Kth Record Research Articles

On maximum likelihood estimates of a proportional hazard rate model parameters based on record values

The proportional hazard rate models have wide acceptance in modeling complex data from different engineering, reliability, and survival applications. The hazard rate function of such models is able to model data with a monotonic or a bathtub-shape hazard rate. In this paper, we propose a subclass of proportional hazard rate models with three-parameters for which we have proved the existence and uniqueness of the maximum likelihood estimators based on upper kth record values. The proposed inference is found especially useful in some real illustrations within mechanical and medical analysis. Finally, several remarks are presented in the final stage of this paper.

Conditions for finiteness and bounds on moments of record values from iid continuous life distributions

We consider the standard and kth record values arising in sequences of independent identically distributed continuous and positive random variables with finite expectations. We determine necessary and sufficient conditions on the type of record k, its number n and moment order r so that the rth moment of the n value of kth record is finite for every parent distribution. Under the conditions we present the optimal upper bounds on these moments expressed in the scale units being the respective powers of the first population moment. The theoretical results are illustrated by some numerical evaluations.

Moments of order statistics and k-record values arising from the complementary beta distribution with application

The order statistics (OSs) arise in both practical and theoretical aspects including goodness-of-fit tests, characterizations of probability distributions and some estimation approaches such as the L-moments method. Most of these instances are connected with moments of OSs. The records and k-record statistics, also called kth record values in the literature, are other important topics related to the OSs. This kind of data appears in some practical circumstances, e.g. shock models, industrial stress testing and meteorological data analysis. This paper deals with deriving recurrence relations for the single and the product moments of OSs as well as k-record values from the complementary beta (CB) distribution defined by Jones (2002). The CB model, as a distribution on the unit interval, is more flexible than the well-known beta distribution in problems involving moments of OSs. This fact suggests that the former distribution can be employed rather than the latter when OSs are of the main interest. The results obtained are applied on four real data sets for the issues of parameter estimation as well as providing a visual inspection for goodness-of-fit of the CB model.

A METHOD FOR CONSTRUCTING AND INTERPRETING SOME WEIGHTED PREMIUM PRINCIPLES

AbstractWe present a method for constructing and interpreting weighted premium principles. The method is based on modifying the underlying risk distribution in such a way that the risk-adjusted expected value (or premium) is greater than the expected value of some conveniently chosen function of claims, which defines the insurer’s perception of the risk. Under some assumptions on the function of claims, the method produces distortion premium principles. We provide several examples under different assumptions on the claim arrival process and different functions of claims, including record claims andkth record claims.

Limiting behavior of the ratio of [formula omitted]th records

Let R0(k),R1(k),… be kth record values derived from samples consisting of independent identically distributed discrete random variables. In the present paper, using the theory of regular variation, we discuss the asymptotic behavior of Rn+m(k)∕Rn(k) as n→∞.

Optimal upper bounds on expected kth record values from IGFR distributions

ABSTRACTThe paper concerns the optimal upper bounds on the expectations of the kth record values () centred about the sample mean. We consider the case, when the records are based on the infinite sequence of the independent identically distributed random variables, which distribution function is restricted to the family of distributions with the increasing generalized failure rate (IGFR). Such a class can be defined in terms of the convex orders of some distribution functions. Particularly important examples of IGFR class are the distributions with the increasing density and increasing failure rate. Presented bounds were obtained with use of the projection method, and are expressed in the scale units based on the standard deviation of the underlying distribution function.

On MLEs of the parameters of a modified Weibull distribution based on record values

ABSTRACTThe modified Weibull distribution can be used quite effectively to model complex data from mechanical engineering or survival analysis studies that posses a monotonic or a bathtub-shape hazard rate. In this paper, we study the MLEs of the parameters of a modified Weibull distribution model in the presence of upper kth record values. The existence and uniqueness of the MLEs are proven in this case. Real data analysis is performed for illustrative purposes.

Characterizations of geometric distributions based on kth record values

ABSTRACTLet be a sequence of kth records arising from the geometric distribution. We show that for and the random variables and are independent or the regression of given is constant iff n=0. Moreover, we prove two characterizations of the geometric distribution based on some properties of kth records.

Upper and lower bounds on the variances of linear combinations of the kth records

ABSTRACTWe describe a method of determining upper bounds on the variances of linear combinations of the kth records values from i.i.d. sequences, expressed in terms of variances of parent distributions. We also present conditions for which the bounds are sharp, and those for which the respective lower ones are equal to zero. A special attention is paid to the case of the kth record spacings, i.e. the differences of consecutive kth record values.

