Marshall-Olkin Geometric Distribution

Transcription

1 Chapter 4 Marshall-Olkin Geometric Distribution 4.1. Introduction Here, we consider a new famil of distributions b adding an additional parameter to geometric distribution b using the MO scheme. We call it Marshall-Olkin geometric (MOG) distribution. In the third section, the nature of epectation and entrop is mentioned. In the fourth section, hazard function, log-conveit, i.d. and g.i.d. are discussed. It is shown that the MOG distribution possesess the propert of new better than used in section 5. In the sith section, the distributions of minimum of sequence of i.i.d r.vs are derived. An AR (1) model with the MOG distribution is discussed in the net section. The MLE of the parameters of MOG is found in section 8. An application of MOG is discussed in the last section. The contents of this chapter have appeared in Sandha and Prasanth [51]. 75

2 4.2. Marshall-Olkin Geometric Distribution We consider a discrete r.v X following Geometric distribution with parameter p with p.m.f., f() = (1 p) 1 p, with (p + q = 1), = 1, 2, 3,... and with the d.f F () = 1 q, then b the MO method, we have the new s.f, G(, θ), b substituting, F () = (q ) in (1.1.3). That is, we have, G(, θ) = θ q /[1 (1 θ) q ]. (4.2.1) The r.v with this s.f is termed as Marshal Olkin Geometric (MOG) r.v. This is denoted b X MOG(q, θ) Stabilit Propert of the New Famil. If we appl the same method again into the new famil, that is, let us add a new parameter α (α > 0) to the reliabilit function we get G() = αθq /[1 (1 αθ)q ]. It is the reliabilit function of the new MOG famil with parameters (q, αθ) The p.m.f. of MOG Distribution. The p.m.f. of MOG (q, θ), g(, θ) = G( 1, θ) G(, θ) = {θq 1 /[1 (1 θ)q 1 ]} {θq /[1 (1 θ)} i.e., g(, θ) = θq 1 (1 q)/[1 (1 θ)q ( 1) ][1 (1 θ)q ], = 1, 2,... (4.2.2) 76

3 We numericall evaluate (Table 4.1 and Table 4.2) and plot the graph of the p.m.f. of, with different values for the parameters q, θ (Figure 4.1) with θ > 1 and (Figure 4.2) with θ < 1. Table 4.1. The p.m.f of X MOG(q, θ) for different q & θ q = 0.5 q = 0.5 q = 0.75 q = 0.75 q = 0.75 q = 0.75 q = 0.75 θ = 0.1 θ = 10 θ = 2 θ = 10 θ = 0.5 θ = 5 θ = e e e e e e e e e e e e e e e e e e e e e e e e e e e

4 Table 4.2. The p.m.f of X MOG(q, θ) for different q & θ q = 0.1 q = 0.1 q = 0.1 q = 0.9 q = 0.9 q = 0.9 q = 0.9 θ = 0.5 θ = 2 θ = 1 θ = 5 θ = 0.5 θ = 2 θ = e e e e e e e e e e e e e e e-08 9e e e-09 9e e e-10 9e e e-11 9e e e-12 9e e e-13 9e e e-14 9e e e-15 9e e e-16 9e e e-17 9e e e-18 9e e e-19 9e e e-20 9e e e-21 9e e e-22 9e e e-23 9e e e-24 9e We plot the graph of p.m.f. with different values for the parameters q, θ (Figure MOG1) with θ > 1 and (Figure MOG2) with θ < 1. 78

5 Figure 4.1. The p.m.f. of X MOG(q, θ) with θ > 1 X ~ MOG (0.9, 5) X~ MOG (0.75, 250) X ~ MOG (0.75, 2) X ~ MOG (0.1, 2) X~ MOG (0.5, 10) Figure 4.2. The p.m.f. of X MOG(q, θ) with θ < 1 X ~ MOG (0.9, ) X ~ MOG (0.75, 5) X ~ MOG (0.75, 0.5)

