EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 157-166 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Nonlinear Least Squares Estimation of the Shifted Gompertz Distribution Dragan Jukić1, Darija Marković1,∗ 1 Department of Mathematics, J.J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, HR-31 000 Osijek, Croatia Abstract. The focus of this paper is the existence of the best nonlinear least squares estimate for the shifted Gompertz distribution. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the lp norm (1 ≤ p <∞). 2010 Mathematics Subject Classifications: 65D10, 65C20, 62J02 Key Words and Phrases: Shifted Gompertz distribution, nonlinear least squares, least squares estimate, lp-norm estimate, existence problem 1. Introduction The shifted Gompertz distribution was introduced by Bemmaor [3] in 1994 as a model of adoption of innovations. The cumulative distribution function (CDF) of the random variable T having the shifted Gompertz distribution is given by F (t; a, b) = { (1− e−bt) e−a e −bt , t > 0 0, t ≤ 0. (1) The parameters a > 0 and b > 0 are called the shape parameter and the scale parameter, respectively. More information on statistical properties of the shifted Gompertz distribu- tion can be found in Bemmaor [3] and Jiménez and Jodrá [12]. Note that the shifted Gompertz distribution can be interpreted as the distribution of the maximum of two independent random variables, one of which has an exponential distribution with parameter b > 0 and the other one has a Gumbel distribution with parameters a > 0 and b > 0. In practice, the unknown parameters of the shifted Gompertz distribution are not known in advance and must be estimated from a random sample. There is no unique way to estimate the unknown parameters and many different statistical methods have been proposed in the literature, such as the maximum likelihood method, the method of ∗Corresponding author. Email addresses: jukicd@mathos.hr (D. Jukić),darija@mathos.hr (D. Marković) http://www.ejpam.com 157 c© 2017 EJPAM All rights reserved. D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 158 moments, the method of percentiles and the Bayesian method. But, since each of these methods has some advantages and disadvantages, several other methods are proposed to estimate the unknown parameters in the shifted Gompertz distribution. For example, maximum likelihood method is very efficient for large samples, but not so efficient with small samples. A very popular method for parameter estimation is the least squares (LS) method. This method usually gives very good estimates even for small data sets. Numerical methods for solving the nonlinear LS problem are described in Dennis and Schnabel [9] and Gill et al. [10]. Before starting an iterative procedure, one should ask whether an LS estimate exists. In the case of nonlinear LS problems, it is still extremely difficult to answer this question (see [2, 5, 19, 20]). Results on the existence of the LS estimate for some special classes of functions other than the shifted Gompertz distribution can be found in [6, 7, 8, 14, 16, 17, 21]. In this paper, we consider the nonlinear weighted LS parameter estimation problem for the shifted Gompertz distribution. Our focus is on the existence of the corresponding best LS estimate. To the best of our knowledge, there is no paper focused on this existence problem. In Section 2, we briefly describe the LS method and show that it is possible that the LS estimate for the shifted Gompertz distribution does not exist (Proposition 1). As our main results, we present two theorems (Theorem 1 and Theorem 2) on the existence of the LS estimate for the shifted Gompertz distribution, as well as their generalizations (Theorem 3 and Theorem 4) in the lp norm (1 ≤ p <∞). This paper is motivated by the paper of Jiménez Torres [13], where LS estimation for the log-transformed shifted Gompertz distribution was considered. 