Estimation of Generalized / ' / \ Negative Binomial Distribution

Transcription

1 C H A P T E R - I I I Estimation of Generalized / ' / \ Negative Binomial Distribution

2 3.1: Introduction Fish e r(l 9 2 2, ) laid the foundation o f the theory o f estimation, although m uch o f the attention w a s directed to approxim ation m easures appropriate to large sam ple. In the present age o f com puter sophistication, com putational difficulty is no lo n ger a justification for se eking alternative and inefficient estimation procedures particularly w hen there are only tw o or three parameters to be estimated. Q u ite often there are m any com peting m ethods o f estimation and specific m ethod m ay be superior to other method. The G N B D is a tri-parameter discrete distribution, w h ich have to be estimated w hen the m odel is used to describe a p h ysical situation. M u c h w o rk has been done in the direction o f estim ation o f param eters o f G N B D. V a rio u s authors have suggested v a rio u s m ethods o f estim ations o f parameters. These m ethods have been critically exam in ed in this chapter. T h e cla ssica l technique o f estim ation by the m ethod o f m axim um likelihood requires a lot o f calculation in generalized distribution, as likelihood equations do not appear to be straight forw ard to be solved. A n alternate m ethod for estim ating generalized negative binom ial distribution ( G N B D ) has been presented w here the non-zero frequency o f a

3 variable is observed o n ly upon a finite value. The m ethod is found very sim ple and q u ic k to apply. T h e m ethod, requires the estim ation o f o n ly tw o parameter 0 and P and other is obtained by linear relationship am ong the parameters by counting the non-zero frequency classes. U s in g this m ethod o f estimation, G N B D has been fitted to som e observed distribution and fit have been found quite satisfactory, [H a ssa n et al. (2004)]. In addition, w e have presented a method o f estim ation for estim ating the parameters o f size biased G N B D and zero truncated G N B D by e xplo itin g the conditions given w ith its probability m ass function. 3.2: Method Of Moments T h e generalized negative binom ial distribution contains three parameter with conditions m > 0, p = 0, or p > 1, 0 < 0 < 1, 0 P < l. F o r estim ating these parameters b y the m ethod o f m om ents, three m om ents o f G N B D are required. Su p p o se a random sam ple o f size n is taken from G N B D m odel (1.6.1), and let fj be the observed frequencies corresponding to the value o f Xj for 1 = 0, 1,2,..., k where, x k is the largest observation in sample. Let Si, s 2 and s 3 denotes the m ean, variance and third central m om ent from sample. C o n su l and F a m o ye ( ) and Fam o ye (1 997) gave fo llo w in g three different m ethods for estim ating the parameters o f G N B D u sin g moments. i) Three moments methods U s in g relation (2.3.2) (3.2.1) (3.2.2) T h is gives 82

4 p = e -i (3.2.3) Rewriting relation (2.3.7) as: ( i - e p ) 2 11 ~ e P (2 ~ 9)] Substituting the values o f (1 0p) 2 from (3.2.2) vve get ^3 = 7 M-2 Hi (1-9 ) [3(i - e) - (2-9)(i - ep)] (3.2.4) Substituting the value o f (I - ep) = [^',(1-0) a; ]'/2in (3.2.4) and replacing population m om ents b y sam ple m oments. W e get after sim plification: (2 - e ) - i - e S 1S 3-3 s : s ls 2 = k (say) (2-0 )2 = k(l - 0) (k - 4)0 - (k - 4) = 0 (3.2.5) The tw o roots o f equation (3.2.5) are - (k - 4) ± '(k - 4 ) - 4 ( k - 4 ) = , 2, V - k Since the range o f 0 is between 0 and 1, the adm issible value o f 0 is thus (3.2.6) the relation (3.2.3) w h ic h w h e n substitute w ith value o f 0 from (3.2.6) gives estimate o f p, as A A P = (0) -i 83

5 then u sin g 0 and P in (3.2.1) one get the estimate o f m, as s,[g~' - p ] ii) First two moments and proportion o f zero (MOZE) Let P 0 be the probability o f zero class in G N B D, then P = ( l - 9 ) m = = f (3.2.7) l o g f o = m l o g ( l - 0 ) (3.2.8) F ro m relation (2.3.2), w e h ave p? = ( m 0 ) 3 ( l - 0 P ) 3 (3.2.9) Substituting the value o f (I - Gp) 3 from (2.3.4) in (3.2.9) on sim plification one gets 2 ( i e )u? m =, (3.2.10) 0 2p 2 Sq u a rin g (3.2.8) and replacing population m om ents w ith their corresponding sam ple m om ents, and equating w ith (3.2.10) w e get f,(0) = s 2(log f 0) 2s (1-0)[log(l - 0)]2 = 0 (3.2.11) S o lv in g (3.2.11) iteratively to obtain estim ators 0 based on M O Z E method, the initial value o f 0 is taken as m om ent estimate. From relation (3.2.10) the estim ates o f m is obtained as m = 0 -,[ ( l - 0)s?s-,J ' 2 (3.2.12) A ls o from (2.3.2), the estimate o f p is give n as Hi, First two moments and ratio o f first two frequencies (MORA) L e t Pj be the pro b ability o f the one class in G N B D (1.6.1). T he ratio o f one class to the ze ro class is g ive n b y

6 -!- = m e ( l- e )!-, Po Let fr be the corre sponding sam ples ratio, then w hich g ive s m 0( l - 0) M = f r (3.2.14) f m = ( ) 0(1-0 ) M u sin g relation (3.2.10), replacing population m om ents w ith sam ple moments, and equating w ith (3.2.14), and after rearranging the terms, one gets s 2f r2s ' 3 = ( l - 0) 2M ( ) After a p p ly in g log, w e get f 2 (9) = (2P - 1 ) log(l - 0) - log[s2f r2s r 3 (3.2.17) A l s o fro m relation (3.2.2) p = e _ /2 (3.2.18) u sin g this relation in (3.2.17) w e get f 2(0) = 2_2 0 0 s', (1 e )s2 J /2 lllo g ( l - 9) - log(s2f r2s, 3) so lv in g f2(0 ) iteratively, the estimate o f 0 based on M O R A m ethod is obtained. M O R A estim ator for m and (3 is obtained by u sin g relation (3.2.12) and (3.2.13). 3.3: MINIMUM CHI-SQUARE ESTIMATION Suppose that sample o f size n is divided into a finite (or countably infinite) sets o f groups. Katti and Gurland (1962) suggested the use o f t statistics S = (sj, S2,.. St) for estimating r parameters (j) = (<})i, <t>r) where t > r. T h e m in im u m chi-square ( M C ) estimators for the parameters <f> can be obtained b y m in im iz in g 85