6 X ~ MOG (0.5, 0.1) X ~ MOG (0.9, 0.5) The following verifies the result in Remark in the case of MOG distribution. Remark If X MOG(X, θ), then F () G(, θ) θf () if θ 1 and θf () < G(, θ) < F () if 0 < θ < 1. Proof. If X MOG(X, θ) then F () = q and G(, θ) = θq /[1 (1 θ)q ] So, the first part F () G(, θ) implies q θq /[1 (1 θ)q ] That is q θq /[1 (1 θ)q ] 0. So q [1 (1 θ)q ] θq 0. q (1 θ)q 2 θq 0 q θq (1 θ)q 2 0 q (1 θ) (1 θ)q 2 0 (1 θ)(q q 2 ) 0. This happens onl when θ > 1 since q q 2 0. The proof for the second part is, 80

7 G(, θ) θf () i.e., {θq /[1 (1 θ)q ]} θq < 0. So θq [1 (1 θ)q ]θq < 0. i.e., θq θq + (1 θ)θq 2 < 0, or (1 θ)θq 2 < 0, this is possible onl when θ > 1, since q 2 and θ alwas greater than zero. Hence the proof Epectation, Standard Deviation and Entrop of MOG Distribution We numericall calculate the epectation, sd and Shannon s entrop. For the discrete data, Entrop S = k I p i[ln(p i )] is referred to as Shannon s entrop. i.e., here Shannon s Entrop = θ(1 q) i {q 1 /[1 (1 θ)q 1 ][1 (1 θ)q ] log θq 1 (1 q)/[1 (1 θ)q 1 ][1 (1 θ)q ]}. Table 4.3. Epectation, sd and entrop of X MOG(q, θ) q = 1 p θ E() sd() Entrop Table continues to the net page... 81

8 q = 1 p θ E() Sd() Entrop A closer look with the entrop of X MOG(q, θ) with different q and θ (Table 4.4 and Figure 4.3) Table 4.4. Entrop of X MOG(q, θ) with different q & θ q θ = θ = 0.5 θ = 0.75 θ = 2 θ = 5 θ =

9 Figure 4.3. Entrop of X MOG(q, θ) with different q and θ 3.0 X~MOG(0.9,θ) 3.0 X~MOG(0.75,θ) X~MOG(0.1,θ) It is clear that, for X MOG(q, θ), the epectation and the standard deviation are decreasing with decreasing value of θ and decreasing value of q when θ < 1. The entrop with respect to MOG(q, θ), is also decreasing with decreasing value of θ and decreasing value of q, when θ < 1. The mean, median and mode of the distribution for different q and θ are computed below (Table 4.5) for X MOG(q, θ). 83

10 Table 4.5. Mean, median and mode of the MOG distribution with different q and θ q θ mean median mode From the Table 4.5 it is clear that, Remark The MOG distribution is positivel skewed when θ <1, since the mean is greater than the mode and the median. Also it is found that the distribution is unimodal. 84

11 4.4. Hazard Function, Log-Conveit, Infinite Divisibilit and Geometric Infinite Divisibilit From (4.2.1) and (4.2.2), the hazard function, γ G () = g()/g() = (1 q)/q[1 (1 θ)q 1 ]. (4.4.1) We compute hazard function of MOG distribution, with different values for the parameters q, θ (Table 4.6 and Table 4.7). Table 4.6. Hazard function of X MOG(q, θ) distribution with different q and θ q = 0.9 q = 0.75 q = 0.9 q = 0.5 q = 0.1 q = 0.75 q = 0.75 θ = 2 θ = 2 θ = 5 θ = 10 θ = 2 θ = 10 θ = Table continues to the net page... 85

12 q = 0.9 q = 0.75 q = 0.9 q = 0.5 q = 0.1 q = 0.75 q = 0.75 θ = 2 θ = 2 θ = 5 θ = 10 θ = 2 θ = 10 θ = Table 4.7. Hazard function of X MOG(q, θ) distribution with different q and θ q = 0.9 q = 0.75 q = 0.9 q = 0.5 q = 0.1 q = 0.1 q = 0.75 θ = 0.5 θ = 0.5 θ = θ = 0.1 θ = 0.5 θ = 1 θ =