2. Estimation of the shifted Gompertz distribution In this section, we first formulate the LS fitting problem for the shifted Gompertz distribution and then present two theorems on the existence of the least squares estimate, as well as their generalizations in the lp norm (1 ≤ p <∞). 2.1. LS fitting problem for the shifted Gompertz distribution Suppose we are given the data (wi, ti, yi), i = 1, . . . , n, n > 2, where 0 < t1 < t2 < . . . < tn denote the values of the independent variable (observations of the nonnegative shifted Gompertz random variable T , arranged in their increasing order), 0 < y1 < y2 < . . . < yn < 1 are the respective estimators of the empirical CDF, and wi > 0 are some data weights. Since the shifted Gompertz random variable T is nonnegative and numbers yi denote empirical CDF values, the above two conditions are natural. D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 159 There are many different ways to derive estimators yi for the empirical CDF corre- sponding to the sample data t1 < t2 < . . . < tn. Most commonly used estimators can be expressed in the following form (see [15, 18]): yi = i− c n+ 1− 2c , i = 1, . . . , n, for some real number c, 0 ≤ c < 1. Some alternatives are as follows: yi = i n+1 (mean rank estimator, c = 0), yi = i−0.5 n (median rank estimator, c = 0.5), yi = i−0.3 n+0.4 (Benard’s median rank estimator, c = 0.3). The goal of the LS method (see e.g. [2, 5, 10, 19, 20]) is to choose the unknown parameters of the shifted Gompertz distribution (1) such that the weighted sum of squared distances between the model and the data is as small as possible. More precisely, the unknown parameters a and b have to be estimated by minimizing the functional S(a, b) = n∑ i=1 wi[F (ti; a, b)− yi]2 (2) on the set (parameter space) P := { (a, b) ∈ R2 : a, b > 0 } . A point (a?, b?) ∈ P such that S(a?, b?) = inf(a,b)∈P S(a, b) is called the least squares estimate (LS estimate), if it exists (see e.g. [5, 8, 10, 19, 20]). The following proposition shows that there exist data such that the LS estimate for the shifted Gompertz distribution (1) does not exist. Proposition 1. Let (wi, ti, yi), i = 1, . . . , n, n ≥ 3, be the data. If the data are such that the points (ti, yi), i = 1, . . . , n, all lie on some exponential curve g(t) = 1− e−b0t, b0 > 0, then the LS estimate does not exist. Proof. Since S(a, b) ≥ for all (a, b) ∈ P, and lim a→0+ S(a, b0) = lim a→0+ n∑ i=1 wi[(1− e−b0ti) e−a e −b0ti −yi]2 = n∑ i=1 wi[(1− e−b0ti)− yi]2 = 0, it is easy to conclude that inf(a,b)∈P S(a, b) = 0. Furthermore, since the graph of any shifted Gompertz distribution (1) intersects the graph of exponential function g(t) = 1− e−b0t in at most two points, and n ≥ 3, it follows that S(a, b) > 0 for all (a, b) ∈ P, and hence the best LS estimate does not exist. 2.2. The LS existence theorem for the shifted Gompertz distribution The following theorem gives a necessary and sufficient condition on the data which guarantee the existence of the LS estimate for the shifted Gompertz distribution. First, D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 160 we introduce one notation. Let E? be an infimum of the weighted sum of squares for the exponential function (distribution) g(t) = 1− e−bt (b > 0), i.e., E? = inf b>0 E(b), where E(b) = n∑ i=1 wi[(1− e−bti)− yi]2. Theorem 1 (Necessary and sufficient condition). Suppose that the data (wi, ti, yi), i = 1, . . . , n, n ≥ 3, satisfy conditions 0 < t1 < t2 < . . . < tn and 0 < yi < 1, i = 1, . . . , n. Then the LS estimate for the shifted Gompertz distribution (1) exists if and only if there is a point (a0, b0) ∈ P such that S(a0, b0) ≤ E?. By using Theorem 3.1 from Jukić [14], it is easy to show that there exists a β? > 0 such that E(β?) = E?. Therefore, in other words, under the assumptions of the theorem, the LS estimate exists if and only if there is at least one shifted Gompertz distribution which is in an LS sense as good as or better than the best exponential distribution. Remark 1. It can be easily shown that if ∂S/∂a < 0 evaluated at a = 0 and b = b0 = argminE(b), then there exists a point (a, b0) ∈ P such that