t3 < t4 . According to Lemmas 1 and 4, the failure rate of the mixture increases within these intervals: (−∞, 0), (t, µ), and (t4 , +∞). The interval (0, t) can be analyzed in the same way: if r (t1 ) ≥ η(t1 ), r (t) increases in this interval; and if r (t1 ) < η(t1 ), r (t) decreases in a certain section of (0, t). However, the case is different for the interval (µ, t4 ). According to Lemma 2 and the properties of η(t), it is
2008
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
(a) p = 0.05, q = 0.95, µ = 6, σ = 5.
(b) p = 0.4, q = 0.6, µ = 6, σ = 5.
Fig. 1. (σ > 1) density function f, η function and the failure rate r (t).
(a) p = 0.5, q = 0.5, µ = 1.3, σ = 0.3.
(b) p = 0.885, q = 0.115, µ = 3, σ = 0.3.
Fig. 2. (0 < σ < 1) density function f, η function and the failure rate r (t).
easy to observe that η(t) becomes very close to its asymptote y = t beginning from t4 . Because of the properties of r (t), namely, r 0 (t) = r (r − η) and r 00 (t) = r [(2r − η)(r − η) − η0 ], it can be assumed that r 0 (t) is constantly nonnegative in this interval (µ, t4 ); and the comparison of the failure rate monotonicity with the monotonicity changes of η(t) in the interval (µ, +∞) proves that if the assumption is true, the failure rate can no longer vary close to its asymptote. This is obviously impossible. In other words, r (t) absolutely decreases in a certain section of (µ, t4 ). When 0 < σ < 1, the following can be concluded in the same way: Corollary 2. If 0 < σ < 1, the failure rate has only two shapes: MBT (r (t1 ) ≥ η(t1 )) and SB (r (t1 ) < η(t1 )). 3. Examples Case I (IFR, σ > 1) p = 0.05, µ = 6 and σ = 5; the mixture is bimodal and the failure rate is in the shape of the IFR (Fig. 1(a)); Case II (MBT, σ > 1) p = 0.05, µ = 12 and σ = 5; the mixture is bimodal and the failure rate is in the shape of the MBT (Fig. 1(b)); Case III (MBT, 0 < σ < 1) p = 0.5, µ = 1.3 and σ = 0.3; the mixture is bimodal and the failure rate is in the shape of the MBT (Fig. 2(a)); Case IV (SB, 0 < σ < 1) p = 0.885, µ = 3 and σ = 0.3; the mixture is bimodal and the failure rate is in the SB shape (Fig. 2(b)).
F. Liu, Y. Liu / Statistics and Probability Letters 78 (2008) 2006–2009
2009
Table 1 Failure rate shape
σ condition
Example
IFR MBT
σ >1 σ >1 0<σ <1 0<σ <1
r (t3 ) ≥ η(t3 ) r (t3 ) < η(t3 ) r (t1 ) ≥ η(t1 ) r (t1 ) < η(t1 )
SB
4. Conclusion The failure rate properties of the mixture model (2.2) can be summarized as in Table 1. Acknowledgements The authors are very grateful to the referees for their valuable remarks and suggestions. References Block, H.W., Li, Y., Savits, T.H., 2005. Mixtures of normal distributions: Modality and failure rate. Statist. Probab. Lett. 74, 253–264. Glaser, R.E., 1980. Bathtub and related failure rate characterizations. JASA 75, 667–672. Robertson, C.A., Fryer, J.G., 1969. Some descriptive properties of normal mixtures. Scand. Actuar. Tidskr. 52, 137–146.
Fig. 1(a) Fig. 1(b) Fig. 2(a) Fig. 1(b)