7 Qmc - ( S ~ 1(S - A) (3.3.1 ) where Q E ( S E (s))(s E (s) is the variance covariance m atrix o f S and Q is consistent estim ator o f Q. A lso, S is a consistent estimator o f X = ( X h X 2,...,^t) where V s are functions o f the V parameters in < >. (0, p, m ) are F o r the G N B D m odel, the estim ating equations for the parameters ({) r &k' n~](s-xy = o (3.3.2) and generalized variance o f the M C estim ators o f < >is given by d<\> dp (3.3.3) E x p re ss S = (k(1), k (2)>k(3),lo g P 0 ), where k (j),l,2,3 are first three factorial cum ulants and lo g P 0 is the logarithm o f the zero frequency. Q = var(c) where var (c) is the variance - convenience m atrix C. In place o f u sin g the logarithm o f the zero frequency, one m ay use the ratio o f the first tw o frequencies, but usually m ethod o f u sin g the logarithm o f the zero frequency is m ore efficient in considerably w ide range o f the parameter space. F a m o ye ( ) com pared param eters o f G N B D in case o f finite sam ple size. F o r this com parison, p seudo - random sam ples o f size n = 200, n = 1000, and n = 10,000 w ere generated from G N B D. T he M O Z E, M O R A and M C m ethods w ere used to estimate the param eters 0, p and m. In order to be able to estimate the bias and the variance o f the estimators, 1000 sam ples were generated fro m each sa m p le size. T h e b ia s w a s com puted b y subtracting the 86

8 actual parameter value from estimated value. Several parameter values were considered and results are sim ila r to those in table It w as found that param eter estimator have large biases w hen the sample size is sm all, as sam ple size increases biases goes down. The value o f parameters 0 are generally overestimated, whereas, the values o f p and m are generally underestim ated. In term s o f bias only, the M O R A estim ators are best. TABLE Bias and Variance (in parenthesis) for Parameter 6 = 0.4, p = 1.2 and m = 2.0 e P m o o<n 1 c M O Z E.2199 (0.0083) (.0077) (.1113) M O R A.2112 (0.0172) (.0488) (.5536) M C.2276 (0.0100) (.0104) (.1266) n = 1000 M O Z E.1714 (0.0080) (.0107) (.1460) M O R A.1567 (0.0161) (.0506) (.6053) M C.1756 (0.0100) (.0129) (.1683) n =10,000 M O Z E.0368 (0.0051) (.0177) (.2039) M O R A.0238 (0.0084) (.0590) (.6675) M C.0341 (0.0056) (.0200) (.2356) In com p a rin g the M O Z E, M O R A and M C m ethods. T h e M O Z E estim ators have the sm allest variance. M O R A estimator has the sm allest bias especially w hen n = 1000, or greater. W h e n both properties o f bias and variance are considered, the M O Z E m ethod seem s to be the best. 3.4: Maximum Likelihood Method of Estimation (MLE) T h e generalized negative b in o m ia l distribution is a tri-param etric m odel w ith param eters m, 0, p. L e t a random sam ple o f size n be taken from the G N B D and let the observed frequencies be fx, x 0,1,2,..., k so that k x=0 ^ 87

9 where k is the largest o f the observed value h a vin g non-zero frequencies. The likelihood function is g ive n by k L = n x=0 m m + px rm + p x x 0 x ( l - 0) m+px" x (3.4.1) v * j - n x=0 f l ( x! ) f» k {m(m + p x - l)(m + P x - 2)- m + p x - (x - 1)0X (1 - Q ) m + P x ~ x )f» k x 1 m ne n x ( i_ e ) mn+n ( M ) x n n (m + px _ j)f«x=2 j=l f l ( x!)f> x=0 The log likelih o o d function is give n by log L = n log m + nx log 0 + [mn + n(p - l)x] log(l - 0) + Z f. x=2 x 1 l o g ( m + p x - j ) - l o g ( x! ) j=i T he m a x im u m likelih o o d estim ator o f 0, p and m are obtained by partially differentiating likelih o o d function w ith respect to 0, P and m and equating to zero. T h e likelih o o d equations are d. T n[x - 0(m + px)] _ 102 J-/ 30 0(1-0) = 0 (3.4.2) k x 1 x f v A l o g L = m x log(l - 0) + X Z n ~ ap x=2 j=l P j = 0 (3.4.3) d. _ n - f, lo g L = dm m (3.4.4) 88

10 T h e eq uation s (3.4.2), (3.4.3) and (3.4.4) are so lve d sim u ltan e o u sly to A A obtain 0, p and m, the M L estim ators o f parameter 0, p and m respectively. G upta ( a ) obtained m axim um likelihood estimates for m odified pow er series distribution (1.3.1) and results were applied to obtain m axim um likelihood estim ators for G N B D. Let X j, X 2,...,X n be random sam ple o f size n from M P S D and let L be the likelihood function, then w here is m ean o f M P S D and x is sam ple mean, the solution o f likelihood equation is then give n by x = *i(0) (3.4.5) so lv in g (3.4.5) for 6 one obtains 0 = ^ ( x ) (3.4.6) provided \x is invertible. T h e in ve rsion o f p. can be justified at least w hen g(0) is increasing because in that case > 0 as variance - ^ 7 % > 0 and hence n (0 ) d0 g (0) d0 is an increasing function o f 0. T h e generalized negative binom ial distribution given by (1.6.1) being a m em ber o f M P S D w ith g (0) - 0( l - 0) M,f(0) = ( l - 0) m and ^ (0) = ^ - ^ u sin g equation (3.4.5), w e get

11 F o r b in o m ia l distribution, p 0, and hence 0 = and for negative m binom ial, p 1 and hence 0 =. T he am ount o f inform ation 1(0) contained in sam ple is 1(0) = - E ( J? L de2 lo g L = n / g (0) (3.4.7) where n 2(0) = ^ is variance o f M P S D, and in case o f G N B D it is equal to m 0( l - 0 ) ( l - e p ) g (6) V ar(0) = [l(0)]_l = %(6) (3.4.8) d0 from (3.4.8), the variance o f 0 in case o f G N B D is V a r(0) = 0(1 0)(1 0P) n In particular for b in o m ial distribution Var(0) = ^ for negative n binom ial distribution, V ar(0) = 0(1-0 ) B ( l - 0) A n d bias o f 0 is obtained as b(0) = by u sin g the relation mn n b(0) = 1 g (0) 2n i2 g '( 0), g ( e ) g '( e ) - ( g '( 0))! M " 3 ( g 'O ))2 where ja3 is third the central m om ents o f G N B D given b y (2.3.7) Amount o f bias in ML estimator using negative moments T h e M L estim ator 0 o f 0 based on the random sam ple X h X 2,...,Xn from G N B D, (1.6.1) w a s g iv e n b y G upta (1 975a) as 0 = 1 m n m + px p p J 1 Y + m n/p_ (3.4.9) 90

