Intenational Jounal of Contempoay Mathematical Sciences Vol. 12, 2017, no. 5, 243-253 HIKARI Ltd, www.m-hikai.com https://doi.og/10.12988/ijcms.2017.7728 Hypothesis Test and Confidence Inteval fo the Negative Binomial Distibution via Coincidence: A Case fo Rae Events Victo Nijimbee School of Mathematics and Statistics, Caleton Univesity Ottawa, ON K1S 5B6, Canada Copyight c 2017 Victo Nijimbee. This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact We conside the null hypothesis H 0 : p 1 = p 2 =, p n = ϱ against the altenative hypothesis H 1 : p i ϱ fo some i, i = 1, 2,, n whee p i is the popotion of successes in each pefomed expeiment, and ϱ is a eal constant. We also conside the case whee ϱ is small (ϱ 1) and the numbe of failues is lage ( ). We show that in this case Poisson distibution is a good appoximation fo the negative binomial distibution and use this fact in establishing a statistical test to examine these hypotheses. We show that if the null hypothesis is not ejected, then most likely the coincidence is expected to occu, and hence we compute a confidence inteval fo ϱ in tems of the genealised hypegeometic function (special function) using the vaiance of the coincidence (o via coincidence). These esults ae also witten in tems of elementay functions using the asymptotic expansion of the hypegeometic function. The obtained esults can, fo example, be used in health cae, quality contol, and so on. Keywods: Coincidence, Hypothesis test, Confidence inteval, Hypegeometic function, asymptotic evaluation
244 Victo Nijimbee 1 Intoduction A discete andom vaiable X having a negative binomial (o Pascal) distibution with paametes and p is denoted as X NB(, p), and its pobability mass function (p.m.f) is P (X = s) = Γ(s + ) s!γ() ps (1 p), s = 0, 1, 2,, (1) whee s is the numbe of successes pio obseving numbe of failues, p is the pobability of successes in each expeiment [7, 9], and Γ is the standad gamma function, see fo example [2]. It is staightfowad to show that the mean of the vaiable X is given by µ = p/(1 p) while the vaiance is given by σ 2 = p/(1 p) 2 [7, 9]. Thee ae two impotant estimatos fo p. One is the minimum vaiance unbiased estimato (MVUE) [9] MVUE(p) = s 1 s + 1, (2) and anothe one, which is a biased estimato fo p, is the maximum likelihood estimato [9] MLE(p) = s s +. (3) The negative binomial distibution (NBD) has many applications in engineeing, health cae, science and social science [1, 3, 4, 5, 8, 12, 13]. Aging eseach in clinical epidemiology [5], thundestom pobability of occuence in meteoology [12], stock contol poblems [13] and accident statistics [3] ae good examples among othe applications of the NBD. Infeence by means of the NBD has to be caefully dawn using suitable statistical methods. Fo instance, is the estimated value of p acceptable to justify the use of the NBD? One should in pinciple pefom statistical tests, and constuct confidence intevals to adequately answe this question. Inteesting computations of confidence intevals fo the paamete p can be found in Khushid et al. [10]. Aban et al. [1] compaed the mean counts of two independent samples. In [1], they investigated the small sample popeties of the likelihood-based tests and compae thei pefomances to those of the t-test and of the Wilcoxon test. Hee, we athe investigate the case whee p is small (p 1) and the numbe of failues is lage ( ). This is a special case of ae events, and an appoximation with Poisson distibution can be applied. Fo this eason, Poisson distibution will be ou impotant tool in the pesent pape (section 2). Fo efeence, a discete andom vaiable X is Poisson distibuted with paamete λ if its pobability mass function (p.m.f) is [7] P (X = s) = e λ λ s, s = 0, 1, 2,. (4) s!
