Computational Statistics and Data Analysis. Estimation for the three-parameter lognormal distribution based on progressively censored data

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1 Computational Statistics and Data Analysis 53 (9) Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: stimation for the three-parameter lognormal distribution based on progressively censored data Prasanta Basak a Indrani Basak a N. Balakrishnan b a Penn State Altoona Altoona PA United States b McMaster University Hamilton Canada a r t i c l e i n f o a b s t r a c t Article history: Received 7 February 9 Accepted 7 March 9 Available online April 9 Some work has been done in the past on the estimation of parameters of the threeparameter lognormal distribution based on complete and censored samples. In this article we develop inferential methods based on progressively Type-II censored samples from a three-parameter lognormal distribution. In particular we use the M algorithm as well as some other numerical methods to determine maximum likelihood estimates (MLs) of parameters. The asymptotic variances and covariances of the MLs from the M algorithm are computed by using the missing information principle. An alternative estimator which is a modification of the ML is also proposed. The methodology developed here is then illustrated with some numerical examples. Finally we also discuss the interval estimation based on large-sample theory and examine the actual coverage probabilities of these confidence intervals in case of small samples by means of a Monte Carlo simulation study. 9 lsevier B.V. All rights reserved.. Introduction Lognormal distribution is one of the distributions commonly used for modeling lifetimes or reaction-times and is particularly useful for modeling data which are long-tailed and positively skewed. It has been discussed extensively by many authors including Cohen (95 988) Hill (963) Harter and Moore (966) Crow and Shimizu (988) Johnson et al. (994) Munro and Wixley (97) and Rukhin (984). It is well-known that there is a close relationship between normal and lognormal distributions. If X = log(y γ ) is normally distributed with mean µ and standard deviation then the distribution of Y becomes a three-parameter lognormal distribution with parameter θ = (γ µ ). The probability density function of such a three-parameter lognormal distribution is } { f (y; θ) = π(y γ ) exp log(y γ ) µ γ < y < > < µ < (.) otherwise. In (.) and µ are the variance and mean of the underlying normal variable X but become the shape and scale parameters of the lognormal variable Y. It is more convenient to use β = exp(µ) and w = exp( ) as the scale and shape parameters of the lognormal variable Y respectively. Also γ is the threshold (location) parameter of the lognormal random variable. Corresponding author. Tel.: ; fax: mail address: i8b@psu.edu (I. Basak) /$ see front matter 9 lsevier B.V. All rights reserved. doi:.6/j.csda.9.3.5

2 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) When the threshold parameter γ is known the parameter estimation can be done using the well-known results for normal distribution by making the transformation from Y to X. stimation methods become more complex when γ is unknown. Adding to that complexity is the fact that quite often the data on lifetimes or reaction-times come with censoring. Censoring occurs when exact survival times are known only for a portion of the individuals or items under study. The complete survival times may not have been observed by the experimenter either intentionally or unintentionally and there are numerous examples for each kind; see for example Nelson (98) and Balakrishnan and Cohen (99). In this article we consider a general scheme of progressive Type-II right censoring. Under this scheme n units are placed on a life-testing experiment and only m (< n) are completely observed until failure. The censoring occurs progressively in m stages. These m stages offer failure times of the m completely observed units. At the time of the first failure (the first stage) R of the n surviving units are randomly withdrawn (censored intentionally) from the experiment R of the n R surviving units are withdrawn at the time of the second failure (the second stage) and so on. Finally at the time of the m-th failure (the m-th stage) all the remaining R m = n m R R m surviving units are withdrawn. This scheme (R R... R m ) is referred to as progressive Type-II right censoring scheme. It is clear that this scheme includes the conventional Type-II