12 where Y denotes the su m o f sam ple values and x is the sam ple mean. The random variable Y has G N B D given by P (Y = y) = mn m n + py mn + Py {9(1-9)m }v (1-0)' y = 0, 1, 2... and zero elsewhere, m > 0, 0 < 0 < 1 and O < p < 0-1. T h e above distribution o f random variable Y is G N B D with only difference that m is replaced by mn. K u m a r and C o n su l u sin g the negative m om ents found the exact am ount o f bias in the M L estimators o f parameters in G N B D. U s in g relation (2.5.4) and (2.5.5) after changin g m by m n for the random variable Y. S in c e k = -^ ~ given = p k - m n - k is negative, thus P u sin g relation (2.5.5) replacing m by mn, w e get Y + m n mn 00 P X i=0 f V mn p \ 0 i+1o - 0) mn/p mn + i +1 p A on substituting in (3.4.9), w e get expected value o f M L estimator 0 o f 0 in G N B D, g ive n as E 0 = mn P 00 ^ m n + i l Qi+in 0 i+1 (1 _ - m 0)' mn/p X p mn t 0 + i + 1 / P (3.4.10) obtained as Subtracting 0 from both sides o f (3.4.10) the bias in M L estimator is b(0) = m n / \ m n + i v p 0 i+i( i - 0) m n/p mn T + i + l (3.4.11) H ow ever, if k = is p o sitive integer, the relation (2.5.3), reduces to 91

13 y + m n = E 1 = I - p y ( -» ' f l - 0 "l i / i\k-l 1^ / 1 a\d r i - e A + ( - 1) l o g ( l - 0)p y + k k f ^ ( k - i ) CD CD (3.4.12) F ro m (3.4.9) and (3.4.12) the expected value o f M L estim ator 9 o f 0 in mn G N B D w hen ^ = k is positive integer is give n as r ^ 1-0 ^ E (e ) = k Z ^ r ( ^ ) ' + ( - l ) k k l o g ( l - 0) N o ( k - 0 e, (3.4.13) T hus, b ias in 0 is g ive n in closed form as k-1 ( / b(0) = (l-0 ) + k]t - i\i 1) M ( k ( l) k log(l - 0) v 0 j G upta ( a ) obtained the m a xim u m likelihood estimation o f decapitated G N B D th rough M P S D as F o r decapitated G N B D give n by (1.6.4) H(0) = (1 0p)[l (1 0)m ] (3.4.14) u sing relation (3.4.5). T h e M L E o f 0 is given by mo x ~ { \ - e p ) [ \ - ( \ - d ) m ] (3.4.15) F o r p = 0,this reduces for truncated binom ial distribution, to x = m 0 (3.4.16) 1 (1 0 )m w hich agree w ith Patil (1962b), and for truncated negative binom ial distribution, m 0 x = ( l- 0 ) [ l- ( l- 0 ) m] (3.4.17) 92

14 W h eneve r the o b serve d num ber o f zero s are m ore than that given by the model, it is recom m ended that the m odel should be m odified accordingly to account for the extra zeros. Gupta, G upta and T riparthi (1 995) obtained the m axim um likelihood estimation o f param eters o f the zero inflated M P S D. The distribution o f inflated m o d ifie d pow er series is given by (1.3.13) Letting X = (1 - a ) 1 - a(0) f (0), the m odel can be written as P ( X = 0) = 1-A. x = 1, 2, 3,... (3.4.18) Let X ], X 2... X n be a random sam ple from (3.4.18) and let nj be the 00 number o f observations su ch that n = ^ n j. T hen likelihood can be written as i=0 L = n [ p ( x = o ] n i=0 00 L = o - x ) n - n i=0 1 - a(x)[g(9)]! a(0) ' f (0) f (0) lo g L = n 0 log(l - X) + ^ n j log i=l 1-00 ^ a(i)[g(9)]1 a(0) f (0) f ( 6) this gives lo g L d X 00 _ n 0 i f - 1-X X 93

15 3, _ -f'(6)(n-n0) log L h ae 5 f (9) - a(0) oo X ini. i=l 8(0) estimator o f X and 0 are g ive n by X = n - n 0 n (3.4.19) 00 Y i n i and i - a(q) n - n, f (0) (3.4.20) where n(0) = g (0) f '( 0) f ( 0) - g '( 0) is mean o f m o d ifie d p o w e r series distribution. F o r generalized negative binom ial distribution g iv e n b y (1.6.1). u sin g relation (3.4.19) and (3.4.20), the mle o f 0 and a for G N B D is w mo (i - ep){\ - (i - o))m «-«(3.4.21) a = - ( i - 0 ) m n i - (i - 0) m (3.4.22) given by F o r b in o m ia l distribution, p 0, therefore m le s 0 and a o f 0 and a are 00 m 0 Z in i i=l 1 - ( 1 0) m n - n 0 a - n, n - ( 1-0 ) m l - ( l - 0 ) m 94

16 T he results o f G upta, G upta and Tripathi (1 995) were extended by M urat and Szynal ( ) to the discrete distributions inflated at any point s. T he p robability m a ss function o f inflated m odified pow er series distribution at point s is g ive n by P(X = x) = a + (1 - a) a (x)(g(0)) f (0) a (x)(g(0))x (1 -c t ) f ( 0) X = s X * s x e N (3.4.23) where 0 < a < 1. N is set o f positive integers. Proceeding on same lines as Gupta, G upta and T rip a th i (1995), they obtained m axim u m likelihood estimator o f a and 0 as n - n, a = n (3.4.24) d Me)fi e)z,. a Le(e)i l, _! _ g in f ( 0) - a ( s ) [ g ( ) ] S n - n s i= li«(3.4.25) where m (0) = m! - f. i s m e a n o f M P S D g '( 0) - f ( 0) 0 can be obtained from (3.4.25) by iterative num erical techniques, thus using g (0 ), f{6 ) and a ( x ), w e have fo llo w in g m le o f 0 and a s!m0r(m + sp - s +1) - smp(m + ps)(l - 0p)0 (1-0) (p-l)s+m 00 1 (1-0P) s!t(m + sp - s +1) - mt(m + Ps)0s (1-0) (P-l)s+m X i n i n - n s i = li«a A a = «5 i r ( m + s/3 - s + 1 ) - nm{m + fls)6s (1-6) $5,!r(/w + sj3 - s + 1) - mt{m + fis)os (1-6) In particular w h e n p = 1, w e obtain the m le o f 0 and a for negative binom ial distribution as 95