S(a, b0) < E?, which according to the Theorem 1 ensure the existence of the least squares estimate for the shifted Gompertz distribution. The next lemma will be used in the proof of Theorem 1. Lemma 1. Suppose we are given the data (ti, yi), i = 1, . . . , n, n > 2, such that 0 < t1 < t2 < . . . < tn and 0 < yi < 1, i = 1, . . . , n. Let wi > 0, i = 1, . . . , n, be some weights. Given any real number τ0 > 0, let Στ0 := ∑ ti<τ0 wiy 2 i + ∑ ti>τ0 wi(1− yi)2. Then there exists a point in P at which functional S defined by (2) attains a value less than Στ0. The summation ∑ ti<τ0 (or ∑ ti>τ0 ) is to be understood as follows: the sum over those indices i ≤ n for which ti < τ0 (or ti > τ0). If there are no such points ti, the sum is empty; following the usual convention, we define it to be zero. Proof. Let τ0 > 0 be given. If τ0 6= ti, for each index i in the range 1 to n, let ξ0 be an arbitrary number from the interval (0, 1), and otherwise, if τ0 = ti for some i, let ξ0 = yi. Define function a : ( 1 τ0 ln 1 1−ξ0 ,∞ ) → R a(b) = ln (1− e−bτ0 ξ0 ) ebτ0 . D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 161 It is easy to verify that a(b) is positive, and therefore (a(b), b) ∈ P. Let us now associate with each real b ∈ ( 1 τ0 ln 1 1−ξ0 ,∞ ) a shifted Gompertz distribution function F (t; a(b), b) = { (1− e−bt) e − ln ( 1−e−bτ0 ξ0 ) e−b(t−τ0) , if t > 0 0, if t ≤ 0. By a straightforward calculation, it can be verified that F (τ0; a(b), b) = ξ0 (3) and lim b→∞ F (t; a(b), b) = { 0, if 0 < t < τ0 1, if t > τ0. Due to this, we may assume that for every sufficiently large b > 0, 0 < F (ti; a(b), b) < yi, if 0 < ti < τ0 (4) yi < F (ti; a(b), b) < 1, if ti > τ0. Hence, for every sufficiently large b > 0 it follows from (3) and (4) that S(a(b), b) = n∑ i=1 wi[F (ti; a(b), b)− yi]2 = ∑ ti<τ0 wi[F (ti; a(b), b)− yi]2 + ∑ ti>τ0 wi[F (ti; a(b), b)− yi]2 < ∑ ti<τ0 wiy 2 i + ∑ ti>τ0 wi(1− yi)2 = Στ0 . This completes the proof of the lemma. Proof of Theorem 1. Assume first that (a?, b?) ∈ P is the best LS estimate, and then show that S(a?, b?) ≤ E?. In order to do this, first note that for all a, b > 0, S(a?, b?) ≤ S(a, b) = n∑ i=1 wi[(1− e−bti) e−a e −bti −yi]2, from where, taking the limit as a→ 0, it follows that S(a?, b?) ≤ n∑ i=1 wi[(1− e−bti)− yi]2. From the last inequality and the definition of E? we obtain that S(a?, b?) ≤ E?, so that it is enough to set (a0, b0) = (a?, b?). D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 162 Let us show the converse of the theorem. Suppose that there is a point (a0, b0) ∈ P such that S(a0, b0) ≤ E?. Since functional S is nonnegative, there exists S? := inf(a,b)∈P S(a, b). It should be shown that the best LS estimate exists, i.e., that there exists a point (a?, b?) ∈ P such that S(a?, b?) = S?. To do this, first note that S? ≤ S(a0, b0) ≤ E? If S? = S(a0, b0), to complete the proof it is enough to set (a?, b?) = (a0, b0). Hence, we can further assume that S? < S(a0, b0) ≤ E?. (5) Let (ak, bk) be a sequence in P, such that S? = lim k→∞ S(ak, bk) = lim k→∞ n∑ i=1 wi[F (ti; ak, bk)− yi]2 = lim k→∞ n∑ i=1 wi[(1− e−bkti) e−ak e −bkti −yi]2. Without loss of generality, in further consideration we may assume that sequences (ak) and (bk) are monotone. This is possible because the sequence (ak, bk) has a subsequence (alk , blk), such that all its component sequences (alk) and (blk) are monotone, and since limk→∞ S(alk , blk) = limk→∞ S(ak, bk) = S?. As each monotone sequence of real numbers converges in the extended real number system R̄, denote a? := lim k→∞ ak, b? := lim k→∞ bk. Note that 0 ≤ a? ≤ ∞ and 0 ≤ b? ≤ ∞, because (ak, bk) ∈ P. To complete the proof, it is enough to show that (a?, b?) ∈ P, i.e., that 0 < a? < ∞ and 0 < b? < ∞. The continuity of the functional S will then imply that S? = limk→∞ S(ak, bk) = S(a?, b?), which will complete the proof of the theorem. Before continuing with the proof, let us note that Lemma 1 implies that S? < Στ0 , (6) for arbitrary τ0 > 0. It remains to show that (a?, b?) ∈ P. The proof will be done in four steps. In