right censoring scheme (when R = R = = R m = and R m = n m) and complete sampling scheme (when n = m and R = R = = R m = ). The ordered lifetime data which arise from such a progressive Type-II right censoring scheme are called progressively Type-II right censored order statistics. For theory methods and applications of progressive censoring readers are referred to the book by Balakrishnan and Aggarwala () and the recent discussion paper by Balakrishnan (7). Suppose n independent lognormally distributed units are placed on a life-testing experiment. Let Y :m:n Y m:m:n denote the above mentioned m progressively Type-II right censored order statistics. For ease in notation let us use Y j (j =... m) to denote these Y j:m:n s. Note that we observe only Y = (Y... Y m ). The purpose of this article is to discuss different estimation procedures for the parameter θ based on the progressively Type-II right censored order statistics Y. Recently Ng et al. () considered two-parameter lognormal distribution (which does not include the threshold parameter) and discussed Newton Raphson algorithm as well as M algorithm for finding the MLs. Inclusion of the third parameter γ introduces an unusual feature in the likelihood function. Hill (963) has shown that there exist paths along which the likelihood function of any ordered sample y... y n from the three-parameter lognormal distribution tends to as (γ µ ) approaches (y ). Thus the global maximization of the likelihood leads to the unreasonable estimate (y ) although in fact the likelihood at the point is zero. But Hill (963) also retained the idea that reasonable estimates could be obtained by solving the likelihood equations. To get these reasonable estimates Cohen (95) Cohen and Whitten (988) and Harter and Moore (966) equated partial derivatives of the likelihood function to zero and solved the resulting equations. These estimates are called local maximum likelihood estimates (LMLs). Harter and Moore (966) and Calitz (973) noted that although these LMLs are not true MLs according to the usual definition they are reasonable estimates and appear to possess most of the desirable properties associated with MLs. However it is noted in Harter and Moore (966) and Cohen and Whitten (988) that sometimes the likelihood function may have no clearly defined local maximum for small samples and so LMLs fail to produce estimates in that case. In the past some work has been done on estimation methods for the three-parameter lognormal distribution based on complete and censored samples; see for example the books by Cohen and Whitten (988) and Balakrishnan and Cohen (99). In Section we describe the Newton Raphson algorithm for determining the MLs of the parameter θ based on a progressively censored sample. The second derivatives of the log-likelihood are required in order to use the algorithm. These computations are complicated when data are progressively censored. Another viable alternative to Newton Raphson algorithm is the well-known M algorithm and in Section 3 we discuss how that can be used to determine the MLs in this case. Asymptotic variances and covariances of the maximum likelihood estimates generated through the M algorithm are given in Section 4. Section 5 describes some simplified estimation methods which yields simple alternative estimators. All these methods discussed in this article are then illustrated with some numerical examples in Section 6. With these illustrative examples we also discuss the interval estimation based on large-sample theory and then examine the actual coverage probabilities in case of small samples through Monte Carlo simulations.. Newton Raphson algorithm One of the standard methods of determining the maximum likelihood estimates is the Newton Raphson algorithm. In this section we describe the Newton Raphson algorithm for finding the MLs numerically when life-times are distributed as a three-parameter lognormal distribution with parameter θ. These MLs are local MLs as mentioned earlier corresponding to the Newton Raphson algorithm and would be denoted here by LML. The log-likelihood function log L(θ) = log L based on the progressively Type-II right censored order statistics Y is log L(θ) = const. m log log(y j γ ) Ψ + { j R j log Φ } Ψ j (.) where Ψ j = log(y j γ ) µ ; see Balakrishnan and Aggarwala (). In (.) and throughout this article φ and Φ = Φ denote the probability density and survival function of the standard normal distribution respectively. Three likelihood