17 m9 - (1-0) / \ 1 - m + s-1 v m+l As (1-0) 0 m + s-1 \ 0 s (1-0 ) m s 00 Z n - ru.,. s 1= 11* S i n ' a = n, ^m + s - 1^ - n "m + s-t ^ s > < s > fm + s - 0 A A n 1-0(1-0)m I s J 0 s (1-0) m 3.5: Maximum Likelihood Estimation of GNBD Using Second Order Derivatives of Likelihood Function A m o n g v a rio u s m ethods o f estimation, the m ethod o f m axim um likelihood is su p e rio r to m a n y others, but som etim es it becom es incredibly difficult to so lve M L equations on account o f intricacies involved, this is true in case o f G N B D also. H a ssa n ( ) b y m a k in g use o f second order partial derivatives o f lik e lih o o d function obtained m le o f parameters o f G N B D Let a random sam ple o f size n be taken from the G N B D and let the k observed frequencies be fx, x = 0,1,2,... k so that ] T f x = n where k is the largest o f the observe d valu e s h a v in g non-zero frequencies. T he likelihood function o f the G N B D can be written as x=0 k x i m (n-fo)0 nx (1 _ 0 ) mn+n(p-l)x (m + p x _ j) x=2 j=l i W * x=0 T h u s, the lo g lik e lih o o d function can be written as log L = (n - f 0) log m + nx log 0 + n[m + (P - l)x]log(l - 0) k x-l + E Z fx log(m + px - j) - log M x=2j=l (3.5.1) 96

18 k where M = J ^ ( x! ) f* x=0 The three likelihood equations are given as, d lo g L _ nx n[m + (p - l)x] 50 ~ 0 (1-0) (3.5.2) S L o g L _ m ^ ^ x f v,3.5.3) ,3.5.4, 5m m x=2h m + Px - J F ro m equation (3.5.2), w e get 6 = * (3.5.5) m + px S o lv in g equation (3.5.3) and (3.5.4), w e have n v k X_1 f -= IE; (3-5-6) m x = 2 j= l ( m + P x - j ) These equations do not seem to be directly solvable and hence some method o f iteration has to be applied. (3.5.4), as T h e second derivatives o f log L can be obtained from (3.5.2), (3.5.3) and d 2 lo g L nx n[m + ( P - l) x ] (3 5 7) dq2 ~ 0 2 (1 0 )2 d2u>gl _ y y ( ) ap2 5^ 2j=i (m + p x - j ) 2 d 2L o g L _ ( n - f 0 ) y y! x (3.5.9) 5m2 m 2 x=2j=i (m + p x - j) 2 97

19 d 2LogL aeap nx (1-0) (3.5.10) a 2L o g L 505m n (1-0) (3.5.11) a 2L o g L _ x f 3 P 3 m X = 2 >? ( m + p x - i ) : (3.5.12) T h e valu e s o f these second order partial derivatives can be put in the fo llo w in g equations in the m atrix form. d2 lo g L d2 lo g L d 2 lo g L dq- aeap aoam a 2 logl a2 logl a2 logl a0ap ap2 apam a2 logl a2 logl a2 logl a0am apam ^m2 0o,Po>m o 0-0 o P-Po rh - m o a lo g L a a logl a logl am 0O Po>mO (3.5.13) A A where 0, p and m are the M L estimators o f 0, P and m respectively and 0 O, po and mo are the trial values o f the parameters. For trial values, the moment estimators can be used or it can be obtained by equating the first three observed relative frequencies to the corresponding theoretical probabilities. This system o f three equations may be used repeatedly till very close estimates o f 0, p and m are obtained. 3.6: B ayesian Estim ation o f GNBD T h e B a y e s estim ators o f a num ber o f param etric functions o f 0 for G N B D w ith k n o w n m and p w a s obtained b y Isla m and C o n su l (1986). Since 0 < 0 < 1, the p rio r inform ation on 0 were sum m arized b y beta distribution B(a,b), w here the param eter a and b are know n, under the squared loss function. I f X u X 2,..., X n be a random sam ple o f size n draw n from G N B D then likelihood function o f X i, X 2,..., X n g ive n 0 is o f the form. 98

20 l(x e)= k enx (i - 9)mn+nnx-nx where K = II m m + Px =, m + (3x i V x (3.6.1) The sum m ation b e ing taken overall x = (x,...xn) such that x, = nx. t= l Sin ce prior inform ation on 0 is give n by beta distribution with probability density function X (0;a,b) = 0 a- ( l - 0) b-> B (a,b) a,b > 0, 0 < 9 < 1 (3.6.2) The posterior distribution o f 0 becom es Qnx+a-l _ 0^mn+b+(P-l)nx-l p(g x) = (3.6.3) B (nx + a, mn + b + (p - l)nx) and B a y e s estim ator o f any function <f)(0) w ith respect to squared error less function is [ 1 <f>(0)0n* + a ~ l (1-0 ) m n + b + ( P - l) n x - l ^(6) = / _ = r B (nx + a, mn + b + (P - l)nx) Bayes estimators of some parametric functions in GNBD * ( 0) m e '( i - 9 ) k B[nx + a + i, (P - l)nx + mn + b + k] I and k non-negative B [nx + a, (p - l)nx + mn + b] integers {0( l - 0)M }k B[nx + k + a, mn + b + (P - l)(nx + k)] B [nx + a, mn + b + (p - l)nx] 0' (nx + a), (pnx + m n + a + b), P(X = k) m fm + pk^ Bfk + nx + a, (p - l)(k + nx) + m (n +1) + b] m + p k ^ k J B[nx + a, ( P - l) n x + mn + b] W h e re x<n) denotes the risin g factorial x (x + l) ( x + 2)...(x + m - 1 ) 99

21 T h e y estim ated the B a y e s relative frequencies by u sin g the estimator o f P ( X - k ).A w id e range o f values from 1 to 100 were considered for a and b and values o f P ( X = k) w ere com puted for k = 0, , and for the follow ing five broad categories: i) a = 1 and b = 1, 10, 20, 30, 40, 50, 100 ii) b = 1 and a = 10, 20, 30, 40, 50, 100 iii) In cre a sin g values o f a and decreasing values o f b. iv) D e c re a sin g values o f a but increasing values o f b. v) E q u a l values o f a and b but increasing from 1 to 50. In each case the values o f estimated B a y e s frequencies were com pared with the sim ulated values. It w as discovered that the estimated B a y e s frequencies w ere quite close to the sim ulated sam ple frequencies, w hen a and b were equal and that the variation in the B a y e s frequencies w as very little as the equal values o f a and b w ere increased. It w as found that % - values between the sim ulated sam ple frequencies and the estimated B a y e s frequencies were the least w hen a = b = Bayesian Estimation in a Decapitated Generalized Negative Binomial Distribution T h e D ecapitated form o f G N B D has been extensively studied over the past 25 years. B a sa l and G an ji ( ) derived B a y e s estimate o f zero truncated G N B D. T h e p ro b ability function o f zero truncated G N B D is given by (1.6.4) as P (x ;m,0,p )= m m -f px ''m + px^ 0 X(1- O ) m + p x -x 1 (1 0) x = 1,2,... (3.7.1) o < e < i Su p p o se 0 is a random variable and m and p are fixed but u nkn ow n constants, sin ce 0 < 0 < 1, p rio r inform ation on 0 m ay be give n by a beta distribution B (a, b) w ith k n o w n hyper-param eters a > 0, and b >