Hypothesis test and confidence inteval fo... 245 Poisson distibution is denoted as X P oi(λ) and has an inteesting popety that both the mean and the vaiance ae equal, and ae given by µ = σ 2 = λ. In this wok, we conside a moe geneal situation with n independent studies in which expeiments ae consideed to follow NBD s, and each with paametes i, p i, i = 1,, n. We ae inteested in establishing a goodness of fit test to evaluate the null hypothesis against the altenative hypothesis H 0 : p 1 = p 2 = = p n = ϱ (5) H 1 : p i ϱ fo some i, i = 1, 2,, n, (6) whee ϱ is some positive eal numbe. Thus, the fist goal of this wok is to establish a statistical test to examine these hypotheses, while the second goal is to constuct a confidence inteval (CI) fo the paamete ϱ if thee is no evidence to eject H 0. Fo small ϱ (ϱ 1), and fo lage ( ), the NBD can be appoximated by Poisson distibution. In that case, we can show (section 2) that the hypotheses in (4) and (5) ae espectively equivalent to the null hypothesis and the altenative hypothesis H 0 : λ 1 = λ 2 = = λ n = θ (7) H 1 : λ i θ fo some i, i = 1, 2,, n, (8) whee λ i, i = 1, 2,, n ae paametes of some Poisson distibutions and θ is some eal positive constant. Using this appoximation, we will then apply known esults fo Poisson distibution (see Nijimbee [10]) to obtain new esults fo the NBD. Fo Poisson distibution, Nijimbee [10] established a χ 2 goodness of fit test to examine the hypotheses in (7) and (8), and constucted a 100(1 α)% confidence inteval (CI) fo θ using the vaiance of the coincidence (o via coincidence) that we ae going to define late (section 3) in the context of the NBD. In this pape, a new goodness of fit test to examine the hypotheses in (5) and (6) is caied out, and hence a new 100(1 α)% CI fo ϱ is obtained using the vaiance of the coincidence as in [10]. 2 Appoximation by Poisson distibution In this section, we pove the appoximation of the binomial distibution by Poisson distibution. And we show that the hypotheses in (5) and (6) and those in (7) and (8) ae equivalent when p is small (p 1) and is lage ( ).
246 Victo Nijimbee Theoem 2.1 Fo p = λ/ 1 and whee λ is a positive constant, P (X = s) = Γ(s + ) s!γ() ps (1 p) e λ λ s, s = 0, 1, 2,, (9) s! and µ = σ 2 λ = p. Poof. Γ(s + ) s!γ() ps (1 p) = And if is lage,, then Γ(s + ) s!γ() ps (1 p) s s! ( + 1) ( + s 1) s! ( ) s ( λ 1 λ ), s = 0, 1, 2,. (10) ( ) s ( λ 1 λ ) = e λ λ s, s = 0, 1, 2,, (11) s! which is Poisson distibution pobability mass function since ( 1 λ ) e λ as. Moeove, since p = λ/ 1, then the mean of X is µ = p/(1 p) p = λ while the vaiance σ 2 = p/(1 p) 2 p = λ. Hence, unde this assumption, the andom vaiable X is Poisson distibuted with paamete λ = p, X P oi(λ = p). This ends the poof. Next, let us conside X i NB( i, p i ), i = 1, 2,. By Theoem 2.1, if i = fo all i, then the hypotheses in (7) and (8) become espectively H 0 : p 1 = p 2 = = p n = θ (12) and H 1 : p i θ fo some i, i = 1, 2,, n. (13) Hence, we have H 0 : p 1 = p 2 = = p n = θ = ϱ (14) and H 1 : p i θ = ϱ fo some i, i = 1, 2,, n, (15) which ae (5) and (6) espectively.