Step 1, we will show that b? 6= 0. In Step 2, we will show that b? 6= ∞, which will imply that 0 < b? < ∞. The proof that a? 6= ∞ will be done in Step 3. Finally, in Step 4 we will show that a? 6= 0. Step 1. Let us first show that b? 6= 0. We prove this by contradiction. Suppose to the contrary that b? = 0. First, note that for all ak, bk ≥ 0 and for all t ≥ 0 0 ≤ (1− e−bkt) e−ak e −bkt ≤ 1− e−bkt . (7) Furthemore, since bk → 0, then 1− e−bkt → 0, D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 163 and thus from inequalities (7) we have F (t; ak, bk) = (1− e−bkt) e−ak e −bkt → 0, for all t ≥ 0. Therefore, we would obtain that S? = lim k→∞ n∑ i=1 wi[F (ti; ak, bk)− yi]2 = n∑ i=1 wiy 2 i > n−1∑ i=1 wiy 2 i = Σtn , which contradicts (6). This means that in this way functional S cannot attain its infimum and we have proved that b? 6= 0. Step 2. Let us now show that b? 6=∞. We prove this by contradiction. Suppose to the contrary that b? = ∞. For each i = 1, . . . , n, let us denote l?i := limk→∞ ake −bkti . Now from the obvious inequalities ake −bkt1 ≥ ake−bkt2 ≥ · · · ≥ ake−bktn ≥ 0, after taking the limit k →∞, we obtain l?1 ≥ l?2 ≥ · · · ≥ l?n ≥ 0. Note that only one of the following subcases can occur: (a) l?i = ∞ for all i = 1, . . . , n, (b) there exists an index i such that 0 ≤ l?i <∞. Subcase (a) If l?i = ∞ for all i = 1, . . . , n, then limk→∞ F (ti; ak, bk) = 0 for all i, and consequently we would have that S? = ∑n i=1wiy 2 i > Σtn . As already shown in Step 1, in this way functional S cannot attain its infimum. Subcase (b) Let i0 := min{i | 0 ≤ l?i <∞}. Then for each i > i0, l?i = lim k→∞ ak e−bkti = lim k→∞ ak e−bkti0 e−bk(ti−ti0 ) = l?i0 lim k→∞ e−bk(ti−ti0 ) = 0. Thus l?i = { ∞, for all i < i0 0, for all i > i0, and consequently lim k→∞ F (ti; ak, bk) = { 0, for all i < i0 1, for all i > i0, from where it follows that S? = lim k→∞ n∑ i=1 wi[F (ti; ak, bk)− yi]2 ≥ ∑ titi0 wi(1− yi)2 = Σti0 . Again, this contradicts (6). Thus we have proved that b? 6=∞. Step 3. In this step, we will show that a? 6= ∞. We prove this by contradic- tion. Suppose to the contrary that a? = ∞. Then, as in Step 1, we would have that D. Jukić, D. Marković / Eur. J. Pure Appl. Math, 10 (2) (2017), 157-166 164 limk→∞ F (t; ak, bk) = 0 for all t ≥ 0, and as we have already shown, in that case func- tional S cannot attain its infimum. Step 4. It remains to show that a? 6= 0. Suppose to the contrary that a? = 0. Then limk→∞ F (t; ak, bk) = 1− e−b ?t for all t ≥ 0. In this case we would have S? = n∑ i=1 wi [ (1− e−b ?ti)− yi ]2 ≥ E?, which contradicts assumption (5). This means that in this way (a? = 0) functional S cannot attain its infimum. Thus we have provided that a? 6= 0, and herewith we have completed the proof of Theorem 1. From the curve fitting point of view, it makes sense to allow parameter a to be zero, i.e., to minimize functional S over the following set of parameters P0 := { (a, b) ∈ R2 : a ≥ 0, b > 0 } . The next theorem tells us that if that is of interest, then the corresponding LS estimate will exist. Theorem 2. Let the points (wi, ti, yi), i = 1, . . . , n, n > 2, be data such that 0 < t1 < t2 < . . . < tn and 0 < yi < 1, i = 1, . . . , n. Then there exists a point (a?, b?) ∈ P0 such that S(a?, b?) = inf (a,b)∈P0 S(a, b). The proof of this theorem is omitted; it is the same for respective parts of the proof of Theorem 1, with the exception that we do not have to prove that a? 6= 0. 