22 0 a-i n g(0; a, b) = < 0 < 1, a, b > B (a,b) I f Xi, x 2,... x n be a random sam ple o f size n draw n from a decapitated G N B D. T h e lik e lih o o d function x, x 2... x n< give n 0, is o f the form L ( x i 0) = n = m + pxj v m + px j x i d - 0) m + P x.-x i i. - { 1- e r f (3.7.2) = C 0 s ( l - 0 ) mn+ps s l - j l - 0 } " ] " m + pxj and s = X x i i=l The posterior distribution o f 0 becom es 7l(0 x) = Qa+s-l (1 _ e )mn+(p-!)s+b-! 1_ }] _ 0 }m -n W (b,s,n ) (3.7.3) where W (b, s,n) = J0 a+s_1 (1-0) 0 m n+(p-l)s+b-l d0 oo ^ - / n + k - 1 k=ov n ' k - l ' = ^T B(a + s,m k + ( p - l) s + b) k=n vn - l, = r ( a + s)d (b,s,n ) where 00 D ( b, s, n ) = ^ k=nvn r(mk + (P - l)s + b) r ( m k + ps + a + b) 7t(0 X ) = 0 3+ s-l ^ _ Q-)mn+(P-l)s+b-l [l _ (j e } m " r ( a + s)d (b,s,n ) 101

23 and Bayes estimate o f 0 with respect to squared error loss function is 0(b,s, n) = Jo 7i (01 x)d0 0 1 Je a+s (i - e )mn+<p-a>s+b- ] [1 _ {! _ e jm ]~n de J _ 0 r ( a + s)d (b,s,n) J0 a+s (1-0 )mn+(p-ds+b-l k=n T(a + s)d(b,s, n) 1V _ Q^m(-n+k) de CO f i r _ i \ 1 Z k=nl n 1 L Mea*s( i - 0)mk* F (a + s)d (b,s,n ) K P -l)s + b-l de o c - i ^ E l. B(a,s + 1,m k + ( p - l ) s + b) k=n vn - l/ T(a + s)d (b,s,n ) = r (a + s + l)r(m k + (P - l)s + b) k=n r (m k + ps + a + b + 1)* T(a + s) * D(b,s, n) k=n\ (a + s)r(m k + (p - l)s + b) r (m k + ps + a + b + l)d(b, s, n) 0 = [(a + s)d (b P,s + l,n)] D (b,s,n ) 3.8 M in im u m V a r ia n c e U n b ia s e d E stim a tio n ( M V T J E ) G upta ( b ) gave the necessary and sufficient conditions for the existence o f m in im u m variance unbiased estimate ( M V U E ) o f the parameters based on sufficient statistics and obtained m in im u m variance unbiased estimates o f param eters o f a M P S D (1.3.1). Theorem : su p p o se Xi, X 2,.*->x n is a random sam ple M P S D. T hen there exists essentially a unique unbiased estim ator o f ^(0) w ith m in im u m variance if and o n ly i f ^(0)[f (0)]n is analytic at the o rig in and has an expansion o f the form 102

24 (O)[f(0)]" = C (z,n )(g (0 )); n W here z ^ X j is a com plete and sufficient for 0 and its probability i=l distribution is giv e n b y P (Z = z) = b(z,n) [g(0)p [f(0)]n T h u s m in im u m variance unbiased estimate o f ^(0) is given by C (z,n ) vj/(z,n) = b(z,n) b(z,n) * 0 (3.8.1 ) The inform ation inequality for the variance o f an unbiased estimator, (0), o f ^(0) based on random sam ple x,, x 2,...xn from M P S D is given by Var M ) > j m U (6 ) ( f ( 6 ))2 ( r ( 6 ))2 \ 2 g '( 0) dja v a r(x ) n g(0) ' d0 (3.8.2) where i(0) is m ean o f M P S D. In order to obtain the M V U E o f ^(0) = 0 r( r > l ) for G N B D Gupta (1977b) em ployed a w e ll k n o w n L a gra n g ia n form ula 00 A ( 6) = A (0 ) + - ZTt z! z=l )Z 1 -(h (e))za'(0) e=o z-1 0 h(0) (3.8.4) to expand A (0 ) = 0 r( l - 0 ) w ith h(0) = ( l - 0 ) -p+i u y er(i-e) = X 1 dz-1 Z=r ^ z! 3 0 z-1 A ( e '( i _ 0)-ra' ) 30 (g(0))! e=o r y i i! l l _ f e r- i ( i - e ) - P z+z- mn] (g (9))2 ^ z! 50 0=0 rrz!3 I 0 z-1 -pz+ z-m n-1 0 r ( i e) 0=0 (g(e))5 103

25 ^ i a z_ r 2 / I A D 7-1 Z=rz! ^ r-1 'r-f* ( - l )1(l - 0)l-Pz+z-mn (g(b))2 zt=0 0=0 ^ i az~l + m n ^ z = rz! ae z-l oo = S Cr(Z nxg(0)): Z=r t=0 w 0=0 (3.8.5) where r r_1 C r(z,n) = Y z! ^ iv (Pz + m n -t-z )! v ~ v r:» z > r (Pz + m n - t - z)! = 0, z < r Repeating the procedure em ployed to obtain C r(z, n) with r = 0, we obtain: b(z,n) = m n(pz + z n - l ) z!(pz + m n - z )! (3.8.6) and M V U E for 0 r is g ive n by b(z,n) for r = 1 _.. 1 (Pz + m n - z )! /n C,(z,n ) = - ^ ( P z + m n z - z ) o f z > l z! (pz + m n - z )! = 0 o f z < 1 T h u s M V U E o f 0 for G N B D is give n by C,(z,n ) (Pz + m n z - z ) b(z, n) m n(pz + m n - 1) (3.8.7) T h u s for P = 1, (3.8.7), g iv e s the M V U E o f parameter for negative binom ial distribution as and G uttm an (1957). z + m n - 1. T h is coincides w ith R o y and M itra (1957) 104