13 From Table 4.6 and Table 4.7 it is clear that for MOG distribution the failure rate is, IFR when θ > 1, DFR when θ < 1 and constant when θ = 1. Figure 4.4. Hazard rate of X MOG(q, θ) for different values for the parameters q, θ X ~ MOG (0.9, 5) X ~ MOG (0.9, 3) X ~ MOG (0.9, 2) X ~ MOG (0.75, 0.5) X ~ MOG (0.9, ) X ~ MOG (0.9, 0.5) X ~ MOG (0.75, 1)

14 Also from the Figure 4.4, we can see that, for MOG the failure rate is: IFR when θ > 1, DFR when θ < 1. Note that for θ = 1, the distribution reduces to geometric distribution and has constant hazard rate. Definition (Log-conveit). For a discrete distribution with infinite support, if [P (X = + 1)] 2 < P (X = )P (X = + 2) then probabilities are log-conve. For X MOG(q, θ), the p.m.f. g(, θ) = P (X = ) = θq 1 (1 q)/ [1 (1 θ)q 1 ][1 (1 θ)q () ], = 1, Then, [P (X = + 1)] 2 = θ 2 q 2 (1 q) 2 /{[1 (1 θ)q ][1 (1 θ)q (+1) ]} 2 and P (X = + 2) = θq +1 (1 q)/[1 (1 θ)q +1 ][1 (1 θ)q (+2) ]. Therefore, the probabilities are log-conve, if [P (X = + 1)] 2 /P (X = )P (X = + 2) = (1 (1 θ)q 1 )(1 (1 θ)q +2 )/(1 (1 θ)q )(1 (1 θ)q +1 ) < 1 i.e., if (θq 1 + θq +2 θq +1 θq )/(q 1 + q +2 q +1 q ) < 1 i.e., if θ < 1. Hence [P (X = + 1)] 2 < P (X = )P (X = + 2) when θ < 1. (4.4.2) i.e., probabilities are log-conve when θ < 1. And it is also seen that [P (X = + 1)] 2 > P (X = )P (X = + 2) when θ > 1. (4.4.3) i.e., probabilities are log-concave when θ > 1. 88

15 Definition (Geometric infinite divisibilit (g.i.d)) A real valued r.v. X is said to have a g.i.d. distribution if for an p (0, 1) there eists a sequence of i.i.d., realvalued r.vs {X (p) i } such that X d = N p i=1 X (p) i where N p has the geometric distribution P (N p = k) = p(1 p) k 1, k = 1, 2, 3..., and N p and {X (p) i } are independent. See Klebanov et al. [33]. Remark In discrete case, following Steutel [69], for distributions on non negative integers, log-conveit implies that the are compound geometric and hence the are i.d. From Klebanov et al. [33], it is clear that compound geometric distributions are g.i.d and that log-conveit implies that the distribution has decreasing hazard rate (DHR) (Johnson et al. [24, p.209]). Thus we have, Theorem If a discrete r.v X MOG(q, θ), = 1, 2, 3,..., then, when 0 < θ < 1, the distribution is i) log-conve ii)compound geometric, iii) i.d., iv) g.i.d, and with v) DHR. Proof of (i) is clear from (4.4.2) and (4.4.3), and the remaining follows from the Remark Remark If a discrete r.v X MOG(q, θ), = 1, 2, Then, when θ > 1, the distribution is log-concave and with IFR. The same can also be stated as follows. 89