3 358 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) equations which need to be solved simultaneously for the required estimate ˆθ = ( ˆµ ˆ ˆγ ) are as follows: log L γ log L µ = log L = Ψj + R j L j y j γ + m Ψ j + R j L j = = m + Ψ + j R j Ψ j L j = y j γ = (.) in which L j = φ(ψ j )/ Φ(Ψ j ) is the hazard function of the standard normal distribution at Ψ j. In the Newton Raphson algorithm the simultaneous solution is obtained through an iterative procedure. In each iterative step the corrections a b c to the previous estimates γ µ produce new estimates ˆγ ˆµ ˆ as ˆγ = γ + a ˆµ = µ + b and ˆ = + c. The iteration method is based on Taylor series expansions of the estimating equations in (.) in the neighborhood of the previous simultaneous estimates. Neglecting powers of a b and c above the first order and using Taylor s theorem we get the following equations which need to be solved for a b c: a log L γ a log L µ γ a log L γ + b log L γ µ + b log L µ + b log L µ + c log L γ + c log L µ + c log L = log L γ = log L µ = log L where the notation A for any partial derivative A means the partial derivative evaluated at (γ µ ). The second derivatives needed in (.3) are as follows: log L γ = (y j γ ) + (Ψ j + R j L j ) (y j γ ) m + R j L j (L j Ψ j ) log L = µ log L = log L γ µ = log L γ = log L µ = m { 3Ψ j + R j(ψ j L j Ψ 3 j L j + Ψ j L j )} Rj Ψ j L j + R j Lj y j γ Ψj + R j (L j Ψ j L j + Ψ j L ) j Ψj + R j (L j Ψ j L j + Ψ j L ) j. y j γ ( R j Ψ j L j + R j L ) j (y j γ ) Using the Newton Raphson method Ng et al. () discussed methods of finding MLs while Balakrishnan et al. (3) discussed the construction of confidence intervals of µ and for the two-parameter lognormal distribution. Mi and Balakrishnan (3) have made use of the fact that the lognormal density is log-concave in order to establish that the MLs of µ and do exist and are unique. Hence in that case the M algorithm and the Newton Raphson algorithm will converge to the same values. In order to obtain LML for the three-parameter lognormal distribution we suggest a variation of the above method which was used by Cohen (95). Calitz (973) found this method of Cohen (95) produced better convergence in the estimation process. Instead of solving all three equations in (.) simultaneously the suggested method is to start with a γ () which is less than y and use the second and third equations of (.) to get µ () = µ(γ () ) and () = (γ () ). The first equation is used as the test equation. If the left hand side of this equation equals zero when (γ () µ () () ) substituted there then no further iteration is required. Otherwise another γ () is chosen which is less than y yielding µ () = µ(γ () ) (.3) (.4)

4 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) and () = (γ () ) and look for the sign change in the left hand side of the first equation of (.). Finally interpolating on γ values first find the final estimate ˆγ and then the final estimates ˆµ = µ( ˆγ ) and ˆ = ( ˆγ ) are obtained using the second and third equations of (.). The final estimates ( ˆγ ˆµ ˆ ) when substituted in the first equation of (.) should produce zero or close to zero. As noted by Cohen and Whitten (988) usually a single root ˆγ will be found. In the event that multiple roots of ˆγ occur Cohen and Whitten (988) suggested to use the root which results in the closest agreement between ȳ and ˆµ+ ˆ ÊY = ˆγ + e. 3. M Algorithm In the case of progressively censored samples an alternative to the Newton Raphson algorithm is the use of M algorithm for numerically finding the MLs. One advantage of the M algorithm is that asymptotic variances and covariances of the M algorithm estimates can be computed which is discussed in Section 4. M algorithm introduced by Dempster et al. (977) is a very popular tool to handle any missing or incomplete data situation; readers are referred to the book by McLachlan and Krishnan (997) for a detailed discussion on M algorithm and its applications. This algorithm is an iterative method which has two steps. In the -step it replaces any missing data by its expected value and in the M-step the log-likelihood function is maximized with the observed data and expected value of the incomplete data producing an update of the parameter estimates. The MLs of the parameters are obtained by repeating the - and M-steps until convergence occurs. Since progressive censoring model can be viewed as a missing data problem M algorithm can be applied to obtain the MLs of the parameters in this case. Let us denote the censored data vector as Z = (Z Z... Z m ) where the j-th stage censored