Hypothesis test and confidence inteval fo... 247 3 Coincidence, pobability and moments The pobability of the coincidence and the moments associated with the coincidence ae deived in tems of the hypegeometic function following Nijimbee [11] in this section. But we fist define the coincidence and the genealized hypegeometic function. Definition 3.1 Let X i NB( i, p i ), i = 1, 2,, n be independent and identically distibuted (iid). An event is said to be a coincidence, if afte counting, the numbe of successes is exactly the same in all n cases. Thus the the coincidence is the event given by C = {X 1 = X 2 = = X n = s}. (16) Definition 3.2. function [2] The genealized hypegeometic function is the special pf q (a 1, a 2,, a p ; b 1, b 2,, b q ; x) = (a 1 ) s (a 2 ) s (a p ) s x n (b 1 ) s (b 2 ) s (b q ) s n!, (17) whee a 1, a 2,, a p and b 1, b 2,, b q ae abitay constants, (ϑ) s = Γ(ϑ + s)/γ(ϑ) fo any complex ϑ, with (ϑ) 0 = 1, and Γ is the standad gamma function [2]. Theoem 3.3 Let i = fo all i = 1, 2,, n. If ϱ is small (ϱ 1) and is lage ( ), then unde H 0, P (C) e nϱ 1F n (1; 1, 1,, 1; (ϱ) n ). (18) Poof. The joint pmf of X 1 = s 1, X 2 = s 2,, X n = s n is P (X 1 = s 1, X 2 = s 2,, X n = s n ) = P (X 1 = s 1 ) P (X n = s n ) Γ(s 1 + 1 ) s 1!Γ( 1 ) ps 1 1 (1 p 1 ) 1 Γ(s n + n ) s n!γ( n ) psn 1 (1 p n ) n = n i=0 If i = fo all i = 1, 2, n, then unde H 0, ( ) P (C) = P {X 1 = X 2 = = X n = s} = lim k Γ(s i + i ) s i!γ( i ) ps i i (1 p i) i. (19) k [ Γ(s + ) s!γ() ] n ϱ ns (1 ϱ) n, (20)
248 Victo Nijimbee whee k = s + is the total numbe of expeiments. By Theoem 2.1 and using (5) and (14), we have P (C) = lim k k [ ] n Γ(s + ) ϱ ns (1 ϱ) n e nθ θ ns s!γ() (s!), (21) n whee θ = ϱ. Following Nijimbee [11] (Theoem 1), we obtain P (C) = lim k k Hence, substituting back θ = ϱ gives which is (18). This ends the poof. [ ] n Γ(s + ) ϱ ns (1 ϱ) n e nθ θ ns s!γ() (s!) n = e nθ 1F n (1; 1, 1,, 1; θ n ). (22) P (C) e nϱ 1F n (1; 1, 1,, 1; (ϱ) n ) (23) Theoem 3.4 Let i = fo all i = 1, 2,, n. If ϱ 1 and, then unde H 0, the γ th moment µ γ associated with the coincidence is given by µ γ e nϱ (ϱ) n 1F n (1; 1,, 1, 2,, 2; (ϱ) n ), γ = 1, 2, 3,. (24) And the vaiance is given by σ 2 C e nϱ (ϱ) n 1F n (1; 1, 1, 2,, 2; (ϱ) n ) e 2nϱ (ϱ) 2n [ 1 F n (1; 1, 2,, 2; (ϱ) n )] 2. (25) Poof. If i = fo all i = 1, 2,, n, by Theoem 2.1 and unde H 0, the γ th moment associated with the coincidence can be appoximated as µ γ = lim k k [ ] n Γ(s + ) s γ ϱ ns (1 ϱ) n e nθ s!γ() s γ θns (s!) n, (26) whee as befoe θ = ϱ. We can now apply Lemma 1 in [11] and obtain µ γ = lim k k Substituting back θ = ϱ gives [ ] n Γ(s + ) s γ ϱ ns (1 ϱ) n e nθ s γ θns s!γ() (s!) n = e nθ θ n 1F n (1; 1,, 1, 2,, 2; θ n ). (27) µ γ e nϱ (ϱ) n 1F n (1; 1,, 1, 2,, 2; (ϱ) n ), s = 1, 2, 3,, (28)