16 Remark Let us define η(t) = 1 [P ( = t + 1)/P ( = t)] and η(t) = η(t + 1) η(t). Then if η(t) > 0, then the distribution is with IFR. If η(t) < 0, it has DFR and if η(t) = 0, it is with constant failure rate (Kemp [32]). i.e., η(t) = η(t + 1) η(t) = [P ( = t + 1)/P ( = t)] [P ( = t + 2)/P ( = t + 1)], is > 0 when P ( = t + 1) 2 > P ( = t)p ( = t + 2). For MOG(q, θ), from (4.4.3) this is true when θ > 1. So η(t) > 0 when θ > 1. Hence, the failure rate of MOG is IFR if θ > 1, DFR if θ < 1 and constant if θ = 1. From remark we get, Corollar Let X MOG(q, θ), for θ > 1, the hazard rate is IFR and for θ < 1, the hazard rate is DFR. Also it is clear that when, the hazard rate of X MOG(q, θ), tends to a constant equal to p/q. i.e., the hazard rate γ G () tends to p/q. i.e., we have γ G () = (1 q)/q[1 (1 θ)q 1 ], when, [1 (1 θ)q 1 ] = 1 since when, q 1 0. Therefore as, γ G () = 1 q = p, for θ > 0. q q 4.5. New Better than Used/New Worse than Used (NBU/ NWU) Let X MOG(q, θ). Also let G, G t and G t+ represent the s.f. of X =, X = t and X = + t respectivel. 90

17 We know that if (G t G /G t+ ) > 1 then the distribution is NBU, and if (G t G /G t+ ) < 1 then it is NWU (Kemp [32]). For X MOG(q, θ), G t G /G t+ = θ[1 (1 θ)q (t+) ]/ [1 (1 θ)q ][1 (1 θ)q t ]. If θ[1 (1 θ)q (t+) ]/[1 (1 θ)q ][1 (1 θ)q t ] > 1 the distribution is NBU. i.e., if θ[1 (1 θ)q (t+) ] [1 (1 θ)q ][1 (1 θ)q t ] > 0 i.e., θ (1 θ)[q (t+) + q + q t ] > 0 the distribution is NBU. This is true when θ > 1. i.e., when θ > 1, it is NBU, and if θ < 1, it is NWU. For distribution with infinite support, (Kemp [32]) if it has IFR then it has IFRA (Increasing Failure Rate on Average). Then it has NBU, implies that the distribution is NBUE (New Better than Used Epectation), hence DMRL (Decreasing Mean Residual Lifetime) too. Remark For MOG distribution, when θ > 1, the distribution is log-conve IFR IFRA NBU NBUE DMRL & when θ < 1, the distribution is log-concave DFR DFRA NWU NWUE IMRL. Theorem Let γ F () and γ H(,θ) are the hazard rates of the geometric distribution and MOG distribution respectivel. Then, (a) Lt α γ G(, θ) = Lt α γ F (, θ) for an θ 0. (b) γ F () /θ γ G(,θ) γ F (), θ 1. 91

18 (c) γ F () γ G(,θ) γ F () /θ, 0 θ 1. Proof. a) From (4.4.1), the hazard function of X MOG(q, θ), γ G (, θ) = (1 q)/q[1 (1 θ)q 1 ]. If, q 1 0. That implies when, γ G (, θ) tends to p/q, which is equal to the hazard function γ F () of geometric distribution. Lt γ G(, θ) = Lt γ F (, θ) for an θ 0. α α b) we have, γ F ()/θ = (1 q)/qθ, γ G (, θ) = (1 q)/q[1 (1 θ)q 1 ] here γ F ()/θ γ G (, θ) when qθ q[1 (1 θ)q 1 ] i.e., when θ [1 (1 θ)q 1 ] i.e., when θ [1 q 1 + θq 1 ] i.e., when θ θq 1 1 q 1 i.e., when θ(1 q 1 ) (1 q 1 ) i.e., when θ 1 also γ G (, θ) γ F () when q[1 (1 θ)q 1 ] q i.e., when [1 (1 θ)q 1 ] 1. This is true onl when θ 1. Hence the proof. c) Through changing the inequalities in the proof of (b) we get γ F () γ G (, θ) γ F ()/θ, when 0 θ 1. 92