data vector Z j is a R j vector Z j = (Z j Z j... Z jrj ) for j =... m. The complete data set is then obtained by combining the observed data Y and the censored data Z. -step of the algorithm requires the computation of the conditional expectation of functions of censored data vector Z conditional on the observed data vector Y and the current value of the parameters. In particular one computes the conditional expectation of the log-likelihood log L(Y Z θ) Y = y as log L(Y Z θ) Y = y = const. n log log(y j γ ) ( ) log(yj γ ) µ R j R j ( ) log(zjk γ ) µ log(z jk γ ) Z jk > y j Z jk > y j. (3.) k= k= The above conditional expectations are obtained using the result that given Y j = y j Z j s have a left-truncated distribution F truncated at y j. More specifically the conditional probability density of Z given Y is given by (see Balakrishnan and Aggarwala ()) R m j f Z Y (z y; θ) = f Zjk Y j (z jk y j ; θ) k= (3.) where f Zjk Y j (z jk y j ; θ) = f (z jk; θ) F(y j ; θ) (3.3) and f (z jk ; θ) is given by (.) and F denotes the corresponding cumulative distribution function. In the M-step of the (h + )-th iteration we will denote the updated estimates of the parameter θ as θ (h+). This θ (h+) maximizes the loglikelihood function involving the observed data Y conditional expectation of the log-likelihood function of censored data vector Z given the observed data vector Y and the h-th iteration value of the parameter θ (h). As a starting value θ () one can use a γ () < y and µ () and () are computed on the basis of the so-called pseudo-complete sample which involves observed data Y and the censored observations at the j-th step Z j all taken to be y j. Thus µ () = µ(γ () ) and () = (γ () ) are then given by µ () = (R j + ) ( ) log y j γ () n () = m (R j + ) ( ). (3.4) log y j γ () µ () n

5 3584 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Starting with the initial estimates in (3.4) the (h + )-th iteration value θ (h+) is obtained using the h-th iteration value θ h as follows: { µ (h+) = m ( ) log y j γ } (h) + R j log(z γ (h) ) Z > y j ; θ (h) n { (h+) = m ( ) log y j γ }. (3.5) (h) + R j log (Z γ (h) ) Z > y j ; µ (h+) (h) γ (h) µ (h+) n The conditional expectations in the above expression (3.5) can be obtained as follows: where log(z γ (h) ) Z > y j ; θ (h) = (h) L j(h) + µ (h) log (Z γ (h) ) Z > y j ; µ (h+) (h) γ (h) = (h) + Ψ j L j + (h) µ (h+) L j + µ (h+) Ψ j(h) = Ψ j (θ (h) ) = log ( y j γ (h) ) µ(h) (h) L j(h) = L j (θ (h) ) = φ(ψ j(h)) Φ(Ψ j(h) ) Ψ j = log ( y j γ (h) ) µ(h+) (h) L j = φ(ψ j ) Φ(Ψ j ). γ (h+) is obtained by solving the following equation for γ : µ(h+) (h+) m y j γ log(y j γ ) + m µ (h+) (h+) R j y j γ Z γ Z > y j; γ θ (h+) log(z γ ) R j Z γ Z > y j; γ θ (h+) = where θ (h+) = (µ (h+) (h+) ). The conditional expectations in the above expression (3.6) can be obtained as follows: Z γ Z > y j; γ θ (h+) µ (h+) = e (h+) P j(h+) (γ ) log(z γ ) Z γ Z > y j; γ θ (h+) µ (h+) = e (h+) (h+) P j(h+) (γ ) + (µ (h+) (h+) )P j(h+)(γ ) where Ψ j(h+) (γ ) = Ψ j (γ θ (h+) ) = log ( y j γ ) µ (h+) (h+) P j(h+) (γ ) = P j (γ θ (h+) ) = Φ(Ψ j(h+) (γ ) + ˆ (h+) ). Φ(Ψ j(h+) (γ )) The final MLs of the parameters are local MLs corresponding to the M algorithm and would be denoted here by LML. In order to obtain LML we suggest a variation of the above method. As mentioned before one can start with a γ () which is less than y and get µ () = µ(γ () ) and () = (γ () ) by using (3.4) and then one gets µ () and () by using (3.5). The suggestion is to treat (3.6) as the test equation. If the left hand side of that equation equals zero when (γ () µ () () ) substituted there then no further iteration is needed. Otherwise the suggested method involves choosing another γ () which is less than y and repeating the procedure looking for the sign change in (3.6). Finally interpolating on γ values one gets the final estimate ( ˆγ ˆµ ˆ ) which when substituted in (3.6) should produce zero or close to zero. 4. Asymptotic variances and covariances of the M algorithm estimates Asymptotic variances and covariances of the MLs when the M algorithm is used can be obtained by using the missing information principle of Louis (98) and Tanner (993). This principle is basically Observed information = Complete information Missing information. Based on this principle Louis (98) developed the procedure of finding the observed information matrix when M algorithm is used to find MLs in an incomplete data situation. We adopt this principle in the situation of progressive } (3.6)