2.3. The lp-norm existence theorem for the shifted Gompertz distribution The LS problem is a nonlinear l2-norm problem. During the last few decades an increased interest in alternative lp-norm has become apparent (see e.g. [1] and [11]). For example, l1-norm criteria are more suitable if there are wild points (outliers) in the data. Thus, instead of minimizing functional S, sometimes a more adequate criterion for estimation of unknown parameters a and b of the shifted Gompertz distribution (1) is to minimize the following functional: Sp(a, b) = n∑ i=1 wi ∣∣F (ti; a, b)− yi ∣∣p, (8) where p (1 ≤ p <∞) is an arbitrary fixed number. A point (a?, b?) ∈ P such that Sp(a ?, b?) = inf (a,b)∈P Sp(a, b) is called the best lp-norm estimate, if it exists. For p = 2, the best l2-norm estimate is the familiar weighted LS estimate. REFERENCES 165 To state the corresponding lp-norm (1 ≤ p <∞) generalizations of Theorems 1 and 2, we need an additional notation. Let E?p := inf b>0 Ep(b), where Ep(b) = n∑ i=1 wi|(1− e−bti)− yi|p. Obviously, E? = E?2 and S = S2. Again, by using Theorem 3.1 from Jukić [14], it is easy to show that there exists a β? > 0 such that Ep(β ?) = E?p . Arguing in a similar way as in proofs of Lemma 1, Theorem 1 and Theorem 2, we can easily show the following lp-norm generalizations of Theorem 1 and Theorem 2. To do this, it suffices to replace the l2 norm by the lp norm. Thereby all parts of the proofs remain the same. Theorem 3 (Necessary and sufficient condition). Suppose that the data (wi, ti, yi), i = 1, . . . , n, n ≥ 3, satisfy conditions 0 < t1 < t2 < . . . < tn and 0 < yi < 1, i = 1, . . . , n.. Then functional Sp defined by (8) attains its infimum on P if and only if there is a point (a0, b0) ∈ P such that Sp(a0, b0) ≤ E?p . Theorem 4. Let the points (wi, ti, yi), i = 1, . . . , n, n > 2, be data such that 0 < t1 < t2 < . . . < tn and 0 < yi < 1, i = 1, . . . , n. Then there exists a point (a?, b?) ∈ P0 such that Sp(a ?, b?) = inf (a,b)∈P0 Sp(a, b). Acknowledgements We would like to thank the referees for the useful comments and suggestions. References [1] A. Atieg, G.A. Watson. Use of lp norms in fitting curves and surfaces to data. ANZIAM J., 45(E):C187-C200, 2004. [2] D.M. Bates, D.G. Watts. Nonlinear regression analysis and its applications. Wiley, New York, 1988. [3] A.C. Bemmaor. Modelling the diffusion of new durable goods: Word-of-mouth effect versus consumer heterogeneity. In G. Laurent, G.L. Lilien, B. Pras, editors, Research Traditions in Marketing., pages 201-229, Boston, 1994., Kluwer. [4] A.C. Bemmaor, J. Lee. The impact of heterogeneity and ill-conditioning on diffusion model parameter estimates. Marketing Sci., 21:209-220, 2002. [5] Å. Björck. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996. [6] E.Z. Demidenko. Criteria for unconstrained global optimization. J. Optim. Theory Appl., 136:375-395, 2008. REFERENCES 166 [7] E.Z. Demidenko. Criteria for global minimum of sum of squares in nonlinear regres- sion. Comput. Statis. Data Anal., 51:1739-1753, 2006. [8] E.Z. Demidenko. Is this the least squares estimate?. Biometrika, 87:437-452, 2000. [9] J.E. Dennis, R.B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996. [10] P.E. Gill, W. Murray, M.H. Wright. Practical Optimization. Academic Press, London, 1981. [11] R. Gonin, A.H. Money. Nonlinear Lp-Norm Estimation. Marcel Dekker, New York, 1989. [12] F. Jiménez, P. Jodrá. A note on the moments and computer generation of the shifted Gompertz distribution. Commun. Stat. Theory Methods, 38:75-89, 2009. [13] F. Jiménez Torres. Estimation of parameters of the shifted Gompertz distribution using least squares, maximum likelihood and moments methods. J. Comput. Appl. Math., 255:867-877, 2014. [14] D. Jukić. A simple proof of the existence of the best estimator in a quasilinear re- gression model. J. Optim. Theory Appl., 162:293-302, 2014. [15] J.F. Lawless. Statistical models and methods for lifetime data. Wiley, New York, 1982. [16] D. Marković, D. Jukić. On parameter estimation in the Bass model by nonlinear least squares fitting the adoption curve. Int. J. Appl. Math. Comput. Sci., 23:145-155, 2013. [17] D. Marković, D. Jukić, M. Benšić. Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start. J. Comput. Appl. Math., 228:304-312, 2009. [18] W. Nelson. Applied life data analysis. Wiley, New York, 1982. [19] G.J.S. Ross. Nonlinear Estimation. Springer, New York, 1990. [20] G.A.F. Seber, C.J. Wild. Nonlinear Regression. Wiley, New York, 1989. [21] V. Srinivasan, C.H. Mason. Nonlinear least squares estimation of new product diffu- sion models. Marketing Sci., 5:169-178, 1986.