19 4.6. MOG distribution as the Distribution of Minimum of a Sequence of i.i.d. Random Variables The following theorem gives characterization of MOG distribution as the minimum of a sequence of i.i.d. variables following geometric distribution. Theorem Let {X i, i 1} be a sequence of i.i.d. r.vs with common s.f, F (). Let N be a geometric r.v independent of {X i, i 1} such that P (N = n) = pq n 1 n = 1, 2, 3,...; 0 < θ < 1. Let U N = min 1 i N (X i). Then {U N } is distributed as MOG distribution if and onl if {X i } Geometric (p), = 1, 2, Proof. Let the s.f of U N is, G() = P (U N > ). = [F ()] n P (N = n). 1 = θf ()/[1 (1 θ)f ()]. = θq /[1 (1 θ)q ]. Which is the s.f of MOG(q, θ). Retracing the steps we get the onl if part. Remark This result can be used to generate r.v following MOG(q, θ). 93

20 4.7. AR (1) Model with MOG Distribution as Innovation Distribution Consider the AR(1) minification process with structure: ɛ n with probabilit θ X n = min(x n 1, ɛ n ) with probabilit (1 θ), 0 θ 1 (4.7.1) where {ɛ n } is a sequences of i.i.d. r.v following geometric distribution with parameter p, independent of {X n 1, X n 2...}. Then the process is stationar with MOG(q, θ) as the marginal distribution. Thus we have, Theorem In an AR (1) process with structure (4.7.1) {X n } is stationar for some 0 < θ < 1 with MOG (q, θ) marginal if and onl if {ɛ n } is distributed as geometric with parameter (p). Proof. From structure (4.7.1) it follows that F Xn () = θf ɛn () + (1 θ) F Xn 1 ()F ɛn (). Under stationarit, this implies that F Xn () = θf ɛ ()/[1 (1 θ)f ɛ ()]. If we take F ɛn () = q, then, F X () = θq /[1 (1 θ)q ], (4.7.2) which is the s.f of the MOG(q, θ). 94

21 Conversel we can show that F ɛn () follows geometric distribution with parameter p. We have, F X () = θf ɛ ()/[1 (1 θ)f ɛ ()], implies that, F ɛ ()] = F X ()/(θ + (1 θ)f X ()). Putting F X () = θq /[1 (1 θ)q ], we get F ɛ ()] = q, which is the s.f of the geometric distribution with parameter p. Hence the proof. Suppose we demand (4.7.1) to be satisfied for each 0 < θ < 1, then (4.7.2) is true for each 0 < θ < 1 with F ɛ () replaced b F ɛ,θ (). Hence X n must be min-g.i.d. Now we have, Theorem The sequence of r.vs {X n } defined in model (4.7.2) is stationar for each 0 < θ < 1 if and onl if X n is min-g.i.d with s.f, F () = 1/(1 log H()) where H() is a s.f. Proof follows from the theorem 3.2 of Satheesh et al. [63] Maimum Likelihood Estimates of the Parameters of MOG Distribution Let 1, 2,..., n be a random sample from MOG(q, θ). Then from the likelihood function of the distribution we can write, (n/θ) q i /[1 (1 θ)q i ] q i 1 /[1 (1 θ)q i 1 ] = 0 (4.8.1) (n/(1 q)) ( i 1)/q + (1 θ) i q i 1 /[1 (1 θ)q i ] + (1 θ)( i 1)q i 2 /[1 (1 θ)q i 1 ] = 0 (4.8.2) 95