6 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Type II right censoring. We will denote the complete observed and missing (expected) information by I(θ) I Y (θ) and I Z Y (θ) respectively. The complete information I(θ) is given by log L(Y Z θ) Y = y I(θ) = (4.) θ in which log L(Y Z θ) Y = y is given by (3.). In (4.) the expectation is taken with respect to both Y and Z. The Fisher information matrix for a single observation which is censored at the time of the j-th failure is given by I (j) (θ) = Z Y log f Zjk Y j (z jk y j ; θ) (4.) θ in which f Zjk Y j (z jk y j ; θ) is given by (3.3). In (4.) the expectation is taken with respect to z jk so that I (j) Z Y (θ) is a function of y j and θ. Then the missing (expected) information is simply I Z Y (θ) = R j I (j) Z Y (θ) (4.3) where I (j) Z Y (θ) is given by (4.). The observed information I Y (θ) is then obtained as follows: I Y (θ) = I(θ) I Z Y (θ) (4.4) where I(θ) and I Z Y (θ) are given by (4.) and (4.3) respectively and are derived below. Finally upon inverting the observed information matrix I Y (θ) in (4.4) one gets the asymptotic variances and covariances of the MLs when M algorithm is used. 4.. Complete information matrix I(θ) The log-likelihood function log L (θ) based on n uncensored observations y i i =... n is given by log L (θ) = n log f (y i ; θ) i= (4.5) where f (y i ; θ) is as given in (.). On differentiating the log-likelihood in (4.5) and equating to zero one obtains the estimating equations given by (.) with m = n and R j = ; j =... m. Negative of the second derivatives of the log-likelihood function log L (θ) are obtained by appropriately differentiating the first derivatives in (.) with m = n and R j = j =... m and are given by (.4) with m = n and R j = j =... m. For the three-parameter lognormal distribution it can be shown that = e µ Y i γ = e ( µ) (Y i γ ) (4.6) log(yi γ ) = (µ )e µ Y i γ log(yi γ ) = (µ )e ( µ). (Y i γ ) One gets the complete information matrix I(θ) in (4.) by using (.4) with m = n R j = j =... m and (4.6) as follows: + e ( µ) e µ e µ I(θ) = n e µ. (4.7) e µ xpressions for the asymptotic variances and covariances of the ML of θ in the uncensored case are obtained by inverting the matrix I(θ) in (4.7). Denoting β = e µ w = e and H = w( + ) ( + ) the asymptotic variances and

7 3586 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) covariances are: V( ˆγ ) = β n w H V( ˆµ) = ( + H) n V( ˆ ) = n ( + H) Cov( ˆγ ˆµ) = β H n w Cov( ˆγ ˆ ) = 3 β H n w Cov( ˆµ ˆ ) = 3 n H. It is worthwhile to mention here that for the uncensored case Cohen (95) has obtained the asymptotic variances and covariances of the MLs of the parameters (γ β ) while those corresponding to the parameters (γ µ ) were obtained by Hill (963). 4.. Missing information matrix I Z Y (θ) The logarithm of the density function of an observation z jk = z censored at y j the time of the j-th failure is given by see q. (3.) { } log(yj γ ) µ log f z yj (z y j ; θ) = const. log log(z γ ) log Φ log(z γ ) µ. (4.8) Differentiating (4.8) with respect to γ µ and one gets log f z yj = log(z γ ) µ + γ z γ z γ log f z yj = log(z γ ) µ L j µ ( log f z yj = log(z γ ) µ y j γ L j ) ( ) + ψ jl j. By using the properties of the left-truncated log-normal distribution it can then be shown that log(z γ ) µ Z > y j ; θ = L j {log(z γ ) µ} Z > y j ; θ = ( + Ψ j L j ) {log(z γ ) µ} 3 Z > y j ; θ = 3 ( + Ψ j) {log(z γ ) µ} 4 Z > y j ; θ = ( ) Ψ j L j + Ψ 3 j j L Z γ Z > y j; θ = e µ LL j ( ) Z γ Z > y j; θ = e ( µ) LLj log(z γ ) Z γ Z > y j; θ = e µ L j + (µ )LL j ( ) log(z γ ) Z γ Z > y j; θ = e ( µ) { Ψ j + (µ ) } L j + { + (µ ) } LL j log(z γ ) (Z γ ) Z > y j; θ = e ( µ) L j + (µ )LL j (log(z γ )) Z γ Z > y j; θ = e µ { Ψ j + (µ ) } L j + { + (µ ) } LL j (log(z γ )) 3 Z γ Z > y j; θ = e µ { Ψ + 3 j 3 (µ ) } L j + { 3(µ ) Ψ j + 6 (µ ) } L j (µ ) 3 LL j + { 3(µ ) + (µ ) } LL j (4.9) (4.)

8 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) in which Ψ j Ψ j Ψ j L j L j L j LL j and LL j are given by ( Ψ j = log y j γ ) µ Ψ j = log(y j γ ) (µ ) Ψ j = log(y j γ ) (µ ) L j = φ(ψ j) Φ(Ψ j ) L j = φ(ψ j) Φ(Ψ j ) L j = φ(ψ j) Φ(Ψ j ) LL j = Φ(Ψ j ) Φ(Ψ j ) LL j = Φ(Ψ j ) Φ(Ψ j ). (4.) The Fisher information matrix based on one observation z jk censored at y j I (j) Z Y (θ) in (4.) can be obtained using (4.9) (4.) as follows: ( ) log ( fz yj e µ) + Ψ j ( = γ (y j γ ) L j L j + e ( µ) + ) LL j e µ (y j γ ) L jl j log fz yj = + Ψj L µ j L j log fz yj = ( ) + Ψj L j Ψj L j + Ψ j ( ) ( ) log fz yj log fz yj = e γ µ 3 µ { } ( + )L j ( + )LL j L j LL j L j + L j y j γ ( ) ( ) log fz yj log fz yj = µ { ( )L j + µ } (4.) L γ j (y j γ ) { } + e µ µ + µ + Ψ j + Ψ jl j + 4µ 3 L 3 j { ( + e µ µ Ψ j µ ) ( µ ) ( 4 + µ µ )} L j ( + e µ µ ) ( ) µ µ + µ LL j + e µ { + (µ ) } ( ) µ µ LL 4 j ( ) ( ) log fz yj log fz yj = Lj + Ψ µ j L j (Ψ j L j ). Using I Z Y (θ) from (4.) and I(θ) from (4.7) the observed information can be obtained from (4.4). Finally one can get the asymptotic variances and covariances of the MLs when the M algorithm is used by inverting this observed information matrix I Y (θ). 