26 3.9: M in im u m V a r ia n c e U n b ia se d E stim a tio n fo r Left T ru n c a te d G N B D in te rm s o f M o d if ie d C - N u m b e r s Jam ( ) m ade a detailed study o f the M V U estimation in left truncated G N B D and M V U estim ators were expressed in terms o f new defined m odified C -n u m b e r. T h e p robability function o f the left truncated G N B D w ith truncation point r(r > 0) is g iv e n by P(X = x) = P ( x : r, 0) m(m + px-l)!{o(l - Q ) ^ 1 r-l x!(m + px - x)!(l- 0)~m- X(-l)k' - ( m Pk + k~'}'fefl- O)^1}X k=o (-m -p x )lk! x = r, r + 1,... (3.9.1) m (m + P x - l)! ( g ( 0 ) ) x x!(m + px -x )!H j(0,-m, r,p ) (3.9.2) where H,(0-m,r,p) = (1-0)-m - ( - l ) k-' m(-m -Pk + k - l)! ( l- 0 ) p ') (3 9 3) (-m-pk)lk! and g(0) = 0 ( l - 0 ) M. (3.9.4) Let Xi, x 2, x n be a random sam ple o f size n from (3.9.2), w hen p and r are k n o w n the distribution o f Z = is give n by 1=1 n P (Z = 2) = d (z n r P )(g (0 ))- z = m, m ( H ^ O - m ^ P ) ) " and w hen r is u n kn o w n, the joint distribution o f (Y, Z ) where Y = m in {x i,x 2...,xn} is g iv e n in Jani (1 977) as = d (Z, n y, P ) - d ( z, n, y + li P ) ( )z (3.9.6) (Hi(0,-m,r,P)) z = n y + n y + l..., y = r, r + l,

27 W here H, (0, -m, r, (5) and g(0) are as given in (3.9.3) and (3.9.4), and d(z, n, r, P) is the coefficient o f (g(0 ))' in expansion o f (H,(0,-m,r,P ))n = ]d (z,n,r,p X g (0))z ( ) z=r n C haralam b ides (1974a) obtained the probability function o f ( Y Z ) as P(y, z; r,0) = C(z>n>y) ~ C(z,n, m, y +1 )0z (1-6)Pz~z hn(0,m,r)z! for y = r, r + 1,..., z = m, m with z > ny, 0 < 0 < 1, 0 p < 1. W here C (z, n, m, r) = V f l m rcm + px,) m! r ( Xi + l)r(m + pxj - x; + 1) C haralam b ides utilized this distribution to obtain M V U estimator o f 0 s, s > 1, w here the truncation point r is assum ed to be know n, and o f rs and 0s w hen r is unknow n. In case r is k n o w n, the M V U estimator 0s is obtained as e s (y) = Zf,r ( Pk Z l )s(y ) L -. C(_z- k,n,m,r) ^ ^ m + s r((p k - k )k -s)k! C(z,n,m,r) In case r is u n k n o w n the M V U estim ators o f rs and 0 s (s > 1) are given respectively by P / =s - S _ C (z,n,m,y + 1) y C(z, n, m, y) - C(z, n, m, y +1) k= ovk, yk (3.9.8) and 0 s fv r (Pk ~ s^ y ^ C ( z - k, n, m, y ) - C ( z - k, n, m, y + l) r ( p k - k ) k - s )! k C ( z, n, m, y ) - C (z,n,m,y + 1) 3.10: Confidence Interval Estimation Fam o ye and C o n su l ( ) considered the case o f both sm all and large sam ples and obtained expression for setting 100( l - o t ) % confidence interval for parameter 0 in case o f G N B D u sin g the general case discussed for M P S D. 106

28 Let a random xj, x 2,..., xn of size n be taken from the MPSD. Its likelihood function is g ive n by (3.10.1) n T h e statistic Y ^ X j com plete and sufficient statistic for parameter and i=l its probability function is g ive n by Py(0) = b(y){g(e)}y {f(0 )}"n where b(y) = ^ a ( x, ) a ( x 2) - a ( x n) Sin ce Y is sufficient statistic for the parameter 0 in (1.3.1), one can equate the su m o f the probabilities o f the distribution o f Y on each side with a/2, thus g iv e s 00 x=y a 2 (3.10.2) Y Xb(x){g(0i)}x{f(ei)}- x=0 a 2 (3.10.3) T h e distribution o f sam ple sum in case o f G N B D is given by using (3.10.2) and (3.10.3), w e get (3.10.4) and (3.10.5) 107

29 F o r any g iv e n set o f m, n, p and Y, (3.10.4) and (3.10.5) can be solved by u sin g N e w to n -R a p h so n iteration method, w hich w ill provide 100(1 - a ) % two sided confidence interval for 0 as (0 ^,0 U). I f a/2 is replaced w ith a in (3.10.4) and (3.10.5) then their solution provide one sided low er ( l - a ) % confidence bound 0, and one sided upper 100( l - a ) % confidence bound 0 u. n F o r large sam ples the sam ple sum Y = ^Tx, is a m odified pow er series w ith m ean n\x and variance n a. T he quantity i=i W = Y -n jj. crvn (x- i)vn a has a lim itin g distribution i.e. norm al w ith mean zero and variance 1. A c c o rd in g ly tw o sided ( l - a ) % confidence interval for 0 is 1- a = P x - Z ng'(6) d 0 \ i /2 g(6) f'(6) - 7 < < x + Z g'(b) f(b) ( g(0) d \i) 1/2" V n g*(6) de J where Z a is the critical value from norm al table. T h e upper bound 0 U and the 2 low er bound 0, are the solutions o f the equation x _ j w. m ± Z a g (8) f(0) f g(0) d n ' 1'2 ng'(6) de = 0 (3.10.6) w here g '( 0) d0 is variance o f M P S D. Since sam ple variance S 2 is given by S 2 = ~ r S ( x i _ x ) 2 n - 1 : A c c o rd in g ly, the tw o sided ( l - a ) % confidence interval can also be obtained fro m 1 - a = P _ z a < ( ^ < z 0 2 J 108

30 = p x - Z f ^S -7 S < i < x + Z a (3.10.7) me O n substituting the value o f u = (i - ep) bounds for param eter 0 o f G N B D are obtained as, the two sided 100(l-a)% confidence x - Z m + p / x - Z \ x + Z (3.10.8) \ m + P v x + Z 3.11: A S im p le a n d Q u ic k E s t im a t o r T h o u g h am o n g va rio u s m ethods o f estimation, the m ethod o f m axim um likelihood is the m ost preferred one. Som etim es the likelihood equations are very com plicated and its explicit algebraic solution is not available. It is to be solved n um e rically in that, by the m ethod o f iteration. In this section w e present a sim ple m ethod for estim ating the parameters o f generalized negative binom ial distribution. T h e m ethod is found very sim ple and quick to apply. In this method, the estim ation o f o n ly tw o parameters 0 and P is needed and o f the other is obtained b y the linear relationship am ong the parameters by counting the num ber o f non-zero frequency classes. T o exam ine the unbiasedness and efficiency o f the estimates o f parameters b y u sin g this method, an intensive M o n te C a rlo technique has been applied. U s in g this m ethod o f estimation, G N B D has been fitted to som e observed distributions and the fits have been foun d quite satisfactory. 109