22 We can find the MLE of q and θ (0 q 1 and θ > 0) numericall from the solution of these two non-linear equations (using Mathematica). Algorithm 1. Through simulation, 1000 random samples were generated from inverse function of probabilit d.f G() for some given value of the parameters θ and q. 2. Then calculate the MLE for the parameters from this with θ < 1 and with θ > For accurac, we repeat the same calculation ten times with same values of θ and a. 4. The mean and SE of these estimates are calculated. 5. Then it is calculated with different values of the parameter θ (Table 4.8) with θ < 1 and (Table 4.9) with θ > The same is found with an increase in sample size 2500 also (Tables 4.10 and 4.11). 7. The mean estimated values for the parameters are tabulated below (Table 4.11, Table 4.12 and Table 4.1). Table 4.8. MLE of θ( θ) and q( q) with (θ < 1, n = 1000) q Mean SE θ q θ q θ q θ q θ q θ ( θ) ( θ) Mean ( q) SE ( q)

23 Table 4.9. MLE of θ( θ) and q( q) with (θ > 1, n = 1000) q Mean θ q θ q θ q θ q θ q θ ( θ) SE ( θ) Mean ( q) SE ( q) Table MLE of θ( θ) and q( q) with (θ < 1, n = 2500) q Mean θ q θ q θ q θ q θ q θ ( θ) SE ( θ) Mean ( q) SE ( q)

24 Table MLE of θ( θ) and q( q) with (θ > 1, n = 2500) q Mean (θe) θ qe θe qe θe qe θe qe θe qe θe SE (θ) Mean (q) SE (q) Table A closer look on SE and mean of MLE of θ and q (let q = 5 for eample) with increase in sample size q = 5 n = 1000 n = 2500 θ q q Mean ( q) SE ( q)

25 Table SE and mean of MLE of θ with increase in sample size (for θ > 1) θ n = 1000 n = 2500 Mean ( θ) SE ( θ) Mean ( θ) SE ( θ) The SE of MLE of the parameter q and θ is decreasing with increase in sample size An Application of MOG Distribution Eample From a pediatrician s clinic, in 50 das (in 2 months eept Sundas) observation, the number of patients who did not consult the doctor, who had alread booked in each da was taken. This is continued 10 times (500 das). The goodness of fit of MOG distribution is tested and compared with the goodness of fit of geometric distribution (Table 4.17 and Table 4.18). We see that the MOG distribution is a better fit. Table Observed frequencies, n = 50 das (12 sample sets from 24 months) Set (Oi) Table continuous to the net page... 99

26 Set (Oi) Assuming MOG distribution, the MLE of the parameters q and θ are calculated (Table 4.15). The SE of these two parameters also found (Table 4.16). Table The MLE of q and θ from the 12 independent set of 50 samples θ q Table The Mean MLE of q and θ n = 50 Mean MLE of θ, θ Mean SE of θ q θ

27 Table Testing the goodness of fit (4 th set) Distribution Observed χ 2 statistics χ 2 table (α = 5) χ 2 α p-value Remark Geometric q = Not a good fit MOG (4 th set) q = Good fit θ = 727 Distribution Table Testing the goodness of fit (12 th set) Observed χ 2 statistics χ 2 table (α = 5) χ 2 α p-value Remark Geometric q = Not a good fit MOG (12 th set) q = Good fit θ = 61 Table 4.17 and Table 4.18 shows that the MOG distribution is a better fit than geometric distribution. Eample From a gnecologist s clinic, in 50 das observation, the number of patients who did not consult the doctor, who had alread booked in each da was taken. This is continued 10 times (500 das). The goodness of fit of MOG distribution is tested and compare with the goodness of fit of geometric distribution (This data is taken from a clinic at Palghat, Kerala, from their dail register throughout 2 ears. That is 10 sample sets from 500 working das of clinic). 101

28 Table Observed frequencies (Oi) Set X (Oi) Assuming MOG distribution, the MLE of the parameters q and θ are calculated (Table 4.20). The SE of these two parameters also found (Table 4.21). Table The MLE of q and θ from the 10 independent set of 50 samples θ q Table The mean and SE of MLE of q and θ n = 50 MLE SE q θ

29 Distribution Observed Table Results of testing the goodness of fit χ 2 χ 2 table (α = 5) χ 2 α p-value Remark Geometric Not a good fit MOG Good fit Comparing the p-values the MOG distribution is better fit than geometric distribution. 103