Hypothesis test and confidence inteval fo... 249 which is (24). To pove (31), we fist appoximate the fist (mean) and second ode moments associated with the coincidence using (24). Thus, the fist ode moment o mean is appoximated by µ 1 = µ γ=1 e nϱ (ϱ) n 1F n (1; 1, 2,, 2; (ϱ) n ), (29) while the second ode moment is appoximated by Hence the vaiance σ 2 C µ 2 = µ γ=2 e nϱ (ϱ) n 1F n (1; 1, 1, 2,, 2; (ϱ) n ). (30) is appoximated by σ 2 C µ 2 µ 2 1 = e nϱ (ϱ) n 1F n (1; 1, 1, 2,, 2; (ϱ) n ) e 2nϱ (ϱ) 2n [ 1 F n (1; 1, 2,, 2; (ϱ) n )] 2. (31) 4 Goodness of fit test and confidence inteval (CI) fo ϱ via coincidence A goodness of fit test to examine the hypotheses in (5) and (6) is established in this section, and 100(1 α)% confidence inteval fo ϱ is computed if the null hypothesis cannot be ejected. Theoem 4.1 If i = fo all i = 1, 2,, n, ϱ 1 and, and thee is no evidence to eject H 0 in (5), then whee P (C) is given by (23). Poof. P (H 0 := tue) P (C), (32) P (H 0 := tue) = P (p 1 = p 2 = = p n = ϱ) ( X1 1 P X 1 + 1 1 = X 2 1 X 2 + 2 1 = = X ) n 1 X n + n 1 = ϱ ( X1 1 = P X 1 + 1 = X 2 1 X 2 + 1 = = X ) n 1 X n + 1 = ϱ. (33) If, then P (H 0 := tue) = P (p 1 = p 2 = = p n = ϱ) ( X1 1 P = X 2 1 = = X ) n 1 = ϱ ( X1 = P = X 2 = = X n = ϱ + 1 = ϱ + 1 ) = P (X 1 = X 2 = = X n = ϱ + 1) = P (C), (34)
250 Victo Nijimbee povided that ϱ + 1 is an intege. This completes the poof. Theoem 4.1 means that we shall expect the coincidence to occu as we pefom moe and moe expeiments if the null hypothesis H 0 cannot be ejected. Moeove, if the null hypothesis H 0 in (5) is not ejected, then most likely the coincidence is expected to happen (Theoem 4.1), and hence the vaiance of X 1, X 2,, X n is that of the coincidence σc 2 given in Theoem 3.4. And if ϱ 10, then, by the Cental Limit Theoem [6], we have X 1, X 2,, X n N(ϱ, σ 2 C), (35) whee σ 2 C is given in Theoem 3.4. Let ˆp 1, ˆp 2,, ˆp n be the sample estimates (MVUE) fo p 1, p 2,, p n, see equation (2). One can also veify that unde H 0 n ˆp i i=1 ˆϱ = n = ˆp 1 + ˆp 2 + + ˆp n n is an unbiased estimato fo ϱ. Theefoe, we have o σ C / (36) N(0, 1), (37) e nϱ (ϱ) n 1F n(1;1,1,2,,2;(ϱ) n ) e 2nϱ (ϱ) 2n [ 1 F n(1;1,2,,2;(ϱ) n )] 2 N(0, 1). (38) Having in mind that Z N(0, 1), a test to examine the hypotheses in (5) and (6) can now be conducted. The null hypothesis H 0 will be ejected if and (o) e nϱ (ϱ) n 1F n(1;1,1,2,,2;(ϱ) n ) e 2nϱ (ϱ) 2n [ 1 F n(1;1,2,,2;(ϱ) n )] 2 < Z α 2 (39) e nϱ (ϱ) n 1F n(1;1,1,2,,2;(ϱ) n ) e 2nϱ (ϱ) 2n [ 1 F n(1;1,2,,2;(ϱ) n )] 2 > Z α 2 (40) We can now constuct a (1 α)% confidence inteval fo ϱ if thee is no evidence to eject H 0. It is actually given by ˆϱ Z α 2 ˆσ C < ϱ < ˆϱ + Z α 2 ˆσ C, (41)
Hypothesis test and confidence inteval fo... 251 whee ˆσ C = e nˆϱ (ˆϱ) n 1F n (1; 1, 1, 2,, 2; (ˆϱ) n ) e 2nˆϱ (ˆϱ) 2n [ 1 F n (1; 1, 2,, 2; (ˆϱ) n )] 2, (42) and ˆϱ is given by (36). Fo n = 2, ˆσ C can be expessed in tems of modified Bessel functions of ode 0 and 1, I 0 and I 1, using Coollay 1 in [11] as ˆσ C = whee befoe ˆϱ = (ˆp 1 + ˆp 2 )/2. e 2ˆϱ (ˆϱ) 2 I 0 (2ˆϱ) e 4ˆϱ (ˆϱ) 4 [I 1 (2ˆϱ)] 2. (43) 5 Asymptotic evaluation of the CI fo ϱ Fo ϱ 1 (ϱ 10), we