δ-exceedance records and random adaptive walks

We study a modified record process where the kth record in a series of independent and identically distributed random variables is defined recursively through the condition with a deterministic sequence called the handicap. For constant and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences , focusing in particular on the case when matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.

On maximum variance of kth records

We consider i.i.d. samples of size n with continuous parent distributions and finite variances. Klimczak and Rychlik (2004) provided sharp bounds for the variances of kth record values in population variance units. The bounds are expressed in terms of the maximum of an analytic function over interval (0, 1). We prove that there exists a point x = x(k, n) such that the function strictly increases in (0, x) and strictly decreases in (x, 1). Consequently the maximum point is unique and it is the zero of the function derivative. We solve this problem for and n ⩾ 1 applying the variation diminishing property of the power functions.

Pitman Closeness of kth Records Based on a Two-Sequence Case

Suppose upper kth records were observed from an X-sequence of iid continuous random variables, and kth upper records from another independent Y-sequence of iid variables from the same distribution are to be observed. The Pitman closeness probabilities of these statistics are derived. For symmetric distribution, the Pitman closeness probabilities of kth record statistics to the population median, are also examined and it is shown that these probabilities are distribution free. Numerical computations are conducted to illustrate the results developed here.

Distances between models of generalized order statistics

The concept of generalized order statistics is a distribution theoretical set-up, which contains a variety of models for ordered data as particular cases, such as common order statistics, sequential order statistics, progressively type-II censored order statistics, record values, kth record values, and Pfeifer record values. In order to quantify the structure of generalized order statistics, distances between different respective models are measured by means of explicit expressions for divergences and distances applied to joint densities of ordered random variables. The results are exemplarily utilized to find a closest common order statistics model to some given model of sequential order statistics. Moreover, statistical applications in reliability are shown.

On characteristic functions of kth record values from the generalized extreme value distribution and its characterization

On characteristic functions of kth record values from the generalized extreme value distribution and its characterization

Distribution of Record Statistics in a q-Factorially Increasing Population

The probability function and binomial moments of the number N n of (upper) records up to time (index) n in a q-factorially increasing population are obtained in terms of the non central signless q-Stirling numbers of the first kind. As a corollary, the probability function of the time T k of the kth record is also expressed in terms of the non central signless q-Stirling numbers of the first kind. The mean of T k is obtained as a q-series with terms of alternating sign. Finally, the probability function of the inter-record time W k = T k − T k−1 is obtained as a sum of a finite number of terms of q-numbers. The mean of W k is expressed by a q-series. As k increases to infinity the distribution of W k converges to a geometric distribution with failure probability q. Additional properties of the non central q-Stirling numbers of the first kind, which facilitate the present study, are derived.

A new model for record values

A new mathematical model of record values, named confirmed records, is constructed. It is based on the notion of kth record values, which, in turn, is a generalization of ordinary mathematical records and extremal order statistics. Confirmed records are considered for arbitrary k = 1, 2, ... under the assumption that the original random variables X1, X2, ... are independent and have the same continuous distribution function. For new record values, two representations are obtained in the most important special cases where the original variables have the exponential and the uniform distribution. For uniform and exponential confirmed records, means and variances are found.

Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.

Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number N n of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q being the inverse of the proportion λ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables N n , n = 1 , 2 , … , are deduced. As a corollary the probability function of the time T k of the kth record is also expressed in terms of the signless q-Stirling numbers of the first kind. The mean of T k is obtained as a q-series with terms of alternating sign. Finally, the probability function of the inter-record time W k = T k - T k - 1 is obtained as a sum of a finite number of terms of q-numbers. The mean of W k is expressed by a q-series. As k increases to infinity the distribution of W k converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.

Tests Resulting from Characterizations Using Record Values

We continue the construction of goodness-of-fit tests resulting from characterizations via moments of the kth record values presented in [4] and [6]. Here we use U -statistics to estimate expectations appearing in the characterizations. We first state some characterizations. Let {Xn, n ≥ 1} be a sequence of independent and identically distributed (i.i.d.) random variables with continuous distribution function (c.d.f.) F and probability distribution function (p.d.f.) f . For a fixed integer k ≥ 1 we define the sequence Uk(1), Uk(2), . . . of kth (upper) record times of {Xn, n ≥ 1} as follows:

Sharp Bounds on the Error in Approximating the Means of Record Statistics by Inverse Hazard Functions

The Moriguti's greatest convex method is applied to obtain sharp distribution-free lower and upper bounds on the error in approximating the expected values of kth record statistics via inverse hazard functions.