31 M o s t often in practice the observed distributions are obtained only up to a finite value o f a variable i.e. up to finite num ber o f classes h avin g non-zero frequencies. Let us consider that 4t is the highest observed value h avin g nonzero frequency i.e. (t+1) is the low est value h avin g zero frequency. F ro m one o f the conditions o f the p m f o f the G N B D, w e have m + pt - 1 < 0 for x > t considering the equality, w e get m + pt - 1 = 0 (3.11.1) w hich give s m = t(l - P) (3.11.2) o b viously here P < 1 as m > 0. C o n su l and G u p ta s (1980) condition that G N B D is a true probability distribution for p=0, and for all values o f p in 1< p< 6~x is o b vio u sly for the range o f x as 0(1 )oo. In case the range o f G N B D is 0(1) t, the value o f p < 1. It can be seen in Jain and C o n su l (1971) and C o n su l and Fam oye (1995) that the estimates o f p b y the m ethod o f m om ents and ML m ethod are com in g to be less than one. Substituting the value o f m from (3.11.2) in the m ean defined in (2.3.2) o f G N B D, w e get _ t(l-p)0 ^ (i-ep) to top i- e p l- e p L e t 1-0 p = a (3.11.3) w hich g ive s, t0 t( l- a ) ^ 1 = (3.11.4) 110

32 Using (2.3.2) and (2.3.4) of GNBD, we define / 2 _ 0-0 ) (1-0 ) A (1 -e/3)2 a2 Substituting the value from (3.11.4) in (3.11.5), we get (3.11.5) t o ( 1 - <? ) ( > - / /, ') 2 t2(\-e)2 - M l ) 2 t 2( \ - 0 ) 2 After a little sim plification, w e get g {t-y [)2y[ A.. (3.11.6) Substituting the value o f 6 from (3.11.6) in (3.11.4), w e get Mi R e p la c in g the population m om ents i', and ji2 by the respective sample m om ents x and s 2, w e get the estimate o f 0 as and estimate o f a as T h u s estimate o f P is obtained by u sin g relation (3.11.3) (3.11.9) Substituting the value o f p from (3.11.9) in (3.11.2), the estimate o f m is obtained as «= /( l-p) Bias and Efficiency o f suggested estimators using Monte Carlo Technique F o r evaluating the am ount o f bias and the efficiency o f the estimators o f m, 0 and p, so obtained, an intensive M o n te C a rlo T echnique has been applied as the m athem atical e xpressio ns for their m eans and variances are not know n. 111

33 Fifty random sam ples each o f size 100 have been drawn from the G N B D on com puter taking different com binations o f values o f the parameters, m, 0 and p. W h ile studying the efficiency o f the B ayesian estimators o f the G N B D, u sin g M o n te C a rlo technique, Isla m and C o n su l (1 986) selected the values o f m, 0 and p as 7, 0.3 and 2 respectively. Selecting the same com binations o f values o f m and 0 as 7 and 0.3, w e selected various different values o f p as 0.1(0.1)0.9. A fte r generating such sam ples, the estimates o f the parameters were obtained u sin g the suggested m ethod for each o f the sam ples for each com bination o f the values o f the parameters. The m ean and variance o f each o f the estimates o f each o f the parameters were computed, w hich indicates about the biased ness and efficiency o f the parameters. T he estimates o f m, 0 and p for n=7, 0 = 0.3 and p = 0.1(0.1)0.9 o f one sam ple and m ean and variance o f 50 sam ples for each com bination o f values o f parameters have been sh o w n in fo llo w in g tables. T a b le : T h e estim ates o f m, 0 a n d p fo r m =7, 0 = 0.3 a n d p = 0.1(0.1)0.9 p t E stim a te s o f m E stim a te s o f 0 E stim a te s o f P 112

34 Table: M e a n a n d v a ria n c e o f the estim ates o f m =7, 0 = 0.3 a n d p=0.1(0.1)0.9 p E stim a te s o f E stim a te s o f E stim a te s of M e a n V a r ia n c e M e a n V a ria n c e M e a n V a ria n c e The estimates w ere found alm ost unbiased and quite efficient. 3.12: G o o d n e ss o f F it A n attempt has been m ade to fit G N B D to num ber o f actual data sets, observed by other researchers, and for w h ich they had either used other m odels or the G N B D. D a ta set I is a sports data used by S in h a (1 984) for fitting the negative b in o m ia l distribution, and gives observed num ber o f w ickets taken by Sobers in 158 com plete in n in g s in test cricket. T he data set I I contains the runs scored b y V ish w a n a th in 142 com plete innings. T h e expected frequencies obtained b y the m ethod o f m om ents for G N B D have also been give n so that quick com p ariso n can be made. 113

35 Table Wickets Taken By Sobers in 158 Complete Innings in Test Cricket Data set I Wickets Observed Expected Frequency taken Frequency Methods of Moments Suggested method Total Mean j x d.f 1 2 a a p a m Table Runs Scored By Vishwanath in 142 Complete Innings Data set II Runs Observed Expected Frequency (unit of 30) Frequency Methods of Moments Suggested method J J Total 142 Mean X/V 2 d.f a 0 A s r A m \ 8.13 ~l

36 By a comparison o f two sets of expected frequencies with observed ones and a lso b y their x, va lue s, it is evident that the frequencies com puted by the suggested m ethod for G N B D are m ore or less close r to the observed ones, though the value o f % are sligh tly higher in case o f suggested method than in case o f m om ent method. H ow e ver, these differences do not seem to be m uch significant. Further, the suggested method has one definite advantage over other m ethods in certain situations. It can be applied w hen the method o f m om ents fail to g ive estim ates o f the parameter. M ore over, it is relatively very quick and so it m ay be preferred to others. 3.13: Estimation of Parameters of Size-Biased and Zero Truncated Generalized Negative Binomial Distribution T h e param eters o f size biased generalized negative binom ial distribution ( S B G N B D ) and zero truncated generalized negative binom ial distribution ( Z T G N B D ) are bit difficult to be estimated and so they are not m uch in com m on practice. In this section a sim ple and quick m ethod o f estim ating the parameters o f S B G N B D and Z T G N B D has been suggested. L e t us consid er that t* is the highest observed value h aving non-zero frequency i.e. (t+1) is the low est value h a v in g zero frequency. F ro m one o f the conditions o f the S B G N B D m odel, w e m ay have m + pt - t < 0 for x > t considering the equality, w e get this give s m + t(p - 1) = 0 (3.13.1) m = t (1 - p) (3.13.2) F ro m (2.12.4) and (2.12.6), putting 1-0 p = a, w e get (3.13.3) F ro m (3.13.2) and (3.13.3), w e have 115