can evaluate the vaiance σc 2 in tems of elementay functions athe than in tems of special functions [11]. This is called asymptotic evaluation. In the case n > 2, the asymptotic expessions fo σc 2 is given by equation (5.39) in Theoem 6 in [11]. Substituting θ = ϱ in (5.39) in [11], and then substituting the esulting expession fo σc 2 in (37), we obtain simple expessions fo (39) and (40). Theefoe, we should eject the null hypothesis H 0 if and (o) 2(2π) 1/2 n/2 n 1/2 (ϱ) 5/2 n/2 4n 1 (2π) 1 n (ϱ) 3 n < Z α 2 (44) 2(2π) 1/2 n/2 n 1/2 (ϱ) 5/2 n/2 4n 1 (2π) 1 n (ϱ) 3 n > Z α 2. (45) If by this test, H 0 cannot be ejected, then a 100(1 α)% CI can be constucted using (41), whee ˆσ C 2(2π) 1/2 n/2 n 1/2 (ˆϱ) 5/2 n/2 4n 1 (2π) 1 n (ˆϱ) 3 n, (46) and as befoe ˆϱ is given by (36). In the case n = 2, the asymptotic expessions fo σc 2 is given by equation (A.73) in Theoem 8 in [11]. Substituting θ = ϱ in (A.73) in [7], and then substituting the esulting expession fo σc 2 in (37), simple expessions fo (39) and (40) ae obtained. Hence, the null hypothesis H 0 shoud be ejected if 1 (ϱ) 3/2 ϱ 2(π) 1/2 4π < Z α 2 (47)
252 Victo Nijimbee and (o) 1 (ϱ) 3/2 ϱ 2(π) 1/2 4π > Z α 2, (48) whee as befoe ˆϱ = (ˆp 1 + ˆp 2 )/2. If thee is no evidence to eject the null hypothesis H 0, one may compute a 100(1 α)% CI fo ϱ using (41), and whee (ˆϱ) ˆσ C 3/2 ˆϱ 2(π) 1/2 4π. (49) 6 Discussion and conclusion We have defined the coincidence (Definition 3.1) in the context of the NBD and appoximated its pobability of occuence in tems of the hypegeometic function by means of Poisson distibution (Theoem 3.3). We have expessed the vaiance and moments associated with the coincidence in tems of the hypegeometic function (Theoem 3.4). We futhe showed that when the numbe of failue is lage the pobability that H 0 is tue equals that of the coincidence (Theoem 4.1). This means that if the null hypothesis is tue (cannot be ejected), then moe coincidences will occu as we keep pefoming the expeiment many times. In that case, the vaiance of X i, i = 1, 2, is given by the vaiance of the coincidence C. In this case, one may use the CLT to establish a statistical test as descibed in sections 4 and 5. Fo simplification pupose, asymptotic expansions of the hypegeometic functions wee used to expess the vaiance of the coincidence C in tems of elementay functions (section 5). The outcomes of this wok can, fo instance, be applied to achieve bette esults in health cae, quality contols, meteoology and so on. Refeences [1] I.B. Aban, G.R. Cutte, N. Mavinga, Infeences and powe analysis concening two negative binomial distibution with application to MRI Lesion counts data, Computational Statistics and Data Analysis, 53 (2008), 820-833. https://doi.og/10.1016/j.csda.2008.07.034 [2] M. Abamowitz, I.A. Stegun, Handbook of Mathematical Functions: with Fomulas, Gaphs and Mathematical Tables, National Bueau of Standads, 1964.
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