5. Alternative estimators As an alternative estimator one can use modified ML (MML). MMLs use first order statistic for estimating γ and are therefore easier to compute than the MLs. For small samples when LMLs fail to converge these MMLs (which always exist) produce reasonable estimates. For progressively censored data one can have two versions of MMLs. One version corresponds to the Newton Raphson algorithm and the other version corresponds to the M algorithm and they will be denoted by MML and MML respectively. In both versions the likelihood equation log L = is replaced by γ γ + e µ+ Φ n+ = y (5.) in which n = m+ m R j is the total sample size. q. (5.) is used as a test equation and it replaces the first equation in (.) for the Newton Raphson algorithm version and q. (3.6) for the M algorithm version. One starts with a first approximation γ and make the transformation log(y i γ ). One then calculates conditional estimates µ = µ(γ ) and = (γ ) by using the second and third equations of (.) for the Newton Raphson algorithm and (3.5) for the M algorithm. The values (γ µ ) are substituted in q. (5.). If the test equation is satisfied then no further iteration is required. Otherwise a

9 3588 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Table Progressively censored data for xample. i Y i R i 4 Table stimates and standard deviations of estimates for γ µ and based on data in Table. γ µ stimator ˆγ ˆγ ˆµ ˆµ ˆ ˆ LML Table 3 Progressively censored data for xample. i Y i R i Table 4 stimates and standard deviations of estimates for γ µ and based on data in Table 3. γ µ stimator ˆγ ˆγ ˆµ ˆµ ˆ ˆ LML second approximation γ is selected and the cycle of calculations as described above is repeated. The iterations are continued until two sufficiently close values γ i and γ i+ are found such that the following is satisfied: γ i + e µ i+ i Φ n+ < (>)y < (>)γ i+ + e µ i++ i+ Φ n+. Thus the final MML γ of γ is obtained using which the final MMLs µ of µ and are then obtained. 6. Illustrative examples To illustrate the computational methods presented in this article we use three examples given in Cohen and Whitten (988). We modified these original examples to consider progressively censored data. We then assume that each datum comes from a three-parameter lognormal distribution and use them to carry out all the estimation procedures discussed in the preceding sections in order to see whether they produce similar results or not. Also in Section 7 we compare the coverage probabilities of confidence intervals based on LML and LML using a simulation study. We used same γ value for obtaining LML and LML. For the determination of MML as discussed in Section 5 Φ =.67 for xamples and 3 since n = in both these cases. For xample n+ Φ =.34 since n = n+ here. Proceeding as explained in Section 5 we get two versions of MML viz. MML and MML (for the Newton Raphson algorithm and M algorithm respectively). xample. The maximum flood levels (in millions of cubic feet per second) for four-year periods from 89 to 969 in the Susquehanna river at Harrisburg Pennsylvania are given in Cohen and Whitten (988) and were also used by Dumonceaux and Antle (973). We modified these data to make it progressively censored with m = stages and these progressively censored data are presented in Table. The LML and their standard deviations are presented in Table. For the data in Table the LML are given by ˆγ =.8 ˆµ =.9 and ˆ =.36 the MML are given by γ =.58 µ =.3 and =.79 and the MML are given by γ =.563 µ =.75 and =.73. xample. This example was used by McCool (974) and is also given in Cohen and Whitten (988). The data are fatigue lives (in hours) of bearings of a certain type. We modified these data to make it progressively censored with m = 7 stages and these progressively censored data are presented in Table 3. LML did not converge for this sample. The LML and their standard deviations are presented in Table 4. The MML are given by γ =.785 µ = 3.93 and =.893 while MML are given by γ = 9.73 µ = and =.836.