37 from (3.13.4), we have (2.12.4) as x*' _ t(0--l + a) 1-6 t(q-\ + a)a + (\-Q) M, _ a ^ = ^ ~ (3-13.5) T h is g iv e s 1-0 = (M; - t)a: T ^ T (3.13.6) J D, ( M ;-t) a 2 m T (3-13-7) Substituting the value o f m and ( 1-0 ) from (3.13.2) and (3.13.6) and a = 1-0 p in ( ), w e get ( M ', - t ) M o = ( l - t a ) 2a ta -tm Ja +M; - ta - Mja + ta 2 + M',a-1 T h is g ive s M 2t2a 3 - [2tM2 - (m; - 1)2 - tm; (m; - t)ja2-2(m; - t)m;«+ (m; - 1) = o Put t - M J = R (S a y ) f(a) = M2t 2a 3 - (2tM2 - R 2 - tm',r)a2+ 2RMJa- R = 0 (3.13.8) R e p la c in g the population m om ents M', and M 2 by respective sam ple m om ents x and s 2 w e get f(a) = s2t2a 3 -(2ts2- r2- r tx ) a 2+ 2 rx a -r = 0 (3.13.9) where r = t - x Sin ce equation (3.13.9) is not directly solvable for a, som e iteration m ethod such as the N e w to n -R a p h so n m ethod can be used for estim ating a. Putting this va lu e o f a in (3.13.7) the estimate o f 0 can be obtained, substituting the estim ate o f a and 0 in (3.13.3) the estimate o f p can be obtained and fin ally from (3.13.2) an estimate o f m can be obtained.

38 Estimation o f Parameters o f Zero-truncated GNBD T he probability m ass function o f zero-truncated generalized negative binom ial distribution ( Z T G N B D ) is given by P3[X = x] = m m + px m + px v x j 0 X ( i _ 0 ) m+px-x p ^ _ e r p x = 1,2,3,... ( ) 0 < 0 < 1, 0 p < 1, m > 0 = 0, for x > t such that m + pt < t T he Z T G N B D probability m ass function as quoted in ( ) clearly assumes the value zero as m entioned in the conditions for all values o f x = t resulting into m + pt < t. T a k in g the highest observed value h aving non-zero frequency as n and c o n sid e rin g m + pn = n, w e obtain m in terms o f n and p as m = n (1 - P) ( ) Since m > 0, n > 0, the value o f p w ill alw ays lie between 0 and 1. T he first and second m om ents about origin o f Z T G N B D is given respectively b y t ; = m 0 ( i - e p ) ( i - ( i - e r ] ( ) and v 2 m9(l - 9) m?92 (i - op)3 [l - (i - e)m] (i - ep)3[i - (i - 9)3] ( ) From ( ) w e get T ' = T,' 1-0 m 0 + (1-9(3) (1 9p) J putting T I 1-9 m9 1 - e p = a, w e g e t ^ = + V ( ) w hich on sim p lific a tio n u sin g ( ) give s T' (1-0)(1 - an) - n = j t; a 2 L = (1-0)(1 - an) ( ) a 117

39 T ' where L = i - n ( ) From ( ), w e get 1-0 = 0 = 1 L a 2 (1 - a n ) L a a n ( ) A g a in from ( ) w e have, 0 + a - l = a 1 - a n 1+ (n + L )a ( ) and 0 + a - l a [ l - ( n + L )a ] a n - L a ( ) Substituting the values from ( ), ( ) and ( ) in ( ), w e get T = (1 - a n ) 1 - n [ l - ( n + L )a ] r La2 /, r 2. a n - ( n + L ) a n 1- n a / 1- a n - L a ' w hich give s on sim p lificatio n a function o f a, say f(a ) as f(a ) = L o g 1 - n[l - ( n + L )a ] (1 - na)t,' an[l-(n + L)a]Log_La 2 =0 (3, 3 2Q) 1 - a n - L a (1 - a n ) F o r estim ating the parameters, the population m om ents T,' and are replaced b y sam ple m om ents x and s'2 thus the equation becom es f (a ) = L o g 1 - n[l - (n + ^)a] (1 - n a )x a n [ l - ( n + Q a ] Lo ( a 1 - a n - t o 2 (1 - n a) = 0 w here = ^ - n ( ) x T h is is an equation in a w h ic h does not appear to be solved directly for a. It can be v e rifie d e a sily that one o f the roots o f this equation lies at a = ^ (n + K ) w h ic h im p lie s that a 1 0 and so p is alw ays unity. O b viously, 118

40 this root w ill not solve our purpose and so, we need another root o f equation. F o r this som e iteration m ethods preferably the R e gula -F a lsi method can be used. U s in g the R e g u la -F a lsi m ethod that the value o f a is obtained for w hich the value o f the function is sufficiently near to zero. Putting this value o f a in ( ) an estimate o f 9 can be obtained. U sin g the estimate o f 0 and a an estimate o f f>can be obtained and finally u sin g ( ) an estimate o f m can be obtained. 3.14: G o o d n e ss o f F it A n attempt has been m ade to study the adequacy o f the zero-truncated G N B D to describe the distribution o f the num ber o f households w hen the num ber o f h ouseholds w ith no m igrant is not available. It has also been tried to fit an alternative to zero truncated G N B D ( Z T G N B D ) and size-biased G N B D ( S B G N B D ) to the sam e data sets. O n fitting, it is found that the S B G N B D describes the distribution o f num ber o f households according to the num ber o f m igrants w hen the num ber o f households with no m igrants is not available, in a better w a y than the Z T G N B D. T h is m ay be probably because o f the general nature o f m igration, w h ich occurs in batches and it increases in proportion to its sizes. Theoretically, S B G N B D is best suited for the m igration data w hen the house h old w ith no m igrants is not know n. T he results have been show n below. 119

41 Table Observed and expected number of households of size group 10 and above having at least one migrant. Number of migrants Observed number of households Expected number of households SBGNBD ZTGNBD ^ 6.86 ^ ^ 9.95 > J 0.50 J Total M e a n x d.f 2 2 P(x2) A A P A m

42 Table Observed and expected number of households of size group 16 and above having at least one migrant. Number of Observed Expected number of households migrants number of SBGNBD ZTGNBD households ' 4.26 > 4.56 " r 6 > 7.31 > > , Total M e a n 2.56 x d.f 2 2 P (x 2) p ^ O b v io u sly, the S B G N B D provides better fits to the observed distributions than to Z T G N B D. M ore o ve r, it is relatively very quick to be obtained and so it m ay be preferred to others w here very quick results are required. 121