10 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Table 5 Progressively censored data for xample 3. i Y i R i 4 Table 6 stimates and standard deviations of estimates for γ µ and based on data in Table 5. γ µ stimator ˆγ ˆγ ˆµ ˆµ ˆ ˆ LML Table 7 Progressively censored data for xample 3 with PCS-. i Y i R i 4 Table 8 Progressively censored data for xample 3 with PCS-. i Y i R i 4 Table 9 stimates and standard deviations of estimates for γ µ and for data in Table 7. γ µ stimator ˆγ ˆγ ˆµ ˆµ ˆ ˆ LML Table stimates and standard deviations of estimates for γ µ and for data in Table 8. γ µ stimator ˆγ ˆγ ˆµ ˆµ ˆ ˆ LML xample 3. In this example the complete data are given by Cohen and Whitten (988) and were also used by Cohen (95). The complete data consists of observations from a three-parameter lognormal distribution with γ = µ = 3.93 and =.4. We modified these data to make it progressively censored with m = stages and these progressively censored data are presented in Table 5. Although Y i values are reported up to one decimal place in Tables 5 7 and 8 we used all three decimal places for these Y i values in Cohen (95) for our computations. The LML and their standard deviations are presented in Table 6. For the data in Table 5 the LML are given by ˆγ =.3 ˆµ = and ˆ =.4 the MML are given by γ = µ = 4.49 and =.95 and the MML are given by γ = µ = 3.97 and =.378. In order to examine the effect of delayed censoring we considered two schemes of progressive censoring. In one scheme the censoring was delayed than the other. Keeping the number of stages m = the same as in Table 4 for the delayed censoring scheme we considered PCS-: R = R = R 3 = R 4 = R 5 = R 6 = R 7 = R 8 = R 9 = R = R = R = 4. For the other scheme we took PCS-: R = 4 R = R 3 = R 4 = R 5 = R 6 = R 7 = R 8 = R 9 = R = R = R =. The progressively censored data obtained under these two schemes are presented in Tables 7 and 8 respectively. The LML estimates and their standard deviations for the data in Tables 7 and 8 are presented in Tables 9 and respectively. For the data in Table 7 the LML are given by ˆγ = 4.46 ˆµ = 3.99 and ˆ =.373 the MML are given by γ = µ = 4.5 and =.98 and the MML are given by γ = µ = 4.7 and =.35. For the data in Table 8 the LML are given by ˆγ = ˆµ = 4.4 and ˆ =.43 the MML are given by γ = 7.5 µ = 4.3 and =.9 and the MML are given by γ = 7.93 µ = 4.37 and =.368. It is

11 359 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Table 95% confidence intervals for xample. stimators γ µ LML ( ) ( ) ( ) LML (.7.45) ( ) (.37.43) MML (.5.574) ( ) ( ) MML (.65.43) ( ) ( ) Table 95% confidence intervals for xample. stimators γ µ LML ( ) ( ) (.33.33) MML ( ) ( ) (.43.48) MML ( ) ( ) (.43.3) Table 3 Coverage probabilities of 95% confidence intervals for γ based on Monte Carlo simulations. n m R LML LML (7... ) (... 7) (5... ) (... 5) (3... ) (... 3) (... ) (... ) (4... ) (... 4) (... ) (... ) (6... ) ) (... ) (... ) clear from Tables 6 9 and that the estimates of γ are affected not only by censoring but also by the pattern of censoring while the estimates of µ and remain more or less the same in all the situations. 7. Simulation study In this section we report results from two simulation studies that we carried out. In the first simulation study we constructed 95% confidence intervals based on randomly generated progressively censored samples corresponding to different estimates of γ µ and for each of the estimation methods discussed in Section 6. We used the same progressive censoring schemes as given in those examples. We provide the confidence intervals from this simulation study in Tables and for xamples and respectively for different estimators. In the second simulation study we compared the performance of LML and LML in terms of coverage probabilities of 95% confidence intervals for the parameters γ µ and for different sample sizes and different degrees of censoring. samples were simulated from the lognormal distribution with γ = µ = 3.93 and =.4 with sample size n = 4. For sample size n = we considered m = stages of censoring and for sample size n = 4 we considered m = stages. In each case we considered two censoring schemes with one reflecting comparatively delayed censoring than the other. The coverage probabilities of 95% confidence intervals from this simulation study for the parameters γ µ and are presented in Tables 3 5 respectively. It is observed from these tables that the coverage probabilities are better when the proportion of uncensored data is larger. The coverage probabilities seem to be almost the same for the two different censoring schemes. The confidence intervals for the parameter γ seems to be most sensitive to the censoring pattern while the confidence intervals for the parameters µ and seem to be stable and quite satisfactory and close to the nominal level of 95%. In a similar tone it is known that different censoring schemes do not change the estimates for the two-parameter exponential distribution (see for example Balakrishnan and Sandhu (996)). The sensitivity of estimation of γ may be due to the fact that γ is the threshold parameter of the lognormal distribution and the pattern of censoring applied to the sample might have affected the nature of the sample observations as far as the estimation of the threshold parameter is involved. Also LML and LML seem to have nearly the same coverage probabilities.

12 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Table 4 Coverage probabilities of 95% confidence intervals for µ based on Monte Carlo simulations. n m R LML LML (7... ) (... 7) (5... ) (... 5) (3... ) (... 3) (... ) (... ) (4... ) (... 4) (... ) (... ) (6... ) ) (... ) (... ) Table 5 Coverage probabilities of 95% confidence intervals for based on Monte Carlo simulations. n m R PC for LML PC for LML (7... ) (... 7) (5... ) (... 5) (3... ) (... 3) (... ) (... ) (4... ) (... 4) (... ) (... ) (6... ) (... 6) (... ) (... ) Concluding remarks In this article we have discussed the M algorithm for the maximum likelihood estimation based on progressively Type-II censored samples from a three-parameter lognormal distribution. We have also considered the traditional Newton Raphson method for this purpose. Additionally we have discussed two versions of modified maximum likelihood estimators. The M algorithm and the Newton Raphson method produced similar results in all three examples considered as well as in terms of coverage probabilities determined from a Monte Carlo simulation study. The coverage probabilities for both methods are better and closer to the nominal level of 95% when the proportion of uncensored data is larger. Acknowledgments The authors are thankful to the referees for their valuable comments which led to a considerable improvement in the presentation of this article. References Balakrishnan N. 7. Progressive censoring methodology: An appraisal (with discussions). Test 96. Balakrishnan N. Aggarwala R.. Progressive Censoring: Theory Methods and Applications. Birkhäuser Boston. Balakrishnan N. Cohen A.C. 99. Order Statistics and Inference: stimation Methods. Academic Press San Diego. Balakrishnan N. Kannan N. Lin C.T. Ng H.K.T. 3. Point and interval estimation for Gaussian distribution based on progressively Type-II censored samples. I Transactions on Reliability Balakrishnan N. Sandhu R.A Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type-ii censored samples. Sankhya Series B Calitz F Maximum likelihood estimation of the parameters of the three-parameter lognormal distribution A reconsideration. Australian Journal of Statistics Cohen A.C. 95. stimating parameters of logarithmic-normal distributions by maximum likelihood. Journal of the American Statistical Association 46 6.

13 359 P. Basak et al. / Computational Statistics and Data Analysis 53 (9) Cohen A.C stimation in the lognormal distributions. In: Crow.L. Shimizu K. (ds.) Lognormal Distributions: Theory and Applications. Marcel Dekker New York. Cohen A.C. Whitten B.J Parameter stimation in Reliability and Life Span Models. Marcel Dekker New York. Crow.L. Shimizu K. (ds.) 988. Lognormal Distributions: Theory and Applications. Marcel Dekker New York. Dempster A.P. Laird N.M. Rubin D.B Maximum likelihood from incomplete data via the M algorithm. Journal of the Royal Statistical Society Series B Dumonceaux R. Antle C Discrimination between the log-normal and the Weibull distributions. Technometrics Harter H.L. Moore A.H Local-maximum-likelihood estimation of the parameters of the three-parameter lognormal populations from complete and censored samples. Journal of the American Statistical Association Corrections: 6 (966) 47; 6 (967) 59 5; 63 (968) 549. Hill B.M The three-parameter lognormal distribution and Bayesian analysis of a point-source epidemic. Journal of the American Statistical Association Johnson N.L. Kotz S. Balakrishnan N Continuous Univariate Distributions vol. second ed. John Wiley & Sons New York. Louis T.A. 98. Finding the observed information matrix when using the M algorithm. Journal of the Royal Statistical Society Series B McCool J.I Inferential techniques for Weibull populations. Aerospace Research Laboratories Report ARL TR 74-8 Wright-Patterson AFB Ohio. McLachlan G.J. Krishnan T The M Algorithm and xtensions. Marcel Dekker New York. Mi J. Balakrishnan N. 3. xistence and uniqueness of the MLs for normal distribution based on general progressively Type-II censored samples. Statistics & Probability Letters Munro A.H. Wixley R.A.J. 97. stimators based on order statistics of small samples from a three-parameter lognormal distribution. Journal of the American Statistical Association Nelson W. 98. Applied Life Data Analysis. John Wiley & Sons New York. Ng H.K.T. Chan P.S. Balakrishnan N.. stimation of parameters from progressively censored data using M algorithm. Computational Statistics & Data Analysis Rukhin A.L Improved estimation in lognormal models. Technical Report No Department of Statistics Purdue University West Lafayette Indiana. Tanner M.A Tools for Statistical Inference: Observed Data and Data Augmentation Methods second ed. Springer-Verlag New York.